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Inverse Trig Functions

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

Trig equations usually have repeated solutions. Students use inverse trig, the unit circle, identities, intervals, and general-solution notation to list every angle that satisfies the equation.

What you'll learn

Why it matters: Repeated motion and waves often hit the same value more than once, so complete solution sets matter.

Worked example

Problem. Inverse Trig Functions: Solve cos(x) = 0 on 0 <= x < 2pi.

  1. Cosine is the x-coordinate.
  2. It is zero on the y-axis.
  3. The solutions are pi/2 and 3pi/2.

Answer: pi/2, 3pi/2

Practice problems

1. Inverse Trig Functions: Evaluate arcsin(1/2).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Inverse Trig Functions: Evaluate arcsin(1/2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. arcsin returns a principal angle.
  4. sin(pi/6) = 1/2.
  5. So arcsin(1/2) = pi/6.
  6. Check the result by substituting or estimating: the response should match pi/6 and make sense in the original problem.

Answer: pi/6

2. Inverse Trig Functions: Evaluate arccos(-1/2).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Inverse Trig Functions: Evaluate arccos(-1/2).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. arccos returns an angle from 0 to pi.
  4. cos(2pi/3) = -1/2.
  5. So arccos(-1/2) = 2pi/3.
  6. Check the result by substituting or estimating: the response should match 2pi/3 and make sense in the original problem.

Answer: 2pi/3

3. Inverse Trig Functions: Solve sin(x) = 1/2 on 0 <= x <= 2pi.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Inverse Trig Functions: Solve sin(x) = 1/2 on 0 <= x <= 2pi.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Sine is positive in Quadrants I and II.
  4. The reference angle is pi/6.
  5. The solutions are pi/6 and 5pi/6.
  6. Check the result by substituting or estimating: the response should match pi/6, 5pi/6 and make sense in the original problem.

Answer: pi/6, 5pi/6

4. Inverse Trig Functions: Solve cos(x) = 0 on 0 <= x < 2pi.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Inverse Trig Functions: Solve cos(x) = 0 on 0 <= x < 2pi.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. Cosine is the x-coordinate.
  4. It is zero on the y-axis.
  5. The solutions are pi/2 and 3pi/2.
  6. Check the result by substituting or estimating: the response should match pi/2, 3pi/2 and make sense in the original problem.

Answer: pi/2, 3pi/2

5. Inverse Trig Functions: Write the general solution to tan(x) = 1.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Inverse Trig Functions: Write the general solution to tan(x) = 1.
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  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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