Math topic中文

Identities

use of:

$\dfrac{\sin A}{\cos A} = \tan A$ , $\dfrac{\cos A}{\sin A} = \cot A$ , $\sin^{2}A + \cos^{2}A = 1$ , $\sec^{2}A = 1 + \tan^{2}A$ , $\operatorname{cosec}^{2}A = 1 + \cot^{2}A$
the expansions of $\sin(A \pm B)$ , $\cos(A \pm B)$ and $\tan(A \pm B)$
the formulae for $\sin 2A$ , $\cos 2A$ and $\tan 2A$
the expression of $a\cos\theta + b\sin\theta$ in the form $R\cos(\theta \pm \alpha)$ or $R\sin(\theta \pm \alpha)$

Practice questions

1Solve the equation $2 \cot^2 2x = 1 - \frac{5}{\sin 2x}$ for $-180^\circ \le x \le 180^\circ$ .
2已知 $\sin (α+ \dfrac {π}{12} )= \dfrac { \sqrt {5}}{5}$ ，则 $\sin ( \dfrac {π}{3} -2α)= (\:\:\:\:)$ A. $\dfrac {4}{5}$ …
3 $8.$ 设 $\triangle ABC$ 的内角 $A$ ， $B$ ， $C$ 所对的边分别为 $a$ ， $b$ ， $c$ ，若 $b\cos C+c\cos B=a\sin A$ ，则 $\triangle ABC$ 的形状为…
4Prove 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 $0 \le x \le 360^\circ$ …
5Given that $\cos x = -\frac{4}{5}$ and $\tan x > 0$ , find $\sin 2x$ , $\cos \frac{x}{2}$ .
6计算：cot44°•cot45°•cot46°= ．
7函数 $f(x)=\sin x-\cos x$ ，则函数 $f(x)$ 的最大值为 $(\quad)$ A. $2$ B. $\sqrt{{2}}$ C. $0$ D. $-2$
8The equation of a curve is given by $y = x + \sin^2 x$ for $0 \le x \le \pi$ . Write down the expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
9Given that $\sin A = -\frac{3}{5}$ and $\cos B = -\frac{8}{17}$ , where angle $A$ and angle $B$ are in the same quadrant,…
10Using $R \cos(\theta - \alpha)$ , where $R > 0$ and $0^\circ \le \alpha \le 90^\circ$ , solve the equation $5 \cos \theta + 8 \sin \theta = 6$ …
11选修 $4-4$ ：坐标系与参数方程在直角坐标系 $xOy$ 中，曲线 $C_{1}$ 的参数方程为 $\begin{cases}x= \sqrt{3}\cos α, \\ y=\sin α\end{cases}(α$ 为参数 $).$ …
12Do not use a calculator in this question. Express $\sin 22.5^\circ$ in the form of $\frac{\sqrt{a - \sqrt{a}}}{a}$ ,…
13Without the use of calculator, show that $\sin 75^\circ = \frac{\sqrt{2} + \sqrt{6}}{4}$ .
14Given that the function $\text{f}(x) = 1 + 2\sin^2 3x + 4\cos^2 3x$ , Express $\text{f}(x)$ in the form $a + b\cos 6x$ . H…
15The diagram shows two chords AB and CD intersect at X such that angle CXB = $90^\circ$ . Given that $AC = 5\ \text{cm}$ ,…
16若 $\tan α=2$ ， $\tan (β-α)=3$ ，则 $\tan (β-2α)$ 的值为 $()$ A. $\dfrac {1}{7}$ B. $- \dfrac {1}{7}$ C. $\dfrac {1}{9}$ …
17The diagram shows two triangles DAB and DBC. It is given that BD = 5 m and BC = 2 m. Angle DAB = angle DBC = angle DEC =…
18Given that $\dfrac{\sin(A-B)}{\sin(A+B)} = \dfrac{2}{3}$ , prove that $\tan A = 5\tan B$ . Hence, or otherwise, solve the…
19已知 $\sin(α-\dfrac{π}{6})=\dfrac{1}{3}$ ，则 $\cos(\dfrac{π}{3}-2α)=(\quad)$ A. $-\dfrac{7}{9}$ B. $-\dfrac{2}{3}$ C.…
