Exact values of inverse trig functions practice problems with step-by-step solutions

Problem 13.61

easy

Find the exact value of \( \displaystyle \arcsin{\left(-1 \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is -1.
Reviewed
\[ \sin{\left(\frac{\left(-1\right) \pi}{2} \right)} = -1 \]
sin(-pi/2) = -1, and -pi/2 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(-1 \right)} = - \frac{\pi}{2} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.62

easy

Find the exact value of \( \displaystyle \arctan{\left(\sqrt{3} \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is sqrt(3).
Reviewed
\[ \tan{\left(\frac{\pi}{3} \right)} = \sqrt{3} \]
tan(pi/3) = sqrt(3), and pi/3 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(\sqrt{3} \right)} = \frac{\pi}{3} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.63

easy

Find the exact value of \( \displaystyle \arctan{\left(0 \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is 0.
Reviewed
\[ \tan{\left(0 \right)} = 0 \]
tan(0) = 0, and 0 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(0 \right)} = 0 \]
So that angle is the value.✓ Proved

Answer \( 0 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.64

easy

Find the exact value of \( \displaystyle \arccos{\left(- \frac{\sqrt{2}}{2} \right)} \).

arccos returns the one angle in [0, π] whose cos is -sqrt(2)/2.
Reviewed
\[ \cos{\left(\frac{3 \pi}{4} \right)} = - \frac{\sqrt{2}}{2} \]
cos(3*pi/4) = -sqrt(2)/2, and 3*pi/4 is in [0, π].✓ Proved
\[ \operatorname{acos}{\left(\frac{\left(-1\right) \sqrt{2}}{2} \right)} = \frac{3 \pi}{4} \]
So that angle is the value.✓ Proved

Answer \( \frac{3 \pi}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.65

easy

Find the exact value of \( \displaystyle \arccos{\left(-1 \right)} \).

arccos returns the one angle in [0, π] whose cos is -1.
Reviewed
\[ \cos{\left(\pi \right)} = -1 \]
cos(pi) = -1, and pi is in [0, π].✓ Proved
\[ \operatorname{acos}{\left(-1 \right)} = \pi \]
So that angle is the value.✓ Proved

Answer \( \pi \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.66

easy

Find the exact value of \( \displaystyle \arctan{\left(\frac{\sqrt{3}}{3} \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is sqrt(3)/3.
Reviewed
\[ \tan{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{3} \]
tan(pi/6) = sqrt(3)/3, and pi/6 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)} = \frac{\pi}{6} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{6} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.67

easy

Find the exact value of \( \displaystyle \arcsin{\left(1 \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is 1.
Reviewed
\[ \sin{\left(\frac{\pi}{2} \right)} = 1 \]
sin(pi/2) = 1, and pi/2 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(1 \right)} = \frac{\pi}{2} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{2} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.68

easy

Find the exact value of \( \displaystyle \arctan{\left(- \frac{\sqrt{3}}{3} \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is -sqrt(3)/3.
Reviewed
\[ \tan{\left(\frac{\left(-1\right) \pi}{6} \right)} = - \frac{\sqrt{3}}{3} \]
tan(-pi/6) = -sqrt(3)/3, and -pi/6 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(\frac{\left(-1\right) \sqrt{3}}{3} \right)} = - \frac{\pi}{6} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{6} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.69

easy

Find the exact value of \( \displaystyle \arccos{\left(- \frac{1}{2} \right)} \).

arccos returns the one angle in [0, π] whose cos is -1/2.
Reviewed
\[ \cos{\left(\frac{2 \pi}{3} \right)} = - \frac{1}{2} \]
cos(2*pi/3) = -1/2, and 2*pi/3 is in [0, π].✓ Proved
\[ \operatorname{acos}{\left(- \frac{1}{2} \right)} = \frac{2 \pi}{3} \]
So that angle is the value.✓ Proved

