Trigonometric Functions and the Unit Circle: math practice, Grade 12 – download the PDF

1 Degrees to radians ★★★

Convert each angle to radians, in terms of \(\pi\): a) \(45^\circ\) b) \(120^\circ\) c) \(270^\circ\) d) \(15^\circ\).

2 Radians to degrees ★★★

Convert to degrees: a) \(\dfrac{5\pi}{6}\) b) \(\dfrac{7\pi}{4}\) c) \(\dfrac{\pi}{10}\) d) \(\dfrac{3\pi}{5}\).

3 Length of an arc ★★★

A circular garden path is an arc of a circle of radius \(8\text{ m}\) with central angle \(\dfrac{\pi}{4}\). Find the exact length of the path and a decimal approximation to the nearest hundredth.

4 Reading the unit circle ★★★

Give the exact value: a) \(\sin\dfrac{\pi}{6}\) b) \(\cos\dfrac{\pi}{3}\) c) \(\tan\dfrac{\pi}{4}\) d) \(\cos\dfrac{\pi}{2}\) e) \(\sin\pi\).

5 Quadrants and signs ★★★

For each angle, name the quadrant and give the signs of \(\sin\) and \(\cos\): a) \(\dfrac{5\pi}{6}\) b) \(\dfrac{4\pi}{3}\) c) \(\dfrac{11\pi}{6}\) d) \(\dfrac{7\pi}{12}\).

6 Right triangle ratios ★★★

A right triangle has legs \(5\text{ in}\) and \(12\text{ in}\). Let \(\theta\) be the acute angle opposite the \(5\)-inch leg. Find the hypotenuse, then \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\).

7 Coterminal angles ★★★

Find the angle between \(0\) and \(2\pi\) that is coterminal with: a) \(\dfrac{13\pi}{6}\) b) \(-\dfrac{\pi}{3}\) c) \(\dfrac{9\pi}{4}\) d) \(-\dfrac{7\pi}{4}\).

8 Reference angles and values ★★★

Give the reference angle, then the exact value: a) \(\sin\dfrac{5\pi}{6}\) b) \(\cos\dfrac{4\pi}{3}\) c) \(\tan\dfrac{7\pi}{4}\) d) \(\cos\dfrac{11\pi}{6}\).

9 Reciprocal functions ★★★

Find the exact value: a) \(\sec\dfrac{\pi}{3}\) b) \(\csc\dfrac{5\pi}{6}\) c) \(\cot\dfrac{3\pi}{4}\) d) \(\sec\dfrac{7\pi}{6}\) (rationalize the denominator).

10 All six from one value ★★★

Suppose \(\sin\theta=\dfrac{3}{5}\) and \(\theta\) is in quadrant II. Find \(\cos\theta\), \(\tan\theta\), \(\csc\theta\), \(\sec\theta\) and \(\cot\theta\).

11 A leaning ladder ★★★

A \(20\)-foot ladder leans against a wall and makes a \(70^\circ\) angle with level ground. How high up the wall does it reach, and how far is its foot from the wall? Round to the nearest hundredth of a foot.

12 Sector of a pizza ★★★

A slice of a large pizza is a sector with radius \(15\text{ in}\) and central angle \(100^\circ\). Find the exact arc length of the crust and the exact area of the slice, then approximate both to the nearest hundredth.

13 True or false? ★★★

Decide whether each statement is true for every \(x\). If false, give a counterexample. a) \(\sin(-x)=-\sin x\) b) \(\cos(-x)=\cos x\) c) \(\tan(x+\pi)=\tan x\) d) \(\sin(x+\pi)=\sin x\).

14 A bicycle wheel ★★★

A bicycle wheel of radius \(13\text{ in}\) turns at \(45\) rpm. Find its angular velocity in radians per second and the linear velocity of a point on the tire, in inches per second and in miles per hour (\(1\text{ mi}=63{,}360\text{ in}\)).

15 Solving sin equals a value ★★★

Solve \(\sin\theta=-\dfrac{\sqrt{3}}{2}\) for \(0\le\theta<2\pi\).

16 Solving a tangent equation ★★★

Solve \(\tan\theta=-\sqrt{3}\) for \(0\le\theta<2\pi\).

17 Large and negative angles ★★★

Find the exact value: a) \(\sin\left(-\dfrac{11\pi}{6}\right)\) b) \(\cos\dfrac{25\pi}{4}\) c) \(\tan\dfrac{17\pi}{3}\) d) \(\sec\left(-\dfrac{13\pi}{3}\right)\).

18 Finding sine and tangent from cosine ★★★

Given \(\cos\theta=-\dfrac{5}{13}\) and \(\tan\theta>0\), find \(\sin\theta\), \(\tan\theta\) and \(\csc\theta\).

19 The Ferris wheel ★★★

A Ferris wheel has diameter \(150\text{ ft}\) and makes one full turn every \(24\) minutes.

Find the angular velocity in radians per minute, the speed of a rider in feet per minute, and that speed in miles per hour (\(1\text{ mi}=5280\text{ ft}\)).

20 Height of a tree ★★★

A surveyor stands \(50\text{ ft}\) from the base of a tree. Her eye is \(5.5\text{ ft}\) above the ground and the angle of elevation to the treetop is \(36^\circ\). How tall is the tree to the nearest tenth of a foot?

21 Using an identity ★★★

Given \(\tan\theta=2\) and \(\theta\) in quadrant III, find \(\sec\theta\), \(\cos\theta\) and \(\sin\theta\) exactly, then check that \(\sin^{2}\theta+\cos^{2}\theta=1\).

22 A car tire ★★★

A car tire has a diameter of \(28\text{ in}\). The car travels at \(60\text{ mph}\) (\(1\text{ mi}=63{,}360\text{ in}\)). Find the tire’s angular velocity in radians per second, then in revolutions per minute (nearest whole number).