# Inverse Trigonometry Exploration

- Author:
- Ben Graber

- Topic:
- Trigonometry

Trigonometry can be used not only to find the measurements of side lengths, but the measurements of angles. Given the values of trigonometric ratios for different reference angles, it is possible to calculate the ratio for a triangle and use the ratio to find a reference angle. Calculate the side length ratios of the triangles below and use this to find the angles of the triangles.
Sin 25° = 0.4226
Sin 30° = 0.5000
Sin 35° = 0.5736
Sin 40° = 0.6429
Sin 45° = 0.7071
Sin 50° = 0.7660
Sin 55° = 0.8192
Sin 60° = 0.8660
Sin 65° = 0.9063

a. Consider the triangle ABC. Calculate the sine ratio of angle A. What is the value of opp/hyp? Which of the sine ratios given above does this equal? What is the measurement of angle A?
b. Calculate the sine ratio of angle C. What is the value of opp/hyp? Which of the sine ratios given above does this equal? What is the measurement of angle C?
c. Consider the triangle DEF. Calculate the sine ratio of angle D. What is the value of opp/hyp? Which of the sine ratios given above does this equal? What is the measurement of angle D? What is the angle measurement of angle F?
d. Consider the triangle HGK. Calculate the sine ratio of angle H. What is the value of opp/hyp? Which of the sine ratios given above does this equal? What is the measurement of angle H? What is the angle measurement of angle K?
e. Consider the triangle IJL. Calculate the sine ratio of angle I. What is the value of opp/hyp? Which of the sine ratios given above does this equal? What is the measurement of angle I? What is the angle measurement of angle L?