8.3 Inverse Trigonometric Functions

Given $\sin \left(\frac{5 \pi}{12}\right) \approx 0.96593 ,$ write a relation involving the inverse sine.

Given $\cos \left(0.5\right) \approx 0.8776,$ write a relation involving the inverse cosine.

Evaluate each of the following.

1. ⓐ $\text{sin}^{- 1} \left(\frac{1}{2}\right)$
2. ⓑ $\text{sin}^{- 1} \left(- \frac{\sqrt{2}}{2}\right)$
3. ⓒ $\cos^{- 1} \left(- \frac{\sqrt{3}}{2}\right)$
4. ⓓ $\tan^{- 1} \left(1\right)$

Evaluate each of the following.

1. ⓐ $\text{sin}^{−1} \left(−1\right)$
2. ⓑ $\tan^{−1} \left(−1\right)$
3. ⓒ $\cos^{−1} \left(−1\right)$
4. ⓓ $\cos^{−1} \left(\frac{1}{2}\right)$

Evaluate $\sin^{- 1} \left(0.97\right)$ using a calculator.

Evaluate $\cos^{- 1} \left(- 0.4\right)$ using a calculator.

Evaluate the following:

1. ⓐ $\sin^{- 1} \left(\sin \left(\frac{\pi}{3}\right)\right)$
2. ⓑ $\sin^{- 1} \left(\sin \left(\frac{2 \pi}{3}\right)\right)$
3. ⓒ $\cos^{- 1} \left(\cos \left(\frac{2 \pi}{3}\right)\right)$
4. ⓓ $\cos^{- 1} \left(\cos \left(- \frac{\pi}{3}\right)\right)$

Evaluate $\tan^{- 1} \left(\tan \left(\frac{\pi}{8}\right)\right) \text{and} \tan^{- 1} \left(\tan \left(\frac{11 \pi}{9}\right)\right) .$

Evaluate $\sin^{- 1} \left(\cos \left(\frac{13 \pi}{6}\right)\right)$

1. ⓐby direct evaluation.
2. ⓑ by the method described previously.

Evaluate $\cos^{- 1} \left(\sin \left(- \frac{11 \pi}{4}\right)\right) .$

Evaluate $\cos \left(\tan^{- 1} \left(\frac{5}{12}\right)\right) .$

Evaluate $\cos \left(\sin^{- 1} \left(\frac{7}{9}\right)\right) .$

Find a simplified expression for $\cos \left(\sin^{- 1} \left(\frac{x}{3}\right)\right)$ for $- 3 \leq x \leq 3.$

Find a simplified expression for $\sin \left(\tan^{- 1} \left(4 x\right)\right)$ for $- \frac{1}{4} \leq x \leq \frac{1}{4} .$

Why do the functions $f \left(x\right) = \sin^{- 1} x$ and $g \left(x\right) = \cos^{- 1} x$ have different ranges?

Explain the meaning of $\frac{\pi}{6} = \arcsin \left(0.5\right) .$

Why must the domain of the sine function, $\sin x ,$ be restricted to $\left[- \frac{\pi}{2} , \frac{\pi}{2}\right]$ for the inverse sine function to exist?

Determine whether the following statement is true or false and explain your answer: $\arccos \left(- x\right) = \pi - \arccos x .$

Evaluate the expressions.

$\sin^{- 1} \left(- \frac{1}{2}\right)$

Evaluate the expressions.

$\cos^{- 1} \left(- \frac{\sqrt{2}}{2}\right)$

Evaluate the expressions.

$\tan^{- 1} \left(- \sqrt{3}\right)$

Evaluate the expressions.

$\tan^{- 1} \left(\sqrt{3}\right)$

Use a calculator to evaluate each expression. Express answers to the nearest hundredth.

$\cos^{- 1} \left(- 0.4\right)$

Use a calculator to evaluate each expression. Express answers to the nearest hundredth.

$\arccos \left(\frac{3}{5}\right)$

Use a calculator to evaluate each expression. Express answers to the nearest hundredth.

$\tan^{- 1} \left(6\right)$

Find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} \left(\sin \left(\pi\right)\right)$

Find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} \left(\sin \left(\frac{\pi}{3}\right)\right)$

Find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} \left(\sin \left(\frac{4 \pi}{3}\right)\right)$

Find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\tan^{- 1} \left(\sin \left(\frac{- 5 \pi}{2}\right)\right)$

Find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\sin \left(\cos^{- 1} \left(\frac{3}{5}\right)\right)$

Find the exact value, if possible, without a calculator. If it is not possible, explain why.

$\cos \left(\tan^{- 1} \left(\frac{12}{5}\right)\right)$

Find the exact value of the expression in terms of $x$ with the help of a reference triangle.

