Trigonometric Functions and the Unit Circle: math practice, Grade 11 – download the PDF

1 Degrees to radians ★★★

Convert each angle to radians. Give exact answers in terms of \(\pi\).

1. \(45^\circ\)
2. \(120^\circ\)
3. \(270^\circ\)
4. \(150^\circ\)

2 Radians to degrees ★★★

Convert each angle to degrees.

1. \(\dfrac{5\pi}{6}\)
2. \(\dfrac{7\pi}{4}\)
3. \(\dfrac{\pi}{9}\)
4. \(\dfrac{11\pi}{6}\)

3 Length of a sprinkler arc ★★★

A lawn sprinkler sprays water 8 inches from its center through a central angle of \(\dfrac{3\pi}{4}\) radians. Find the length of the arc swept by the water at that distance, first in inches (to the nearest hundredth), then in centimeters. (1 inch = 2.54 cm.)

4 Special values on the unit circle ★★★

Give the exact value of each expression.

1. \(\sin\dfrac{\pi}{6}\)
2. \(\cos\dfrac{\pi}{3}\)
3. \(\sin\dfrac{\pi}{4}\)
4. \(\cos\dfrac{\pi}{2}\)
5. \(\tan\dfrac{\pi}{4}\)

5 Quadrants and signs ★★★

For each angle, name the quadrant of its terminal side and tell whether sine and cosine are positive or negative.

1. \(200^\circ\)
2. \(\dfrac{5\pi}{6}\)
3. \(300^\circ\)
4. \(-\dfrac{\pi}{4}\)

6 Amplitude and period ★★★

Give the amplitude and the period of each function.

1. \(y=4\sin x\)
2. \(y=\sin 3x\)
3. \(y=-2\cos\dfrac{x}{2}\)

7 True or false? ★★★

Decide whether each statement is true or false and justify your answer.

1. \(\cos\pi=-1\).
2. \(\tan\dfrac{\pi}{2}=0\).
3. A straight angle measures \(2\pi\) radians.
4. \(\sin^2 20^\circ+\cos^2 20^\circ=1\).

8 Exact values with reference angles ★★★

Find the exact value of each expression. Show the quadrant and the reference angle.

1. \(\sin 150^\circ\)
2. \(\cos 225^\circ\)
3. \(\tan 300^\circ\)
4. \(\sin\dfrac{4\pi}{3}\)

9 Cosine known, quadrant IV ★★★

The angle \(\theta\) lies in quadrant IV and \(\cos\theta=\dfrac{3}{5}\). Find \(\sin\theta\) and \(\tan\theta\).

10 Sine known, quadrant II ★★★

The angle \(\theta\) lies in quadrant II and \(\sin\theta=\dfrac{5}{13}\). Find \(\cos\theta\) and \(\tan\theta\).

11 Key points of a reflected cosine ★★★

Consider \(y=-3\cos(2x)+1\).

1. Give the amplitude, the period and the midline.
2. Give the maximum and the minimum values.
3. Complete the table of key points for \(x=0,\dfrac{\pi}{4},\dfrac{\pi}{2},\dfrac{3\pi}{4},\pi\).

12 A sine equation ★★★

Solve \(2\sin x-1=0\) on the interval \([0,2\pi)\).

13 A cosine equation ★★★

Solve \(2\cos x+\sqrt{2}=0\) on \([0,2\pi)\).

14 The Ferris wheel ★★★

The height in feet of a seat on a Ferris wheel is \(h(t)=25-20\cos\dfrac{\pi t}{6}\), where \(t\) is the time in minutes after the seat leaves the boarding platform. The graph of one turn is shown below.

1. Find the lowest height, the highest height and the time for one turn. Give the highest height in meters too (1 ft = 0.3048 m).
2. Find the height after 2 minutes.
3. At what times during the first turn is the seat at 35 feet?

15 Harbor tide ★★★

In a simplified model, the water depth at the end of a pier is \(d(t)=8+3\sin\dfrac{\pi t}{6}\) feet, where \(t\) is the number of hours after midnight.

1. Find the period, the maximum depth and the minimum depth. Give the maximum depth in meters (1 ft = 0.3048 m).
2. Find the depth at 1 a.m.
3. At what times between \(t=0\) and \(t=12\) is the depth exactly 9.5 feet?

16 A shifted sine wave ★★★

Consider \(y=2\sin\left(x-\dfrac{\pi}{3}\right)\).

1. Give the amplitude, the period, the phase shift and the range.
2. Find the x-intercepts in \([0,2\pi)\).
3. Find where the maximum and the minimum occur in \([0,2\pi)\).

17 One branch of a tangent ★★★

Consider \(f(x)=\tan(2x)\).

1. Find the period of \(f\).
2. Find the vertical asymptotes that border the branch containing the origin.
3. Give the points of that branch where \(f(x)=-1\), \(0\) and \(1\).

18 Factoring a trigonometric equation ★★★

Solve \(2\cos^2x-\cos x-1=0\) on \([0,2\pi)\).

19 Using the Pythagorean identity to solve ★★★

Solve \(2\sin^2x+3\cos x=3\) on \([0,2\pi)\). (Hint: rewrite everything in terms of cosine.)

20 Writing an equation from features ★★★

A sinusoidal graph has a maximum value of 5 at \(x=0\), a minimum value of \(-3\), and a period of \(\pi\).

1. Find its amplitude and midline.
2. Write an equation of the form \(y=A\cos(Bx)+D\).
3. Check your equation at \(x=0\) and at \(x=\dfrac{\pi}{2}\).

21 Daily temperature ★★★

In a desert town the temperature in degrees Fahrenheit is modeled by \(T(t)=62-14\cos\dfrac{\pi(t-4)}{12}\), where \(t\) is the number of hours after midnight (\(0\le t<24\)).

1. Find the lowest and highest temperatures and the times they occur.
2. Find the temperature at 10 a.m. and at noon.
3. At what times is the temperature 69°F? Convert 69°F to Celsius to the nearest tenth.

22 Proving and checking an identity ★★★

(a) Show that \(\dfrac{\sin^2x}{1-\cos x}=1+\cos x\) for every \(x\) with \(\cos x\ne1\).

(b) Check the identity numerically for \(x=\dfrac{\pi}{3}\).