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Geometry Unit 5: Right-triangle trigonometry

Trig ratios connect a right triangle's angles to its side lengths, so you can find missing sides or angles.

Covers: Sine, cosine, tangent, solving right triangles, angles of elevation and depression, complementary angles

Study guide

Trigonometry connects angles to side ratios in right triangles. Relative to an acute angle, the opposite side is across from it, the adjacent side touches it (and isn't the hypotenuse), and the hypotenuse is opposite the right angle. SOH-CAH-TOA gives the ratios: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.

To find a missing side, pick the ratio that uses the angle you know, the side you know, and the side you want, then solve the equation. To find a missing angle, use an inverse trig function: if tan A = 3/4, then A = tan⁻¹(3/4). Make sure your calculator is in degree mode.

Solving a right triangle means finding all its sides and angles, using trig ratios, the Pythagorean theorem, and the fact that the acute angles add to 90°.

Angles of elevation are measured up from horizontal, and angles of depression are measured down from horizontal. Because the horizontal lines are parallel, an angle of depression from one point equals the angle of elevation from the other. Draw the picture and label the right angle first.

Because the two acute angles are complementary, sin A = cos B when A + B = 90°. Know the exact values: sin 30° = 1/2, cos 60° = 1/2, tan 45° = 1.

The full set of exact values: sin 30° = cos 60° = 1/2, sin 45° = cos 45° = √2/2, sin 60° = cos 30° = √3/2, tan 30° = √3/3, tan 45° = 1 and tan 60° = √3. If sin A = cos B, then A + B = 90°. So an equation such as sin(2x + 10)° = cos(3x − 5)° is solved by setting (2x + 10) + (3x − 5) = 90.

When an elevation or depression problem has two right triangles sharing a side, solve for that shared length in one triangle first, then use it in the other. Find the distance asked for by adding or subtracting the pieces.

Key ideas

• SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
• Use inverse trig (sin⁻¹, cos⁻¹, tan⁻¹) to find an angle from sides.
• Angle of elevation looks up from horizontal; depression looks down.
• sin x = cos(90° − x).
• Set your calculator to degree mode.

Formulas and rules

Common mistakes

• Labeling opposite and adjacent from the wrong angle.
• Calculator in radian mode.

Words to know

SOH-CAH-TOA

sin = opp/hyp, cos = adj/hyp, tan = opp/adj

tan 45°

1

sin 30°

1/2

Complementary acute angles

In a right triangle, the two acute angles add to 90°, so sin A = cos B.

Inverse trig functions

sin⁻¹, cos⁻¹, tan⁻¹: find an angle from a ratio of sides.

Sine

Opposite ÷ hypotenuse.

Cosine

Adjacent ÷ hypotenuse.

Tangent

Opposite ÷ adjacent.

Angle of elevation

Measured upward from horizontal.

Angle of depression

Measured downward from horizontal.

Solving a right triangle

Finding all its sides and angles.

Sample practice questions

37 more questions like these, from level 1 to 5, are in the practice set.