9.1 基本三角关系式 Basic Trigonometric Identities - 独中课程 | 线上补习

高中一 | 高级数学

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9.1 Basic Trigonometric Identities 基本三角关系式

1. Prove that 试证

(a) $cos^4x - sin^4x = cos^2x - sin^2x$

(b)

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(d) $\frac{{sin}^2{x}-{cos}^2{x}}{1+2sin{x}cos{x}}=\frac{tan{x}-1}{tan{x}+1}$

(e) $\frac{cos{(}-x)-1}{sin{(}3\pi+x)}=\frac{1}{cos{e}c(\pi-x)-tan{(}x-\frac{3\pi}{2})}$

(f) $\frac{2(cos{x}+1)}{sec{x}+tan{x}+1}=1-sin{x}+cos{x}$

(g) $\frac{tan{(}\pi+x)+cot{(}\frac{3\pi}{2}-x)}{1+tan{(}2\pi-x)cot{(}\frac{\pi}{2}+x)}=2sin{x}cos{x}$

(2)
If $u=\frac{1+sin{x}}{cos{x}}$ , prove that $\frac{1}{u}=\frac{1-sin{x}}{cos{x}}$

若 $u=\frac{1+sin{x}}{cos{x}}$ , 证明 $\frac{1}{u}=\frac{1-sin{x}}{cos{x}}$

(3)
If $tan^2A - 2tan^2B = 1$ , show that $2cos^2A - cos^2B = 0$

若 $tan^2A - 2tan^2B = 1$ , 证明 $2cos^2A - cos^2B = 0$

(4)
If $sin A + cos A = \frac{1}{2}$ , find the values of
若 $sin A + cos A = \frac{1}{2}$ , 求下列各式的值
(a) sin A cos A
(b) tan A + cot A

Answer 答案：

(a) $-\frac{3}{8}$

(b) $-2\frac{2}{3}$

(5)
If $tan x + cot x = 4\frac{1}{2}$ , then find the value of $tan2x + secx cscx + cot2x$
若 $tan x + cot x = 4\frac{1}{2}$ 则求 $tan2x + secx cscx + cot2x$ 之值

Answer 答案：

$22\frac{3}{4}$

(6)
Given that $sin x + cos x = 1.2$ , find $sin x - cos x$ , if $0 < x < 45^{\circ}$
已知 $sin x + cos x = 1.2$ , 求 $sin x - cos x$ , 当 $0 < x < 45^{\circ}$

Answer 答案：

$-\sqrt{0.56}$

(7)
Given that $msec{\theta}=1+tan{\theta}$ , $nsec{\theta}=1-tan{\theta}$ , prove that $m^2 + n^2 = 2$ .

已知 $msec{\theta}=1+tan{\theta}$ , $nsec{\theta}=1-tan{\theta}$ , 证明 $m^2 + n^2 = 2$ .

(8)
If $\frac{a}{sin{A}}=\frac{b}{cos{A}}$ show that $sin{A}cos{A}=\frac{ab}{a^2+b^2}$

若 $\frac{a}{sin{A}}=\frac{b}{cos{A}}$ 证明 $sin{A}cos{A}=\frac{ab}{a^2+b^2}$