Math topic中文

Special angles

exact values of the trigonometric functions for special angles ( $30^\circ$ , $45^\circ$ , $60^\circ$ ) or ( $\frac{\pi}{6}$ , $\frac{\pi}{4}$ , $\frac{\pi}{3}$ )

Practice questions

1已知 $\left| \overrightarrow{a}\right|=4$ ， $\left| \overrightarrow{b}\right|= \sqrt{2}$ ， $\overrightarrow{a}$ 与 $\overrightarrow{b}$ …
2In the triangle $ABC$ , $AB = x\ \text{cm}$ , $AC = 3x\ \text{cm}$ and angle $BAC = \dfrac{2\pi}{3}$ . If $x$ is increasing…
3The normal to the curve $y = 2\tan 3x - 1$ at point $P\left(\frac{\pi}{3}, -1\right)$ crosses the $y$ -axis at point $A$ …
4The diagram shows a prism in which its cross-section is a regular hexagon of side $(5 + \sqrt{3})$ cm. Find an expressio…
5计算：cot44°•cot45°•cot46°= ．
6The equation of a curve is given by $y = x + \sin^2 x$ for $0 \le x \le \pi$ . Write down the expression for $\frac{\mathrm{d}y}{\mathrm{d}x}$ …
7已知点 $P\left(-1,3\tan\dfrac{5\pi}{6}\right)$ 是角 $\theta$ 终边上一点，则 $\cos\theta$ 的值为 $(\quad)$ A. $\dfrac{1}{2}$ B. $\dfrac{\sqrt{3}}{2}$ …
8已知函数 $f(x)=\cos(ωx+φ)(ω>1,ω\in Z,|φ|< \dfrac {π}{2})$ 在区间 $[\dfrac{π}{3},\dfrac{3π}{4}]$ 上是单调函数，且图象过点 $(\dfrac{π}{12},0)$ …
9已知 $a$ ， $b$ ， $c$ 是 $\triangle ABC$ 中角 $A$ ， $B$ ， $C$ 的对边， $S$ 是 $\triangle ABC$ 的面积.若 $a^{2}+c^{2}=b^{2}+ac$ ， $($ Ⅰ…
10In $\Delta ABC$ , $AC = 12\text{ cm}$ , $\angle ABC$ is a right angle, $\angle ACB = \frac{\pi}{6}$ radians and $M$ is the…
11Do not use a calculator in this question. Express $\sin 22.5^\circ$ in the form of $\frac{\sqrt{a - \sqrt{a}}}{a}$ ,…
12 $\triangle ABC$ 的边 $BC$ 在平面 $α$ 内， $A$ 在 $α$ 内的射影是 $A_{1}$ ，设 $\triangle ABC$ 的面积为 $2$ ，平面 $ABC$ 和平面 $α$ 所成的二面角的大小为 $45°$ …
13The diagram shows a chocolate bar in the form of a triangular prism and the cross-section of the chocolate bar is an iso…
14Without the use of calculator, show that $\sin 75^\circ = \frac{\sqrt{2} + \sqrt{6}}{4}$ .
15在平面直角坐标系 $xOy$ 中，曲线 $C$ 的参数方程为 $\begin{cases} \overset{x=4\cos a+2}{y=4\sin a}\end{cases}(a$ 为参数 $)$ ，以 $O$ 为极点，以 $x$ 轴…
16Given 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…
17Given that $\dfrac{\sin(A-B)}{\sin(A+B)} = \dfrac{2}{3}$ , prove that $\tan A = 5\tan B$ . Hence, or otherwise, solve the…
18在 $\triangle ABC$ 中，若 $\left|2\cos A-1\right|+{\left(\sqrt{3}-\tan B\right)}^{2}=0$ ，则 $\angle C=$ ___.
19It is given that $f'(x)=6\cos 3x-12\sin 3x$ and $f\left(\frac{\pi}{3}\right)=0$ . Find $f(x)$ . Find the exact value of $f''\left(\frac{\pi}{12}\right)$ …
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. $\dfrac {7 \sqrt {2}}{10}$ …
21在 $\triangle ABC$ 中，若 $b^{2}=ac$ ， $B=60^{\circ}$ ，则 $A=$ ( ) A. $120^{\circ}$ B. $60^{\circ}$ C. $45^{\circ}$ D.…
22A particle, P, starts moving in a straight line with a velocity of 8 m/s at a displacement of 3 m from a fixed point O.…
23已知数列 $\{a_{n}\}$ 为等差数列，且 $a_{1}+a_{7}+a_{13}=π$ ，则 $\tan (a_{2}+a_{12})$ 的值为 $()$ A. $\sqrt {3}$ B. $- \sqrt {3}$ …
24A conical vessel, whose vertical angle is $30^\circ$ as shown in the diagram, has water poured into it at such a rate th…
25It 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…
26一个三角形用斜二测画法画出来的直观图是边长为 $2$ 的正三角形，则原三角形的面积是 $()$ A. $2 \sqrt {6}$ B. $4 \sqrt {6}$ C. $\sqrt {3}$ D. 都不对
27Show that $\frac{d}{dx}(\tan^{3} 2x) = 6\sec^{4} 2x - 6\sec^{2} 2x$ . Hence, find the exact value of $\displaystyle \int_{0}^{\pi/8} \sec^{4} 2x\, dx$ …
28A farmer uses 160 m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a recta…