Answer \( \frac{2 \pi}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.72

easy

Find the exact value of \( \displaystyle \arccos{\left(\frac{\sqrt{3}}{2} \right)} \).

arccos returns the one angle in [0, π] whose cos is sqrt(3)/2.
Reviewed
\[ \cos{\left(\frac{\pi}{6} \right)} = \frac{\sqrt{3}}{2} \]
cos(pi/6) = sqrt(3)/2, and pi/6 is in [0, π].✓ Proved
\[ \operatorname{acos}{\left(\frac{\sqrt{3}}{2} \right)} = \frac{\pi}{6} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{6} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.73

easy

Find the exact value of \( \displaystyle \arctan{\left(- \sqrt{3} \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is -sqrt(3).
Reviewed
\[ \tan{\left(\frac{\left(-1\right) \pi}{3} \right)} = - \sqrt{3} \]
tan(-pi/3) = -sqrt(3), and -pi/3 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(- \sqrt{3} \right)} = - \frac{\pi}{3} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.74

easy

Find the exact value of \( \displaystyle \arcsin{\left(- \frac{1}{2} \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is -1/2.
Reviewed
\[ \sin{\left(\frac{\left(-1\right) \pi}{6} \right)} = - \frac{1}{2} \]
sin(-pi/6) = -1/2, and -pi/6 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(- \frac{1}{2} \right)} = - \frac{\pi}{6} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{6} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.173

easy

Find the exact value of \( \displaystyle \arcsin{\left(\frac{\sqrt{3}}{2} \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is sqrt(3)/2.
Reviewed
\[ \sin{\left(\frac{\pi}{3} \right)} = \frac{\sqrt{3}}{2} \]
sin(pi/3) = sqrt(3)/2, and pi/3 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} = \frac{\pi}{3} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.174

easy

Find the exact value of \( \displaystyle \arctan{\left(-1 \right)} \).

arctan returns the one angle in (−π/2, π/2) whose tan is -1.
Reviewed
\[ \tan{\left(\frac{\left(-1\right) \pi}{4} \right)} = -1 \]
tan(-pi/4) = -1, and -pi/4 is in (−π/2, π/2).✓ Proved
\[ \operatorname{atan}{\left(-1 \right)} = - \frac{\pi}{4} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.176

easy

Find the exact value of \( \displaystyle \arccos{\left(\frac{1}{2} \right)} \).

arccos returns the one angle in [0, π] whose cos is 1/2.
Reviewed
\[ \cos{\left(\frac{\pi}{3} \right)} = \frac{1}{2} \]
cos(pi/3) = 1/2, and pi/3 is in [0, π].✓ Proved
\[ \operatorname{acos}{\left(1 \cdot \frac{1}{2} \right)} = \frac{\pi}{3} \]
So that angle is the value.✓ Proved

Answer \( \frac{\pi}{3} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 13.180

easy

Find the exact value of \( \displaystyle \arcsin{\left(- \frac{\sqrt{2}}{2} \right)} \).

arcsin returns the one angle in [−π/2, π/2] whose sin is -sqrt(2)/2.
Reviewed
\[ \sin{\left(\frac{\left(-1\right) \pi}{4} \right)} = - \frac{\sqrt{2}}{2} \]
sin(-pi/4) = -sqrt(2)/2, and -pi/4 is in [−π/2, π/2].✓ Proved
\[ \operatorname{asin}{\left(\frac{\left(-1\right) \sqrt{2}}{2} \right)} = - \frac{\pi}{4} \]
So that angle is the value.✓ Proved

Answer \( - \frac{\pi}{4} \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Exact values of inverse trig functions practice problems

16 worked problems

Problem 13.61

Problem 13.62

Problem 13.63

Problem 13.64

Problem 13.65

Problem 13.66

Problem 13.67

Problem 13.68

Problem 13.69

Problem 13.72

Problem 13.73

Problem 13.74

Problem 13.173

Problem 13.174

Problem 13.176

Problem 13.180

More exact values of inverse trig functions practice