Math topic中文

Amplitude periodicity

amplitude, periodicity and symmetries related to sine and cosine functions

Practice questions

1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. $-2$
2Using $R \cos(\theta - \alpha)$ , where $R > 0$ and $0^\circ \le \alpha \le 90^\circ$ , solve the equation $5 \cos \theta + 8 \sin \theta = 6$ …
3State the range of principal values of $\tan^{-1} x$ . The curve $y = a + b \cos c x$ is shown below for $0 \le x \le \tfrac{3\pi}{2}$ …
4The diagram below shows a portion a plastic fibre cable, which allows light waves to pass through. The path of the light…
5The diagram shows a wind turbine with blade $25\text{ m}$ in length. The centre of the wind turbine is $35\text{ m}$ fro…
6The diagram shows two chords AB and CD intersect at X such that angle CXB = $90^\circ$ . Given that $AC = 5\ \text{cm}$ ,…
7The diagram shows the curve $y = a\cos bx + c$ for $0 \le x \le \tfrac{3\pi}{2}$ radians. A minimum point $(\tfrac{\pi}{2}, -2)$ …
8A clock is set on the vertical wall on the roof of a building. The distance from the centre of the clock, $O$ , to the ti…
9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 $f( \dfrac {θ}{2}+ \dfrac {π}{12})= \dfrac {6}{5}$ …
10A particle moves in a straight line such that its velocity, $v\text{ m/s}$ , is given by $v = 9\sin\left(\frac{t}{2}\right)$ …
11The diagram below shows two horizontal lines $AB$ and $CD$ , where $L$ is the vertical distance between the lines $AB$ an…
12The diagram shows the front view of a rectangular block $ABCE$ , with dimensions 14 cm by 8 cm, placed on a ramp, $VE$ , t…
13The function f is defined by $f(x)=1+2\cos 2x$ for $0\le x\le\pi$ . State the amplitude and the period of $f(x)$ . State t…
14The 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…
15In the diagram, BE intersects AD at F. AF = 7 cm and FD = 4 cm. $\angle ABF$ , $\angle ACD$ and $\angle DEF$ are right an…
16The diagram shows a pair of similar triangles $YUX$ and $VUW$ . Point $Y$ lies on the straight line $VU$ such that $VY = 12\text{ cm}$ …
1712 Solve $0.5\sin 2x = 3\sin x$ for $0 < x < 2\pi$ . Solve, for values of $\theta$ between $0^\circ$ and $180^\circ$ , $\csc\left(\frac{\theta}{2}-40^\circ\right)+5=0.$ …
18The 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…
19The diagram shows an area that is enclosed by glass panels at $AB$ , $BC$ and $AD$ . $AB = 10\text{ m}$ , $AD = 30\text{ m}$ …
20The 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…
21The function $f$ is defined, for $x \ge 0^\circ$ , by $f(x) = 3\cos\left(\frac{2x}{3}\right) - 1$ . State the maximum and…
22The expression $8 \sin \theta - 3 \cos \theta$ is defined for $0^\circ \le \theta \le 180^\circ$ . Using $R \sin(\theta - \alpha)$ …
23The height of water in a harbour at time $t$ hours after midnight on a certain day is modelled by the formula $h = 4.8 \sin k\pi t + 5.1$ …
24The figure shows a piece of cardboard in the shape of a trapezium in which $\angle BAD$ and $\angle ABC$ are right angle…
25The diagram shows part of the graph of $y = a\tan(bx + c)$ . The graph has vertical asymptotes at $x = -4\pi$ and $x = 4\pi$ …
26The equation of a curve is $y = 3 - 4\sin 2x$ . State the minimum and maximum value of $y$ . Sketch the curve $y = 3 - 4\sin 2x$ …
27In the diagram, $OC = 4\text{ m}$ , $OD = 7\text{ m}$ and angle $CBO = \text{angle } DOC = \text{angle } DAO = 90^\circ$ .…
28The equation of a curve is $y = 4 - 3\cos 2x$ . State the period and amplitude of $y$ . Sketch the graph of $y = 4 - 3\cos 2x$ …
29The function $f$ is defined by $f(x) = 4 \cos ax + b$ for $0 \le x \le \pi$ , where $a$ and $b$ are constants. The period…
30设向量 $\overrightarrow{a}=\left( \cos \left( 2x-\dfrac{\pi }{3} \right),2 \right),\overrightarrow{b}=\left( 1,{{\cos }^{2}}x \right)$ …