29The diagram shows the curve $y = a\cos bx + c$ for $0 \le x \le \tfrac{3\pi}{2}$ radians. A minimum point $(\tfrac{\pi}{2}, -2)$ …
30已知函数 $f(x)= \sqrt{3}\sin (ωx- \dfrac{π}{6})+b(ω > 0)$ ，且函数图象的对称中心到对称轴的最小距离为 $\dfrac{π}{4}$ ，当 $x∈[0, \dfrac{π}{4}]$ …
31The 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 $\pi$ …
32Differentiate $3x \sin 2x$ with respect to $x$ . Hence, evaluate $\int_{0}^{\frac{\pi}{4}} x \cos 2x \text{ d}x$ .
33Given 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 $k$ …
34Prove that $3\cos^2 x + \sin^2 x = 2 + \cos 2x$ . Hence, using your result from part (a), Hence, using your result from p…
35求函数 $f(x)=\tan x$ ， $x\in[-\dfrac{\pi}{4},\dfrac{\pi}{3}]$ 的最大值和最小值.
36在直角坐标系 $xOy$ 中，圆 $C$ 的方程为 $(x- \sqrt {3})^{2}+(y+1)^{2}=9$ ，以 $O$ 为极点， $x$ 轴的非负半轴为极轴建立极坐标系． (1)求圆 $C$ 的极坐标方程； (2)直线 $OP$ …
37The diagram shows part of the curve $y = \sin x$ . A is a point on the curve that lies on the line $x = \frac{\pi}{3}$ . T…
38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…
39已知向量 $\overrightarrow{a}=(-2,1)$ ， $\overrightarrow{b}=(3,-4)$ ． (1)求 $( \overrightarrow{a}+ \overrightarrow{b})⋅(2 \overrightarrow{a}- \overrightarrow{b})$ …
40Prove that $\csc 2x + \cot 2x = \cot x$ . Hence, deduce the value of $\cot 15^\circ$ in surd form. Using part (i), solve…
41Prove the trigonometric identity: $\dfrac{\csc^{2}A + 2\cot A}{(\cos A + \sin A)^{2}} = \csc^{2}A$ Hence solve the equ…
42直线 $\sqrt{3}x{-}y{-}3{=}0$ 的倾斜角为 $({ })$ A. $30^{{∘}}$ B. $60^{{∘}}$ C. $\ 120^{{∘}}$ D. $\ 150^{{∘}}$
43Without using the calculator, simplify the following trigonometric functions. $\sin \left(x + \frac{4\pi}{3}\right)$ $\cos 75^\circ$ …
44Evaluate $\int_{0}^{\frac{\pi}{3}} \left( \frac{1}{2} \sin 3x - \cos^2 x \right) dx$ , correct to 2 decimal places.
45It 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…
46A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all…
47计算： $(π- \sqrt {5})^{0}+ \sqrt {4}+(-1)^{2015}- \sqrt {3}\tan 60^{\circ}$ ．
48已知函数 $g(x)= \dfrac {1}{x\cdot \sin \theta }+\ln x{在}[1,+∞)$ 上为增函数，且 $θ∈(0,π),f(x)=mx- \dfrac {m-1+2e}{x}-\ln x,m∈R$ ． (1…
49已知函数 $y=\cos x$ 与 $y=\sin (2x+φ)(0\leqslant φ\leqslant π)$ ，它们的图象有一个横坐标为 $\dfrac {π}{3}$ 的交点，则 $φ=()$ A. $\dfrac {π}{6}$ …
50计算： ${\left(1- \sqrt{2}\right)}^{0}-\tan 60^{\circ}+{\left(- \dfrac{1}{2}\right)}^{-1}+ \dfrac{2}{ \sqrt{3}-1}$ ；
51测量塔高 $AB$ 时，选取与塔底 $B$ 在同一水平面内的两个测量点 $C$ 与 $D$ ，现测得 ${\rm∠}BCD=\dfrac{5π}{12}$ ， ${\rm∠}BDC=\dfrac{π}{3}$ ， $CD=100$ ，在点 $C$ …
52计算： $|-2|-(π-2)^{0}+(\dfrac{1}{3})^{-1}-4\tan 45°.$
53The diagram shows the curve $y = p \sin \frac{x}{q} + r$ for $0 \le x \le 8\pi$ radians. The curve has a minimum point a…
54Show that $\dfrac{d}{dx}(\cot x) = -\csc^{2} x$ . Hence find the value of $\displaystyle \int_{\pi/4}^{\pi/2} -\cot^{2} x\, dx$ …
55(1)计算： ${{(-\dfrac{1}{3})}^{-2}}+{{(\pi -\tan 60{}^\circ )}^{0}}-2\sqrt{3}\cos 30{}^\circ (2)$ 解方程： $x(x-2)=3x+1$
56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 $\triangle ABC$ …
57The diagram shows part of the curve $y = 2 \sin 2x + 1$ . $P$ is the maximum point of the curve and $Q$ is the point on t…
58已知 $θ∈( \dfrac {π}{2},\;π)$ ，且 $\sin θ= \dfrac {3}{5}$ ． $($ Ⅰ $)\tan θ=$ ___ ； $($ Ⅱ $)$ 求 $\cos (θ+ \dfrac {π}{3})$ …
59Show that $\cos(A + B) \cos(A - B) = \cos^2 A + \cos^2 B - 1$ . Hence, determine the value of $\cos 15^\circ \cos 75^\circ$ …
60Without using a calculator, show that $\cot 15^\circ = 2 + \sqrt{3}$ . Hence show that $\csc^2 15^\circ = 4 \cot 15^\circ$ …