20已知 $α$ 是 $\triangle ABC$ 的一个内角， $\tan α= \dfrac {3}{4}$ ，则 $\cos (α+ \dfrac {π}{4} )$ 等于 $(\:\:\:\:)$ A. $\dfrac {7 \sqrt {2}}{10}$ …
21The diagram below shows a circle with chords $PQ$ and $RS$ intersect at the point $X$ . $P, Q, R$ and $S$ are points on t…
22Solve the simultaneous equations $\begin{aligned} y &= x^2 - 6, \\ y &= 2x - 3. \end{aligned}$ Prove that $\frac{3 \cos \theta + \cos 2\theta - 1}{\cos^2 \theta + 2 \cos \theta} = 2 - \sec \theta$ …
23已知 $0 < α < \dfrac{π}{2} < β < π$ ， $\overset{→}{a}=(\cos α,3) \overset{→}{b}=(-4,\sin α)$ ，且 $\overset{→}{a}⊥ \overset{→}{b}$ …
24Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
25The equation of a curve is $y = e^{-2x}\tan x$ . Show that $\dfrac{dy}{dx} = e^{-2x}(1 - \tan x)^2$ . Find, in terms of $\pi$ …
26已知函数 $f(x)=\sin ωx+ \sqrt {3}\cos ωx$ 的最小正周期为 $π$ ， $x∈R$ ， $ω > 0$ 是常数． (1)求 $ω$ 的值； (2)若 $f( \dfrac {θ}{2}+ \dfrac {π}{12})= \dfrac {6}{5}$ …
27Prove that $3\cos^2 x + \sin^2 x = 2 + \cos 2x$ . Hence, using your result from part (a), Hence, using your result from p…
28已知不等式 $a+ \sqrt{3}b\leqslant \left|{x}^{2}-1\right|$ 对于满足条件 ${a}^{2}+{b}^{2}=1$ 的任意实数 $a$ ， $b$ 恒成立，实数 $x$ 的取值范围是．
29The diagram shows a trapezium $ABCD$ . The point $X$ lies on line $AB$ such that $DX = 12\text{ cm}$ and $CX = 9\text{ cm}$ …
30已知锐角 $α$ ， $β$ 满足： $\cos α= \dfrac {1}{3}$ ， $\cos (α+β)=- \dfrac {1}{3}$ ，则 $\cos (α-β)=()$ A. $- \dfrac {1}{2}$ B.…
31已知 $α$ 是第二象限的角，且 $\sin (π+α)=- \dfrac {3}{5}$ ，则 $\tan 2α$ 的值为 ___ ．
32Prove the identity $\frac{\cos x}{\text{cosec } x - 1} + \frac{\cos x}{\text{cosec } x + 1} = 2 \tan x$ . Solve the equat…
33The diagram below shows two horizontal lines $AB$ and $CD$ , where $L$ is the vertical distance between the lines $AB$ an…
34在平面直角坐标系中，曲线 $C_{1}$ 的参数方程为 $\begin{cases}x=-1+\sqrt{5}\cos α\\ y=-2+\sqrt{5}\sin α\end{cases}(α$ 为参数 $).$ 以坐标原点 $O$ 为极点…
35The diagram below shows an isosceles triangle $ABC$ , where $AB = AC = 2\text{ cm}$ , and a square $BCDE$ . $M$ is the midp…
36Prove that $\csc 2x + \cot 2x = \cot x$ . Hence, deduce the value of $\cot 15^\circ$ in surd form. Using part (i), solve…
37Prove the trigonometric identity: $\dfrac{\csc^{2}A + 2\cot A}{(\cos A + \sin A)^{2}} = \csc^{2}A$ Hence solve the equ…
38Without using the calculator, simplify the following trigonometric functions. $\sin \left(x + \frac{4\pi}{3}\right)$ $\cos 75^\circ$ …
39Evaluate $\int_{0}^{\frac{\pi}{3}} \left( \frac{1}{2} \sin 3x - \cos^2 x \right) dx$ , correct to 2 decimal places.
40It is given that $\mathrm{f}(x)$ is such that $\mathrm{f}'(x) = 3 \sin x \cos x$ and $\left(\frac{\pi}{2}, 1\right)$ is…
