Math topic中文

Simplification

simplification of trigonometric expressions

Practice questions

1Solve the equation $2 \cot^2 2x = 1 - \frac{5}{\sin 2x}$ for $-180^\circ \le x \le 180^\circ$ .
2Prove that $\sin^{3} x\sec^{2} x + \sin x = \tan x\sec x$ . Solve the equation $\sin^{3} x\sec^{2} x + \sin x = 5$ for $0 \le x \le 360^\circ$ …
3计算：cot44°•cot45°•cot46°= ．
4Do not use a calculator in this question. Express $\sin 22.5^\circ$ in the form of $\frac{\sqrt{a - \sqrt{a}}}{a}$ ,…
5Solve the simultaneous equations $\begin{aligned} y &= x^2 - 6, \\ y &= 2x - 3. \end{aligned}$ Prove that $\frac{3 \cos \theta + \cos 2\theta - 1}{\cos^2 \theta + 2 \cos \theta} = 2 - \sec \theta$ …
6It is given that $f(x)$ is such that $f'(x) = \sin 2x + \cos 3x$ . Given also that $f\left(\frac{\pi}{6}\right) = 0$ , sho…
7A curve has an equation $y = 2x\sqrt{2x+1}$ . Show that $\frac{dy}{dx} = \frac{ax+b}{\sqrt{2x+1}}$ , where $a$ and $b$ are…
8Given that $y = 2\cos x \sin 2x - \sin x \cos 2x$ , show that $\dfrac{dy}{dx} = k\cos x \cos 2x$ , stating the value of $k$ …
9Prove that $3\cos^2 x + \sin^2 x = 2 + \cos 2x$ . Hence, using your result from part (a), Hence, using your result from p…
10Prove the identity $\frac{\cos x}{\text{cosec } x - 1} + \frac{\cos x}{\text{cosec } x + 1} = 2 \tan x$ . Solve the equat…
11Prove that $\csc 2x + \cot 2x = \cot x$ . Hence, deduce the value of $\cot 15^\circ$ in surd form. Using part (i), solve…
12Prove the trigonometric identity: $\dfrac{\csc^{2}A + 2\cot A}{(\cos A + \sin A)^{2}} = \csc^{2}A$ Hence solve the equ…
13Without using the calculator, simplify the following trigonometric functions. $\sin \left(x + \frac{4\pi}{3}\right)$ $\cos 75^\circ$ …
14Prove the identity $\dfrac{\cos^{2} x - \sin^{2} x}{1 + 2\sin x\cos x} = \dfrac{1 - \tan x}{1 + \tan x}$ . Solve the equa…
15Show that $\cos(A + B) \cos(A - B) = \cos^2 A + \cos^2 B - 1$ . Hence, determine the value of $\cos 15^\circ \cos 75^\circ$ …
16Prove the identity $\csc 4x + \cot 4x = \cot 2x$ . Hence, find, for $-180^\circ \le \theta \le 180^\circ$ , the values of…
17Prove the identity $\frac{1+\cos \theta}{\sin \theta} + \frac{\sin \theta}{1+\cos \theta} = 2\csc \theta$ . Hence, find a…
18State the principal value of $\sin^{-1}\left(-\dfrac{\sqrt{3}}{2}\right), the values between which the principal value of$ …
19A and B are acute angles, such that $\cos A = \frac{3}{5}$ and $\tan B = \frac{12}{5}$ .\n\nFind the exact values of $\tan 2A$ …
20Prove that $\dfrac{\sin \theta}{(\sec \theta - \tan \theta)(\sin \theta + 1)} = \tan \theta$ . Find all the values of $\theta$ …
21Show that $\frac{(\sin\theta-\cos\theta)(1+\sin\theta\cos\theta)}{\cos^{3}\theta}=\tan^{3}\theta-1.$ Hence, for $0<\theta<2\pi$ …
22The point $(a, 2)$ is a stationary point on the graph $y = x^3 - x^2 - x + b$ , where $a$ and $b$ are positive constants.…
23Prove the identity $\frac{\sin 2A + \cos A}{1 - \cos 2A + \sin A} = \cot A$ Hence, solve, for $-\pi \le x \le \pi$ , the…
24Factorise $\sin^3 x + \cos^3 x$ completely. Show that $\frac{\sin^3 x + \cos^3 x}{\sin x + \cos x} = 1 - \frac{1}{2}\sin 2x$ …
25Do not use a calculator in this question. Given that $\tan\theta=\frac{3}{5}$ and that $\theta$ is non-acute, find the e…
26Given that $y = \sec x$ , show that $\frac{dy}{dx} = \sec x \tan x$ . Hence show that $y^4 - y^2 = \left(\frac{dy}{dx}\right)^2$ …
27The diagram shows a triangle $PQR$ . Point $M$ lies on $QR$ such that $QM = 1$ unit and $MR = 3$ units. Angle $PRQ = 90^\circ$ …
28Prove the identity $\frac{\tan^2 y - \sin^2 y}{\sec^2 y (\csc^2 y - 1)} = \tan^2 y \sin^4 y$ . Hence solve the equation…
29 $A$ and $B$ are acute angles such that $\sin(A - B) = \frac{1}{4}$ and $\sin A \cos B = \frac{3}{5}$ . Without using a ca…
30A calculator must not be used in this question. Show that $\sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}$ . Hence, fi…
31Prove that $\dfrac{\tan x}{1+\sec x}+\dfrac{1+\sec x}{\tan x}=2\csc x$ . Hence, solve $\dfrac{\tan 2x}{1+\sec 2x}+\dfrac{1+\sec 2x}{\tan 2x}=3\sec x$ …
