Lý thuyết Công thức lượng giác (Kết nối tri thức 2024) Toán 11

Lý thuyết Toán lớp 11 Bài 2: Công thức lượng giác

A. Lý thuyết Công thức lượng giác

1. Công thức cộng

$\begin{array}{c}\sin \left(a+b\right)=\sin a\cos b+\cos asinb\\ sin\left(a-b\right)=\sin a\cos b-\cos asinb\\ \cos \left(a+b\right)=\cos a\cos b-\sin asinb\\ \cos \left(a-b\right)=\cos a\cos b+\sin asinb\\ \tan \left(a+b\right)=\frac{\tan a+\tan b}{1-\tan a\tan b}\\ \tan \left(a-b\right)=\frac{\tan a-\tan b}{1+\tan a\tan b}\end{array}$$\begin{array}{c}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\\ \mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\\ \mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\\ \mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}\\ \mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\frac{\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}}{\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}}\\ \mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; spanUnknown\; node\; type:\; span\frac{\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}}{\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}\mathrm{Unknown\; node\; type:\; span}Unknown\; node\; type:\; span\mathrm{Unknown\; node\; type:\; span}}\end{array}$

2. Công thức nhân đôi

$\begin{array}{l}\sin 2a=2\sin a\cos a\\ \cos 2a={\cos }^{2}a-{\sin }^{2}a=2{\cos }^{2}a-1=1-2{\sin }^{2}a\\ \tan 2a=\frac{2\tan a}{1-{\tan }^{2}a}\end{array}$

Suy ra, công thức hạ bậc:

${\sin }^{2}a=\frac{1-\cos 2a}{2},{\cos }^{2}a=\frac{1+\cos 2a}{2}$

${\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">,</span></span>{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$3. Công thức biến đổi tích thành tổng

cosacosb=12[cos(a+b)+cos(a−b)]sinasinb=12[cos(a−b)−cos(a+b)]sinacosb=12[sin(a+b)+sin(a−b)]$\begin{array}{l}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">[</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">(</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">)</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">(</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">)</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">]</span></span>\\ \mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">[</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">(</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">)</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">(</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">)</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">]</span></span>\\ \mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">[</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">(</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">)</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">(</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">)</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">]</span></span>\end{array}$

4. Công thức biến đổi tổng thành tích

cosa+cosb=2cosa+b2cosa−b2cosa−cosb=−2sina+b2sina−b2sina+sinb=2sina+b2cosa−b2sina−sinb=2cosa+b2sina−b2$\begin{array}{l}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\\ \mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\\ \mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\\ \mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;"></span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\end{array}$

B. Bài tập Công thức lượng giác

Bài 1. Tính sin2a và tan2a biết cos a = $\frac{1}{4}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}$ và $\frac{3\pi }{2}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$$\pi$$\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}$.

Hướng dẫn giải

Vì $\frac{3\pi }{2}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$$\pi$$\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}$nên sina < 0.

Ta có:

sin2a + cos2a = 1 ⇒ sin2a = 1 – cos2a = 1 -  = $\frac{15}{16}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">16</span></span>}}$

⇒ sina = $-\frac{\sqrt{15}}{4}$$<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}$.

Ta có: sin2a = 2sina cosa = 2..$\frac{1}{4}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}$ = $-\frac{\sqrt{15}}{8}$$<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}$

Ta có: tana = $\frac{\sin a}{\cos a}=-\sqrt{15}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}}$

⇒$\tan 2a=\frac{2\tan a}{1-{\tan }^{2}a}$$\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">tan</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">tan</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">tan</span></span>}}^{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}$==$=\frac{-2\sqrt{15}}{-14}=\frac{\sqrt{15}}{7}$$<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}}}{<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">14</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">7</span></span>}}$.

Bài 2. Tính

a) sin biết sin a = $\frac{3}{4}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}$ và 0 < a < $\frac{\pi }{2}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$;

b) cos$\frac{3\pi }{8}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}$.cos$\frac{\pi }{8}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}$ + sin$\frac{3\pi }{8}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}$.sin$\frac{\pi }{8}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}$.

Hướng dẫn giải

a) Vì 0$\frac{\pi }{2}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$ nên cosa > 0.

Ta có: sin2a + cos2a = 1 ⇒ cos2a = 1 – sin2a = 1-=$\frac{7}{16}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">7</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">16</span></span>}}$

⇒ cosa = null.

Vậy sin$=\sin a\cos \frac{\pi }{3}-\cos a\sin \frac{\pi }{3}=\frac{3}{4}.\frac{1}{2}-\frac{\sqrt{7}}{4}.\frac{\sqrt{3}}{2}=\frac{3-\sqrt{21}}{8}$$<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">.</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">7</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">.</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">21</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}$ .

Suy ra: $\cos \frac{3\pi }{8}.\cos \frac{\pi }{8}+\sin \frac{3\pi }{8}.\sin \frac{\pi }{8}=\frac{\sqrt{2}}{4}+\frac{\sqrt{2}}{4}=\frac{\sqrt{2}}{2}$$\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">.</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">cos</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">3</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">.</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">8</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$.

Bài 3. Tính

a) cos(–15°) + cos255°;

b) sin$\frac{13\pi }{24}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">13</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">24</span></span>}}$sin$\frac{5\pi }{24}$$\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">5</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">24</span></span>}}$.

Hướng dẫn giải

a) Ta có:

cos(-15o) + cos255o = 2.cos$\frac{-15°+255°}{2}$$\frac{<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">°</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">255</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">°</span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$.cos$\frac{-15°-255°}{2}$$\frac{<span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">15</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">°</span></span><span\; style="font-size:18px;"><span\; style="line-height:2;">-</span></span>\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">255</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">°</span></span>}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$

= 2.cos120o.cos(135o) = 2

Vậy cos(–15°) + cos255° = $\frac{\sqrt{2}}{2}$$\frac{\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}$.

b) Ta có:

Vậy $\sin \frac{13\pi }{24}\sin \frac{5\pi }{24}=\frac{1+\sqrt{2}}{4}$$\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">13</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">24</span></span>}}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">sin</span></span>}\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">5</span></span>}\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">�</span></span>}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">24</span></span>}}<span\; style="font-size:18px;"><span\; style="line-height:2;">=</span></span>\frac{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">1</span></span>}<span\; style="font-size:18px;"><span\; style="line-height:2;">+</span></span>\sqrt{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">2</span></span>}}}{\mathrm{<span\; style="font-size:18px;"><span\; style="line-height:2;">4</span></span>}}$.

Video bài giảng Toán 11 Bài 2: Công thức lượng giác - Kết nối tri thức