Solve Equations & Model with Principal Values

Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7202 | Total Attempts: 9,524,167
| Questions: 20 | Updated: Nov 10, 2025
Please wait...
Question 1 / 20
0 %
0/100
Score 0/100
1) Solve sin(x) = 1/2 for the principal value.

Explanation

sin(π/6) = 1/2.

The range of arcsin is [−π/2, π/2].

Hence, the principal value is x = π/6.

Submit
Please wait...
About This Quiz
Solve Equations & Model With Principal Values - Quiz

Apply principal values to real-life and modeling scenarios. This quiz connects inverse trigonometric equations to geometric contexts like ladders, ramps, hills, and slopes. Students use arcsin, arccos, and arctan to determine angles of elevation, inclination, and depression, applying definitions such as θ = arctan(opposite/adjacent) or θ = arcsin(opposite/hypotenuse). The problems... see moreintegrate algebraic solving with applied reasoning, building confidence in interpreting inverse trig relationships within practical models. see less

2)
We’ll put your name on your report, certificate, and leaderboard.
2) Solve cos(x) = −√3/2 in [0, π].

Explanation

cos(5π/6) = −√3/2.

Since arccos(x) gives results in [0, π], the correct answer is 5π/6.

Submit
3) Solve tan(x) = 1 for the principal value.

Explanation

tan(π/4) = 1.

Because π/4 lies in the principal range (−π/2, π/2), x = π/4.

Submit
4) A ladder leans against a wall. Height = 12 ft, base = 5 ft. Find θ.

Explanation

In a right triangle, tan(θ) = opposite/adjacent = 12/5.

Therefore, θ = arctan(12/5).

Submit
5) A ramp rises 2 m over 8 m horizontally. Find θ in degrees.

Explanation

tan(θ) = 2/8 = 0.25.

θ = arctan(0.25) ≈ 14.0°.

Submit
6) Solve for x: sin(x) = 0.8 (principal value).

Explanation

For the principal value, x = arcsin(0.8).

It lies in the range [−π/2, π/2].

Submit
7) A roof rises 4 ft for every 12 ft run. Find θ.

Explanation

tan(θ) = 4/12 = 1/3.

θ = arctan(1/3) ≈ 18.4°.

Submit
8) Solve tan(2x) = √3 for principal x.

Explanation

2x = arctan(√3) = π/3.

Dividing by 2 gives x = π/6.

Submit
9) A hill rises 30 m over 100 m horizontally. Find the incline angle θ.

Explanation

tan(θ) = 30/100 = 0.3.

θ = arctan(0.3) ≈ 16.7°.

Submit
10) A boat climbs a wave where y = A sin(kx), A = 2, y = 1. Find kx.

Explanation

y = 1 and A = 2 → sin(kx) = 1/2.

Hence, kx = arcsin(1/2) = π/6.

Submit
11) Solve 3cos(x) = 1 for principal x.

Explanation

cos(x) = 1/3.

The principal value of x is x = arccos(1/3).

Submit
12) A drone climbs at 50 m altitude over 200 m horizontally. Find θ in degrees.

Explanation

tan(θ) = 50/200 = 0.25.

θ = arctan(0.25) ≈ 14°.

Submit
13) A 10 m ramp rises 2.5 m. Find θ.

Explanation

sin(θ) = 2.5/10 = 0.25.

θ = arcsin(0.25) ≈ 14.5°.

Submit
14) A 25 m building casts a 50 m shadow. Find the sun angle.

Explanation

tan(θ) = 25/50 = 0.5.

θ = arctan(0.5) ≈ 26.6°.

Submit
15) A slope climbs 9 m for every 40 m horizontal. Find θ.

Explanation

tan(θ) = 9/40 = 0.225.

θ = arctan(0.225) ≈ 12.7°.

Submit
16) A ladder of 8 m reaches 7 m up. Find θ.

Explanation

sin(θ) = opposite/hypotenuse = 7/8.

Hence, θ = arcsin(7/8).

Submit
17) If sin(x) = −0.5, find x in (−π/2, π/2).

Explanation

sin(π/6) = 1/2.

For a negative sine, the angle is below the x-axis, so x = −π/6.

Submit
18) A slope rises 10 m for every 30 m horizontal. Find θ.

Explanation

tan(θ) = 10/30 = 1/3.

θ = arctan(1/3) ≈ 18.4°.

Submit
19) A car ascends a hill with rise/run = 1/8. Find θ.

Explanation

tan(θ) = 1/8 = 0.125.

θ = arctan(0.125) ≈ 7.1°.

Submit
20) A tower 40 m high is seen from 100 m away. Find the angle of elevation.

Explanation

tan(θ) = 40/100 = 0.4.

θ = arctan(0.4) ≈ 21.8°.

Submit
View My Results
Cancel
  • All
    All (20)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
Solve sin(x) = 1/2 for the principal value.
Solve cos(x) = −√3/2 in [0, π].
Solve tan(x) = 1 for the principal value.
A ladder leans against a wall. Height = 12 ft, base = 5 ft. Find θ.
A ramp rises 2 m over 8 m horizontally. Find θ in degrees.
Solve for x: sin(x) = 0.8 (principal value).
A roof rises 4 ft for every 12 ft run. Find θ.
Solve tan(2x) = √3 for principal x.
A hill rises 30 m over 100 m horizontally. Find the incline angle θ.
A boat climbs a wave where y = A sin(kx), A = 2, y = 1. Find kx.
Solve 3cos(x) = 1 for principal x.
A drone climbs at 50 m altitude over 200 m horizontally. Find θ in...
A 10 m ramp rises 2.5 m. Find θ.
A 25 m building casts a 50 m shadow. Find the sun angle.
A slope climbs 9 m for every 40 m horizontal. Find θ.
A ladder of 8 m reaches 7 m up. Find θ.
If sin(x) = −0.5, find x in (−π/2, π/2).
A slope rises 10 m for every 30 m horizontal. Find θ.
A car ascends a hill with rise/run = 1/8. Find θ.
A tower 40 m high is seen from 100 m away. Find the angle of...
Alert!

Back to Top Back to top
Advertisement