31The graph of $y = 5 \sin 3x + c$ , where $c$ is a constant, passes through the point $(50^\circ, \frac{1}{2})$ . Show that…
32A ferris wheel with a radius of 15 m completes one rotation every 3 minutes. Passengers get onto the ferris wheel at the…
33The diagram below shows the side view of a warehouse $ABCD$ such that $BC = 10\text{ m}$ , $CD = 6\text{ m}$ , $\angle BCD = \angle BAD = 90^\circ$ …
34State the amplitude and period of $2\sin 3x + 1$ . Sketch the graph of $y = 2\sin 3x + 1$ for $0 \le x \le \pi$ . By drawi…
35In the diagram, $AC = 35\text{ cm}$ , $EC = 15\text{ cm}$ and angle $ACB = \theta$ radians. $ED$ and $AB$ are perpendicul…
36The diagram shows the curve $y = a\cos bx + c$ for $0 \le x \le \dfrac{2\pi}{3}$ radians, where $a$ , $b$ and $c$ are int…
37The diagram shows the curve $y = a + 6\cos bx$ for $0 \le x \le \frac{2\pi}{3}$ , where $a$ and $b$ are positive constant…
38The depth of water, $h$ metres, in a shallow port on a particular day is modelled by the formula $h = 7 + 2\sin\left(\frac{\pi}{6}t\right),$ …
39It 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…
40The function f is defined by $f(x)=4\cos ax + b$ for $0 \le x \le 3\pi$ , where $a$ and $b$ are constants. The maximum va…
41A competitive platform diving competition involves one or two divers performing spins, twists and somersaults into a poo…
42Solve the equation $4\cos 2A = 3 - 2\sin A$ for $0 \le A \le 2\pi$ . It is given that $\text{f}(x) = 2\cos 6x - \frac{1}{2}$ …
43The diagram shows part of the graph of $y = p\sin(qx) + r$ . Find the value of each of the constants $p$ , $q$ and $r$ . Gi…
44The diagram shows the curve $y = 4 - 2\cos 2x$ for $0 \le x \le \frac{3}{2}\pi$ radians and a straight line passing thro…
45Sketch, on the axes below, the graphs of $y = -3\sin x$ and $y = 1 - \frac{2x}{\pi}$ for $0 \le x \le 2\pi$ . Find the nu…
46State the amplitude and period of the graph of $y=3\sin\frac{x}{2}+1$ . Sketch the graph of $y=3\sin\frac{x}{2}+1$ for $0 \le x \le 4\pi$ …
47The diagram shows a figure $ABCD$ , in which $BC=13\ \text{m}$ , $CD=6\ \text{m}$ , angle $BAD = 90^\circ$ and angle $ADC =$ …
48It is given that $y_1 = \sin x - 2$ and $y_2 = -3 \cos 2x$ . State the amplitude and period, in degrees, of (a) $y_1$ , (b…
49The diagram above shows the side view of a bus stop shelter $BCD$ such that $BC = 4\text{ m}$ , $CD = 1\text{ m}$ , angle…
50Show 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…
51The figure shows two circles $C_1$ and $C_2$ which touch each other and lie in the $xy$ -plane as shown below. $C_1$ has…
52It is given that $f(x)=2\sin\frac{x}{2}$ and $g(x)=3\cos x+1$ , where $0 \le x \le 2\pi$ . State the period of $f(x)$ . Sta…
53The graph below shows the vertical displacement of sea tidal waves, $y$ m, represented by $y = 2 - \frac{3}{2} \cos pt$ ,…
54Show that $\tan x + \frac{\cos x}{1 + \sin x} = \sec x$ . Explain, without the use of calculator, why the equation $\tan x + \frac{\cos x}{1 + \sin x} = -\frac{1}{3}$ …
55It is given that $\mathrm{f}(x) = 2x^3 + ax^2 + x + b$ . Find the value of $a$ and of $b$ for which $2x^2 + x - 1$ is a f…
56Sketch the graphs of $y=-2\cos 3x+1$ and $y= -\dfrac{2}{\pi}x+1$ on the same axes, for $0 \le x \le \pi$ . Explain how th…
57Solve the equation $2\sin^4 x + 7\cos^2 x = 4$ for $0^\circ \le x \le 360^\circ$ .
58In Singapore, Alternating Current (AC) power is provided by utility companies to households at a frequency of 50 Hz, or…
59Given that $y = p \sin 2x + q$ , where $p$ and $q$ are positive integers. The maximum and minimum values of $y$ are 8 and…
60The function $f$ is defined, for $0 \le x \le 2\pi$ , by $f(x) = 1 - 2 \cos 2x$ . State the period and amplitude of $f$ . S…