Right Triangle Trigonometry

Trigonometry begins with a simple idea: the ratios of sides in a right triangle depend only on the angles, not on the triangle's size. This insight lets you calculate distances you cannot measure directly -- the height of a building, the width of a canyon, or the distance to a star.

The Three Trigonometric Ratios

For a right triangle with an acute angle θ:

sin(θ) = opposite / hypotenuse cos(θ) = adjacent / hypotenuse tan(θ) = opposite / adjacent

SOH-CAH-TOA

This mnemonic captures all three ratios:

Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

"Opposite" and "adjacent" are always relative to the angle you are working with. The hypotenuse is always the longest side, opposite the right angle.

Finding Missing Sides

When you know one acute angle and one side, you can find any other side using the appropriate ratio.

Worked Example 1 -- Finding a Side

A right triangle has an angle of 35° and the hypotenuse is 20 cm. Find the side opposite the 35° angle.

Identify: we want the opposite side, and we know the hypotenuse. Use sine.
sin(35°) = opposite / 20.
opposite = 20 · sin(35°) ≈ 20 × 0.5736 ≈ 11.47 cm.

Finding Missing Angles

When you know two sides, use the inverse trigonometric functions (sin^-1, cos^-1, tan^-1) to find the angle.

Worked Example 2 -- Finding an Angle

A right triangle has legs of length 7 and 10. Find the angle opposite the side of length 7.

tan(θ) = opposite/adjacent = 7/10 = 0.7.
θ = tan^-1(0.7) ≈ 34.99° ≈ 35.0°.

Real-World Applications

Worked Example 3 -- Height of a Tree

You stand 40 meters from the base of a tree. Looking up to the top, the angle of elevation is 28°. How tall is the tree?

The 40 m distance is adjacent to the 28° angle; the tree height is opposite.
tan(28°) = height / 40.
height = 40 · tan(28°) ≈ 40 × 0.5317 ≈ 21.3 m.

Special Right Triangles

45-45-90 Triangle

Sides in ratio 1 : 1 : √2.

sin(45°) = cos(45°) = √2/2 ≈ 0.707

tan(45°) = 1

30-60-90 Triangle

Sides in ratio 1 : √3 : 2.

sin(30°) = 1/2, cos(30°) = √3/2

sin(60°) = √3/2, cos(60°) = 1/2

Common Mistake

Mixing up "opposite" and "adjacent." These labels change depending on which angle you are working with. Always label the triangle from the perspective of your chosen angle before setting up the ratio.

Practice Problems

A right triangle has a 50° angle and the adjacent side is 12 cm. Find the opposite side.
Show Solution

tan(50°) = opp/12. opp = 12 · tan(50°) ≈ 12 × 1.1918 ≈ 14.3 cm.
A ladder 15 feet long leans against a wall at a 70° angle with the ground. How high up the wall does it reach?
Show Solution

sin(70°) = height/15. height = 15 sin(70°) ≈ 15 × 0.9397 ≈ 14.1 feet.
A right triangle has legs 5 and 12. Find all three angles.
Show Solution

Hypotenuse = √(25 + 144) = 13. Angle opposite the 5: sin^-1(5/13) ≈ 22.6°. Angle opposite the 12: sin^-1(12/13) ≈ 67.4°. Third angle = 90°.
From the top of a 60 m cliff, the angle of depression to a boat is 22°. How far is the boat from the base of the cliff?
Show Solution

The angle of depression equals the angle of elevation from the boat. tan(22°) = 60/d. d = 60/tan(22°) ≈ 60/0.4040 ≈ 148.5 m.

Summary

SOH-CAH-TOA defines sine, cosine, and tangent as ratios of sides in a right triangle.
Use trig ratios to find missing sides when you know an angle and a side.
Use inverse trig functions to find angles when you know two sides.
Special triangles (30-60-90 and 45-45-90) give exact values for common angles.