$\tan \left(\sin^{- 1} \left(x - 1\right)\right)$

Find the exact value of the expression in terms of $x$ with the help of a reference triangle.

$\cos \left(\sin^{- 1} \left(\frac{1}{x}\right)\right)$

Find the exact value of the expression in terms of $x$ with the help of a reference triangle.

$\tan \left(\sin^{- 1} \left(x + \frac{1}{2}\right)\right)$

Find the function if $\sin t = \frac{x}{x + 1} .$

$\cos t$

Find the function if $\sin t = \frac{x}{x + 1} .$

$\cot t$

Find the function if $\sin t = \frac{x}{x + 1} .$

$\tan^{- 1} \left(\frac{x}{\sqrt{2 x + 1}}\right)$

For what value of $x$ does $\sin x = \sin^{- 1} x ?$ Use a graphing calculator to approximate the answer.

Suppose a 13-foot ladder is leaning against a building, reaching to the bottom of a second-floor window 12 feet above the ground. What angle, in radians, does the ladder make with the building?

An isosceles triangle has two congruent sides of length 9 inches. The remaining side has a length of 8 inches. Find the angle that a side of 9 inches makes with the 8-inch side.

A truss (interior beam structure) for the roof of a house is constructed from two identical right triangles. Each has a base of 12 feet and height of 4 feet. Find the measure of the acute angle adjacent to the 4-foot side.

The line $y = \frac{- 3}{7} x$ passes through the origin in the x,y-plane. What is the measure of the angle that the line makes with the negative x-axis?

A 20-foot ladder leans up against the side of a building so that the foot of the ladder is 10 feet from the base of the building. If specifications call for the ladder's angle of elevation to be between 35 and 45 degrees, does the placement of this ladder satisfy safety specifications?

Use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y = 12 , 000 + 8 , 000 \sin \left(\right. 0.628 x ),$ where the domain is the years since 1980 and the range is the population of the city.

What is the largest and smallest population the city may have?

Use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y = 12 , 000 + 8 , 000 \sin \left(\right. 0.628 x ),$ where the domain is the years since 1980 and the range is the population of the city.

What are the amplitude, period, and phase shift for the function?

Use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: $y = 12 , 000 + 8 , 000 \sin \left(\right. 0.628 x ),$ where the domain is the years since 1980 and the range is the population of the city.

What is the predicted population in 2007? 2010?

Suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

At time = 0, what is the displacement of the weight?

Suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.

What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?

Find the exact value without the aid of a calculator.

$\cos^{- 1} \left(\frac{\sqrt{3}}{2}\right)$

Find the exact value without the aid of a calculator.

$\cos^{- 1} \left(\frac{1}{\sqrt{2}}\right)$

Find the exact value without the aid of a calculator.

$\sin^{- 1} \left(\cos \left(\frac{\pi}{6}\right)\right)$

Find the exact value without the aid of a calculator.

$\sin \left(\sec^{- 1} \left(\frac{3}{5}\right)\right)$

Find the exact value without the aid of a calculator.

$\tan \left(\cos^{- 1} \left(\frac{5}{13}\right)\right)$

Find the amplitude, period, phase shift, and midline.

$y = \sin \left(\frac{\pi}{6} x + \pi\right) - 3$

Find the amplitude, period, phase shift, and midline.

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming $t$ is the number of hours since midnight, find a function for the temperature, $D ,$ in terms of $t .$

Find the period and horizontal shift of each function.

$g \left(x\right) = 3 \tan \left(6 x + 42\right)$

Find the period and horizontal shift of each function.

If $\sec x = 4 ,$ find $\sec \left(- x\right) .$

Let $f \left(x\right) = \frac{3}{5} \cos \left(6 x\right) .$

What is the largest possible value for $f \left(x\right) ?$

Let $f \left(x\right) = \frac{3}{5} \cos \left(6 x\right) .$

Where is the function increasing on the interval $\left[0 , 2 \pi\right] ?$

Find the exact value.

$\tan^{- 1} \left(\sqrt{3}\right)$

Find the exact value.

$\cos^{- 1} \left(\sin \left(\pi\right)\right)$

Find the exact value.

$\cos \left(\sin^{- 1} \left(1 - 2 x\right)\right)$

Find the exact value.

$\cos \left(\tan^{- 1} \left(x^{2}\right)\right)$

Suppose $\sin t = \frac{x}{x + 1} .$ Evaluate the following expressions.

$\csc t$

Determine whether the equation is true or false.

$\arcsin \left(\sin \left(\frac{5 \pi}{6}\right)\right) = \frac{5 \pi}{6}$

Determine whether the equation is true or false.

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.