EZ Math

Trigonometry · Right triangle trigonometry

How to Find a Missing Angle with Inverse Sine

Problem

sin⁡θ=12,0∘≤θ≤90∘\sin \theta = \frac{1}{2},\quad 0^\circ \le \theta \le 90^\circ

Answer

30∘30^\circ

Inverse sine turns a ratio back into an angle. For an acute angle in a right triangle, one ratio and the unit circle values are enough.

Step-by-step solution

  1. Apply inverse sine to both sides.

    θ=sin⁡−1(12)\theta = \sin^{-1}\left(\frac{1}{2}\right)

    Why: Inverse sine undoes sine and returns the angle.

  2. Recall the unit circle value for sine one half.

    sin⁡30∘=12\sin 30^\circ = \frac{1}{2}

    Why: The 30-60-90 triangle gives this exact value.

  3. Write the angle in the given range.

    θ=30∘\theta = 30^\circ

    Why: The range from 0 to 90 degrees keeps the principal acute answer.

Common mistakes

  • Using the sine button instead of inverse sine and leaving theta on both sides.
  • Using the cosine value. Read the ratio first: this is opposite over hypotenuse.
  • Answering 150 degrees. That angle is outside the given range and its sine is positive too, but 30 is the acute solution.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. sin⁡θ=22,0∘≤θ≤90∘\sin \theta = \frac{\sqrt{2}}{2},\quad 0^\circ \le \theta \le 90^\circ

    Show answer

    45∘45^\circ

  2. sin⁡θ=32,0∘≤θ≤90∘\sin \theta = \frac{\sqrt{3}}{2},\quad 0^\circ \le \theta \le 90^\circ

    Show answer

    60∘60^\circ

  3. sin⁡θ=1,0∘≤θ≤90∘\sin \theta = 1,\quad 0^\circ \le \theta \le 90^\circ

    Show answer

    90∘90^\circ

  4. sin⁡θ=0,0∘≤θ≤90∘\sin \theta = 0,\quad 0^\circ \le \theta \le 90^\circ

    Show answer

    0∘0^\circ

  5. sin⁡θ=12,0∘≤θ≤90∘\sin \theta = \frac{1}{2},\quad 0^\circ \le \theta \le 90^\circ

    Show answer

    30∘30^\circ