32In the diagram, $AC = 35\text{ cm}$ , $EC = 15\text{ cm}$ and angle $ACB = \theta$ radians. $ED$ and $AB$ are perpendicul…
33Prove the identity $\cot \theta - 1 = \dfrac{\cos 2\theta}{\sin \theta (\cos \theta + \sin \theta)}$ . Hence solve the eq…
34Prove the identity $\frac{\sin x}{1-\cos x} + \frac{1-\cos x}{\sin x} = 2\csc x.$ Hence solve the equation $\frac{\sin x}{1-\cos x} + \frac{1-\cos x}{\sin x} = 5$ …
35It is given that $\text{f}(x) = 2 - 3\sin 3x$ and $\text{g}(x) = 4\cos\left(\frac{x}{2}\right) - 1$ . State the least and…
36Given that $\theta$ is an acute angle and $\sin \theta = m$ , express in terms of $m$ , Another acute angle $\alpha$ is su…
37Given that $\frac{\sin(A + B)}{\sin(A - B)} = \frac{1}{5}$ , prove that $2\tan A + 3\tan B = 0$ . Hence, solve the equatio…
38Show that the value of $\sin 105^\circ$ can be expressed as $\tfrac{1}{4}(p+q)$ , where $p$ and $q$ are constants. Given…
39Prove that $\frac{2 - \sec^2 \theta}{\sec \theta (\sin \theta + \cos \theta)} = 1 - \tan \theta$ . Hence solve the equati…
40A and B are acute angles such that $\sin(A - B) = \frac{3}{8}$ and $\sin A \cos B = \frac{5}{8}$ . Without using a calcul…
41Solve the equation $6^{z}(6^{z} - 1) = 12.$ A curve has the equation $y = x e^{3x}$ . Find $\dfrac{dy}{dx}$ . Find the val…
42Prove the identity $\dfrac{\sin\theta}{1+\cos\theta} = \tan \dfrac{\theta}{2}$ . Hence solve the equation $\dfrac{1+\cos\theta}{\sin\theta} = -3$ …
43Show that $4 \sin x + 2 \cos \left( x + \frac{\pi}{6} \right) = a \cos x + b \sin x$ , where $a$ and $b$ are exact values…
44Prove that $\dfrac{\sec^2 x}{1 + \sec^2 x} = \dfrac{2}{\cos 2x + 3}$ . Hence, solve the equation $\dfrac{\sec^2 2x}{1 + \sec^2 2x} = \dfrac{4}{7}$ …
45It is given that $\sin A = 0.3$ , where $A$ is obtuse. Find the following trigonometric ratios. $\sec A$ , $\cos 2A$ , $\tan(A + 45^\circ)$ …
46Show that $\tan x + \frac{\cos x}{1 + \sin x} = \sec x$ . Explain, without the use of calculator, why the equation $\tan x + \frac{\cos x}{1 + \sin x} = -\frac{1}{3}$ …
47 $f(x)$ is such that $f'(x) = \sin \frac{1}{4}x - \cos 4x$ . Given that $f(2\pi) = 1$ , show that $16f''(x) + f(x) = a \sin 4x + b$ …
48Prove the identity $\frac{1 - \sin^4 x}{\sin^2 x} = \cot^2 x + \cos^2 x$ . Hence solve the equation $\cot^2 2x + \cos^2 2x = 0$ …
49Solve the equation $3 \cos 2x = \sin x + 2$ , for $0^\circ \le x \le 360^\circ$ . Find all the angles between 0 and 5 whic…
50Prove the identity $\sin\left(\frac{\pi}{3}+x\right)-\sin\left(\frac{\pi}{3}-x\right)=\sin x$ . Hence solve the equation…
51Show that $\frac{\text{d}}{\text{d}\theta}(\cos^4 \theta + \sin^4 \theta) = p \sin q\theta$ , where $p$ and $q$ are integ…
52Prove that $\csc 2x + \cot 2x = \cot x$ . Hence, solve $\csc x + \cot x = \sqrt{3}$ where $0 \le x \le 2\pi$ .
53It is given that $f(x)$ is such that $f'(x) = \cos 4x - \sin 2x$ . Given also that $f\left(\frac{\pi}{2}\right) = \frac{1}{4}$ …
54It is given that $y = (x - 2)(2x - 5)^3$ . Obtain an expression for $\frac{dy}{dx}$ in the form $(ax + b)(2x - 5)^2$ , whe…
55Show that $\dfrac{\tan x}{1+\sec x}+\dfrac{1+\sec x}{\tan x}=\dfrac{2}{\sin x}$ . Hence solve the equation $\dfrac{\tan x}{1+\sec x}+\dfrac{1+\sec x}{\tan x}=1+3\sin x$ …
56Prove that $\dfrac{\tan 2\theta}{\sec 2\theta - 1} = \cot \theta$ .
57The equation of a curve is $y = x\sqrt{x+1}$ . Show that $\frac{d}{dx}\left(x\sqrt{x+1}\right)=\frac{3x+2}{2\sqrt{x+1}}.$ …
58Prove the identity $\dfrac{\tan A - \cot A}{\tan A + \cot A} = 1 - 2\cos^{2} A$ . Hence solve the equation $\frac{\tan 2\theta - \cot 2\theta}{\tan 2\theta + \cot 2\theta} = \frac{1}{2}$ …
59It is given that $\sin A = 0.3$ , where $A$ is obtuse. Find the following trigonometric ratios. $\sec A$ , $\cos 2A$ . $\tan(A + 45^\circ)$ …
60Given that $y = A - B \cos 4x - \frac{1}{2} \sin 2x$ and $\frac{\text{d}^2y}{\text{d}x^2} + 4y = 3 \cos 4x + 1$ , find th…