Inverse Trigonometric Functions and Trigonometric Equations: Precalculus Math Worksheet – PDF Download

Problem 1: Evaluate arcsin

Find the exact value of each expression.

1. \(\sin^{-1}(\frac{1}{2})\)
2. \(\sin^{-1}(-\frac{1}{2})\)
3. \(\sin^{-1}(\frac{\sqrt{3}}{2})\)
4. \(\sin^{-1}(-1)\)

Problem 2: Evaluate arccos

Find the exact value of each expression.

1. \(\cos^{-1}(\frac{\sqrt{2}}{2})\)
2. \(\cos^{-1}(-\frac{1}{2})\)
3. \(\cos^{-1}(0)\)
4. \(\cos^{-1}(-1)\)

Problem 3: Evaluate arctan

Find the exact value of each expression.

1. \(\tan^{-1}(1)\)
2. \(\tan^{-1}(-1)\)
3. \(\tan^{-1}(\sqrt{3})\)
4. \(\tan^{-1}(-\frac{1}{\sqrt{3}})\)

Problem 4: Graph of arctan

The graph shows \(y=\tan^{-1}x\). State its domain, its range, and the equations of its two horizontal asymptotes. Then find \(\tan^{-1}(0)\).

Problem 5: Restricted cosine

The graph of \(y=\cos x\) is shown with the thick piece used to define \(\cos^{-1}x\). State the interval of the thick piece, then find each value.

1. \(\cos^{-1}(\cos\frac{\pi}{3})\)
2. \(\cos^{-1}(\cos(-\frac{\pi}{3}))\)
3. \(\cos^{-1}(\cos\frac{4\pi}{3})\)

Problem 6: Compositions that cancel

Simplify each expression, or write undefined.

1. \(\sin(\sin^{-1}(0.3))\)
2. \(\cos(\cos^{-1}(-0.8))\)
3. \(\tan(\tan^{-1}(25))\)
4. \(\sin(\sin^{-1}(1.2))\)

Problem 7: Arcsine of a sine

Find each value.

1. \(\sin^{-1}(\sin\frac{\pi}{4})\)
2. \(\sin^{-1}(\sin\frac{3\pi}{4})\)
3. \(\sin^{-1}(\sin\frac{7\pi}{6})\)
4. \(\sin^{-1}(\sin(-\frac{2\pi}{3}))\)

Problem 8: Triangle compositions

The triangle shows an angle \(\theta\) with \(\cos\theta=\frac{3}{5}\). Find the exact value of each expression.

1. \(\sin(\cos^{-1}\frac{3}{5})\)
2. \(\cos(\sin^{-1}\frac{5}{13})\)
3. \(\tan(\cos^{-1}\frac{4}{5})\)
4. \(\tan(\sin^{-1}(-\frac{3}{5}))\)

Problem 9: Calculator practice

With the calculator in radian mode, evaluate each expression to the nearest hundredth.

1. \(\sin^{-1}(0.35)\)
2. \(\cos^{-1}(0.2)\)
3. \(\tan^{-1}(4)\)
4. \(\cos^{-1}(-0.65)\)

Problem 10: Solve a sine equation

Solve \(\sin x=\frac{\sqrt{3}}{2}\) for \(0\le x< 2\pi\).

Problem 11: Solve a cosine equation

Solve \(\cos x=-\frac{\sqrt{2}}{2}\) for \(0\le x< 2\pi\).

Problem 12: Solve a tangent equation

Solve \(\tan x=-1\) for \(0\le x< 2\pi\).

Problem 13: Equation with a unit circle

Solve \(2\cos x+1=0\) for \(0\le x< 2\pi\). The vertical line on the unit circle shows where \(\cos x\) has the needed value.

Problem 14: Sine equation by calculator

Solve \(3\sin x=1.2\) for \(0\le x< 2\pi\). Round to the nearest hundredth.

Problem 15: Cosine equation by calculator

Solve \(\cos x=0.3\) for \(0\le x< 2\pi\). Round to the nearest hundredth.

Problem 16: General solutions

Write all real solutions. Let \(k\) be any integer.

1. \(\sin x=\frac{1}{2}\)
2. \(\cos x=0\)
3. \(\tan x=\sqrt{3}\)

Problem 17: Quadratic in sine

Solve \(2\sin^{2}x-\sin x-1=0\) for \(0\le x< 2\pi\).

Problem 18: Multiple-angle equation

Solve \(\sin 2x=\frac{\sqrt{2}}{2}\) for \(0\le x< 2\pi\).

Problem 19: Wheelchair ramp

A wheelchair ramp is 12 ft long and rises 1 ft. Find the angle \(\theta\) the ramp makes with the ground, in degrees to the nearest hundredth.

Problem 20: Ladder angle

The base of a 16-ft ladder is 4 ft from a wall.

1. Find the angle the ladder makes with the ground, to the nearest tenth of a degree.
2. How far from the wall should the base be to make a \(70^{\circ}\) angle? Round to the nearest tenth of a foot.

Problem 21: Daylight hours

The hours of daylight in a northern city are \(D(t)=12+4\sin(\frac{2\pi}{365}(t-80))\), where \(t\) is the day of the year. Find the days when \(D=14\) and describe when the city has more than 14 hours of daylight.

Problem 22: Spring height

A weight on a spring has height \(h(t)=20+6\cos(\frac{2\pi t}{3})\) inches above the floor, with \(t\) in seconds. Find all times in \(0\le t< 3\) when the weight is 23 in high, and when it is 16 in high (nearest hundredth).

Problem 23: Household voltage

The voltage of a US wall outlet is \(V(t)=170\sin(120\pi t)\) volts, with \(t\) in seconds. Find the first two times in one cycle when \(V=100\) (in milliseconds, nearest hundredth). Then state the length of one cycle.

Problem 24: Sun angle

A 40-ft flagpole casts a 55-ft shadow.

1. Find the angle of elevation of the sun to the nearest tenth of a degree.
2. How long is the shadow when the sun is \(60^{\circ}\) above the horizon? Round to the nearest tenth of a foot.

See every chapter on the Precalculus math page.