Inverse Trigonometry (Right Triangles) - Interactive Activity

Definitions and Main Ideas

Lesson 41 used an angle to find a ratio; this lesson runs the process backwards. When two sides of a right triangle are known, their ratio feeds an inverse trigonometric function — sin⁻¹, cos⁻¹, or tan⁻¹ — to recover the measure of the angle. You will evaluate inverse functions on the calculator, find indicated angles in triangles, and solve word problems, including a ladder against a wall and the angle a steep street makes with the horizontal.

Calculator check: Always confirm the setting of the calculator. Test: sin 30° = 0.5. If you get anything other than 0.5, your calculator is not set to degrees.

Vocabulary

Inverse Trigonometric Functions

The functions sin⁻¹, cos⁻¹, and tan⁻¹, which take a trigonometric ratio as input and return the measure of the angle that produces it.

Reference Angle

The angle being referred to in a trigonometric ratio; the right angle is never used as a reference angle.

Opposite Side

The side directly across the triangle from the reference angle.

Adjacent Side

The side next to the reference angle that is not the hypotenuse.

Hypotenuse

The side opposite the right angle — always the longest side of the right triangle.

Main Ideas

1. Choose the inverse function that matches the two sides you know: opposite and hypotenuse → sin⁻¹; adjacent and hypotenuse → cos⁻¹; opposite and adjacent → tan⁻¹.
2. The setup order matters: form the ratio from the sides first, then apply the inverse function to that ratio — the inverse function undoes the ratio, not the sides.
3. An inverse function returns an angle, so the answer is in degrees; round angle measures as directed, usually to the nearest tenth.
4. The two acute angles of a right triangle are complementary: if one inverse calculation gives 65.1°, the other acute angle is 24.9°.
5. In word problems, draw the right triangle, mark the known sides relative to the angle you want, and only then choose the function.

The Formulas

The three ratios from Lesson 41, written for the reference angle ∠A:

sin ∠A = oppositehypotenuse, cos ∠A = adjacenthypotenuse, tan ∠A = oppositeadjacent.

Each ratio runs backwards through its inverse function, and the angle is the result:

m∠A = sin⁻¹(oppositehypotenuse), m∠A = cos⁻¹(adjacenthypotenuse), m∠A = tan⁻¹(oppositeadjacent).

Ratio-to-Angle Explorer

What to do: Drag the orange handle to change the adjacent side and the purple handle to change the opposite side. Use the buttons to choose which two sides are known. Beside the triangle, the two known sides form their ratio, and the matching inverse function recovers the measure of ∠A from that ratio.

Pick the Inverse

What to do: Each question shows a right triangle with one angle marked and two sides labeled. Decide which inverse function — sin⁻¹, cos⁻¹, or tan⁻¹ — uses those two sides to find the marked angle. The choice you pick is drawn in the diagram, right or wrong.

Build the Angle

What to do: Each problem gives two sides of a right triangle, or a ladder scene, and asks for an angle. Work in three steps: (1) choose the inverse function, (2) choose the equation that forms the ratio, and (3) report the angle, rounded to the nearest tenth. Press New Item for the next problem, in order.