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Trigonometry Final Exam

A free Trigonometry lesson from the “Applications of Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

The final exam is cumulative across the full Trigonometry course: right triangles, radians, the unit circle, graphs, identities, equations, inverse trig, laws of sines and cosines, vectors, and modeling.

What you'll learn

Why it matters: Surveyors, pilots, engineers, architects, and data modelers use trig when right triangles are not enough.

Worked example

Problem. Trigonometry Final Exam: If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).

  1. Use sin^2 + cos^2 = 1.
  2. cos^2 = 16/25.
  3. Cosine is positive, so cos = 4/5.

Answer: 4/5

Practice problems

1. Trigonometry Final Exam: Convert 135 degrees to radians.

Show solution
  1. Final Exam Review: First identify exactly what the question is asking: Trigonometry Final Exam: Convert 135 degrees to radians.
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Use 180 degrees = pi radians.
  4. Compute 135 * pi/180.
  5. The result is 3pi/4.
  6. Check the result by substituting or estimating: the response should match 3pi/4 and make sense in the original problem.

Answer: 3pi/4

2. Trigonometry Final Exam: Evaluate sin(pi/6).

Show solution
  1. Final Exam Review: First identify exactly what the question is asking: Trigonometry Final Exam: Evaluate sin(pi/6).
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Sine is the y-coordinate.
  4. Use the point (sqrt(3)/2, 1/2).
  5. sin(pi/6) = 1/2.
  6. Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.

Answer: 1/2

3. Trigonometry Final Exam: Find the period of y = sin(4x).

Show solution
  1. Final Exam Review: First identify exactly what the question is asking: Trigonometry Final Exam: Find the period of y = sin(4x).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. For sine, period = 2pi/|b|.
  4. Here b = 4.
  5. The period is pi/2.
  6. Check the result by substituting or estimating: the response should match pi/2 and make sense in the original problem.

Answer: pi/2

4. Trigonometry Final Exam: If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).

Show solution
  1. Final Exam Review: First identify exactly what the question is asking: Trigonometry Final Exam: If sin(theta)=3/5 and theta is in Quadrant I, find cos(theta).
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Use sin^2 + cos^2 = 1.
  4. cos^2 = 16/25.
  5. Cosine is positive, so cos = 4/5.
  6. Check the result by substituting or estimating: the response should match 4/5 and make sense in the original problem.

Answer: 4/5

5. Trigonometry Final Exam: Write the general solution to tan(x) = 1.

Show solution
  1. Final Exam Review: First identify exactly what the question is asking: Trigonometry Final Exam: Write the general solution to tan(x) = 1.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. tan(pi/4)=1.
  4. Tangent has period pi.
  5. Add pi*k.
  6. Check the result by substituting or estimating: the response should match x = pi/4 + pi*k and make sense in the original problem.

Answer: x = pi/4 + pi*k

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