41The diagram shows part of the curve $y = \sqrt{4x + 5}$ . The normal of the curve at $P$ meets the $x$ -axis at $T$ . The $x$ …
42The diagram shows the front view of a rectangular block $ABCE$ , with dimensions 14 cm by 8 cm, placed on a ramp, $VE$ , t…
43The equation of a curve is $x^2 + 4y^2 = 20$ and the equation of a line is $y = k - x$ , where $k$ is a constant. Find th…
44Prove 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…
45已知函数 $f(x)=2\sin ωx\cos ωx+2 \sqrt {3}\sin ^{2}ωx- \sqrt {3}(ω > 0)$ 的最小正周期为 $π$ ． (1)求 $ω$ 的值及函数 $f(x)$ 的单调减区间； (2)将函数 $f(x)$ …
46A particle travels in a straight line so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v\text{ m s}^{-1}$ …
47In the diagram, BE intersects AD at F. AF = 7 cm and FD = 4 cm. $\angle ABF$ , $\angle ACD$ and $\angle DEF$ are right an…
48The diagram shows a pair of similar triangles $YUX$ and $VUW$ . Point $Y$ lies on the straight line $VU$ such that $VY = 12\text{ cm}$ …
4912 Solve $0.5\sin 2x = 3\sin x$ for $0 < x < 2\pi$ . Solve, for values of $\theta$ between $0^\circ$ and $180^\circ$ , $\csc\left(\frac{\theta}{2}-40^\circ\right)+5=0.$ …
50Show that $\dfrac{d}{dx}(\cot x) = -\csc^{2} x$ . Hence find the value of $\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx$ …
51The diagram shows an area that is enclosed by glass panels at $AB$ , $BC$ and $AD$ . $AB = 10\text{ m}$ , $AD = 30\text{ m}$ …
52The diagram shows the side view of a 14 cm by 8 cm rectangular block $PQRS$ , placed on a ramp, $VS$ , tilted at an acute…
53已知 $\triangle ABC$ 的内角 $A$ ， $B$ ， $C$ 的对边分别为 $a$ ， $b$ ， $c$ ，满足 $b\cos A+a\cos B=2c\cos B$ ， $b=2\sqrt{3}$ ，则 $\triangle ABC$ …
54已知 $θ∈( \dfrac {π}{2},\;π)$ ，且 $\sin θ= \dfrac {3}{5}$ ． $($ Ⅰ $)\tan θ=$ ___ ； $($ Ⅱ $)$ 求 $\cos (θ+ \dfrac {π}{3})$ …
55Show that $\cos(A + B) \cos(A - B) = \cos^2 A + \cos^2 B - 1$ . Hence, determine the value of $\cos 15^\circ \cos 75^\circ$ …
56设 $\vartriangle ABC$ 的内角 $A$ 、 $B$ 、 $C$ 所对的边分别为 $a$ ， $b$ ， $c$ ，若 $a\sin A=b\cos C+c\cos B$ ，则 $\vartriangle ABC$ 的形状为…
57Given that $\sin A = -\frac{4}{5}$ , $\tan B = -\frac{5}{12}$ and $\cos A > 0$ , where $A$ and $B$ are in different quadra…
58Without using a calculator, show that $\cot 15^\circ = 2 + \sqrt{3}$ . Hence show that $\csc^2 15^\circ = 4 \cot 15^\circ$ …
59The expression $8 \sin \theta - 3 \cos \theta$ is defined for $0^\circ \le \theta \le 180^\circ$ . Using $R \sin(\theta - \alpha)$ …
60下列叙述正确的是 $(\quad)$ A. $y=\tan x$ 的定义域是 $R\:\:\:\:\:\:\:\:\:\:\:\:\:\:$ B. $y=\sqrt{x}$ 的值域为 $R$ C. $y=\dfrac{1}{x}$ …