Graphs of Trigonometric Functions: math practice, Grade 12 – download the PDF

1 Amplitude and period ★★★

State the amplitude and the period of \(y=4\sin(3x)\), then give its maximum and minimum values.

2 A reflected cosine ★★★

For \(y=-2\cos\left(\dfrac{x}{2}\right)\), find the amplitude, the period, and the first \(x\geq 0\) where the graph reaches its maximum.

3 Midline, maximum, minimum ★★★

The function \(y=2\sin x+5\) models the water level of a small pond in inches. Give its midline, maximum and minimum values, and where in \([0,2\pi]\) each extreme occurs.

4 A table of values ★★★

Complete the values of \(y=3\cos x\) for \(x=0,\ \dfrac{\pi}{3},\ \dfrac{\pi}{2},\ \dfrac{2\pi}{3},\ \pi\), then say what kind of curve passes through these points.

5 True or false? ★★★

Say whether each statement is true or false and justify briefly.

1. The period of \(y=\cos x\) is \(2\pi\).
2. The range of \(y=\tan x\) is \([-1,1]\).
3. \(\sec x\) is defined for every real number \(x\).
4. The graph of \(y=\sin x\) is symmetric about the origin.
5. The amplitude of \(y=-4\cos x\) is \(-4\).

6 Asymptotes of a tangent ★★★

For \(y=\tan(2x)\), give the period and list the vertical asymptotes between \(x=-\dfrac{\pi}{2}\) and \(x=\dfrac{\pi}{2}\).

7 Inverse values ★★★

Evaluate exactly, giving each answer in radians.

1. \(\arcsin\dfrac12\)
2. \(\arccos\left(-\dfrac12\right)\)
3. \(\arctan 1\)
4. \(\arcsin\left(-\dfrac{\sqrt2}{2}\right)\)

8 Cosine becomes sine ★★★

The function \(y=\sin\left(x+\dfrac{\pi}{2}\right)\) is graphed. By which transformation of \(y=\sin x\) is it obtained, and which familiar function is it?

9 Factor before you read ★★★

For \(y=-4\cos(3x+\pi)+2\), find the amplitude, period, phase shift (state the direction), midline, maximum and minimum, and compute \(y\) at \(x=0\).

10 Equation of a sine wave ★★★

A sine wave in \(x\) has amplitude 6, midline \(y=2\), period 10, and passes through the midline going up at \(x=0\). Write its equation, then find \(y\) at \(x=2.5\) and \(x=7.5\).

11 A cosine from its extremes ★★★

A function has a maximum value of 7 at \(x=\dfrac{\pi}{4}\) and a minimum value of \(-1\); the next maximum occurs at \(x=\dfrac{5\pi}{4}\). Write it as \(y=A\cos(B(x-C))+D\).

12 Solving trigonometric equations ★★★

Solve on \([0,2\pi)\).

1. \(\sin x=\dfrac12\)
2. \(2\cos x=-\sqrt3\)
3. \(\tan x=-1\)

13 A shifted tangent ★★★

Consider \(y=\tan\left(\dfrac{x}{2}\right)+1\). Find its period, its vertical asymptotes in \((-2\pi,2\pi)\), its values at \(x=0\) and \(x=\dfrac{\pi}{2}\), and its \(x\)-intercept in \((-\pi,\pi)\).

14 A stretched secant ★★★

For \(y=2\sec x\): (a) give the domain and the range; (b) compute \(y\) at \(x=\dfrac{\pi}{3}\) and \(x=\dfrac{2\pi}{3}\); (c) identify the local extremum at \(x=0\) and at \(x=\pi\).

15 Compositions with inverses ★★★

Find the exact value.

1. \(\sin(\arcsin 0.3)\)
2. \(\cos\left(\arcsin\dfrac{8}{17}\right)\)
3. \(\tan\left(\arccos\dfrac{5}{13}\right)\)
4. \(\sin\left(\arctan\dfrac34\right)\)

16 A bicycle pedal ★★★

The height of a pedal above the ground is \(h(t)=12+7\sin(2.5\pi t)\) inches, with \(t\) in seconds. (Radius 7 in is about 17.8 cm.) (a) Find the maximum and minimum heights. (b) How long is one revolution, and how many revolutions per minute is that? (c) Find the height at \(t=0.1\) s.

17 A sound wave ★★★

The pressure of a sound wave is modeled by \(p(t)=0.4\sin(2\pi\cdot262\,t)\), with \(t\) in seconds. Find the amplitude, the period in milliseconds, and the number of full cycles in 0.05 s.

18 Tide model ★★★

The depth of water in a harbor is \(d(t)=6+2.5\cos\left(\dfrac{\pi t}{6}\right)\) feet, where \(t\) is the number of hours after high tide. (a) Give the amplitude, period, maximum and minimum depths. (b) Find \(d(2)\). (c) A boat needs at least 7.75 ft (about 2.4 m) of water. For how long, around one high tide, can it float?

19 Reading an equation from a graph ★★★

The curve below shows one full period of a sinusoidal function.

(a) Find the amplitude, midline and period. (b) Write an equation of the form \(y=A\cos(Bx)+D\). (c) Write another equation of the form \(y=A\sin(B(x-C))+D\) with \(A>0\).

20 Inverse of the function, not the identity ★★★

Evaluate each expression exactly and explain why the result differs from the angle inside.

1. \(\arcsin\left(\sin\dfrac{4\pi}{3}\right)\)
2. \(\arccos\left(\cos\left(-\dfrac{\pi}{4}\right)\right)\)
3. \(\arccos\left(\cos\dfrac{7\pi}{6}\right)\)
4. \(\arctan\left(\tan\dfrac{3\pi}{4}\right)\)

21 An identity for inverse functions ★★★

Prove that \(\arcsin x+\arccos x=\dfrac{\pi}{2}\) for every \(x\) in \([-1,1]\), then check the claim numerically for \(x=0.6\).

22 A decreasing tangent ★★★

Let \(g(x)=-2\tan\left(\dfrac{\pi x}{4}\right)\). (a) Find the period. (b) Find the vertical asymptotes in \((-4,4)\) and the \(x\)-intercepts in \((-2,2)\). (c) Compute \(g(1)\) and \(g(-1)\). (d) Is \(g\) increasing or decreasing between consecutive asymptotes?

23 Daily temperature ★★★

In a desert town the temperature in \(^\circ\)F is modeled by \(T(t)=73+11\cos\left(\dfrac{\pi(t-15)}{12}\right)\), where \(t\) is the number of hours after midnight. (a) Find the hottest and coldest temperatures and when they occur. (b) Find \(T(9)\) and \(T(18)\), and convert \(T(18)\) to \(^\circ\)C. (c) During which hours is it at least 80 \(^\circ\)F? Give the times to the nearest minute.