DWG NO. 08.3 — Lesson 3 of 5

Introduction to Trig Ratios

Unit 8: Right Triangles and Trigonometry · ~15–30 min

Objective

Identify the opposite, adjacent, and hypotenuse sides relative to an acute angle in a right triangle, and compute its sine, cosine, and tangent ratios.

Naming the sides relative to an angle

Special right triangles only get you so far — most right triangles don't have a 45° or 30° angle. Trigonometry gives a ratio-based tool that works for any acute angle. Pick one of the two acute angles, call it θ (theta), and the three sides get new names relative to it:

The hypotenuse is still the side opposite the right angle — the longest side, and it never changes name.
The opposite side is the leg directly across from θ.
The adjacent side is the leg that touches θ (but isn't the hypotenuse).

Switch which acute angle you're looking at, and opposite and adjacent swap — the hypotenuse is the only side whose label never depends on θ.

Relative to θ, the near leg is adjacent, the far leg is opposite, and the hypotenuse stays the same.

The three basic ratios: SOH-CAH-TOA

The sine, cosine, and tangent of θ are each a ratio of two of these sides. The mnemonic SOH-CAH-TOA keeps them straight:

Ratio	Definition
sin θ	opposite ÷ hypotenuse (SOH)
cos θ	adjacent ÷ hypotenuse (CAH)
tan θ	opposite ÷ adjacent (TOA)

These ratios depend only on the angle θ, not on the size of the triangle — any right triangle with the same acute angle gives the same sine, cosine, and tangent.

Worked Example 1 · Computing all three ratios

ProblemA right triangle has legs 3 (opposite θ) and 4 (adjacent to θ), and hypotenuse 5.

1.sin θ = opposite/hypotenuse = 3/5 = 0.6.

2.cos θ = adjacent/hypotenuse = 4/5 = 0.8.

3.tan θ = opposite/adjacent = 3/4 = 0.75.

sin θ = 0.6, cos θ = 0.8, tan θ = 0.75

Worked Example 2 · Using a known ratio to find a side

ProblemA right triangle has a 30° angle and hypotenuse 10. Find the side opposite the 30° angle. (Recall sin 30° = 0.5.)

1.sin 30° = opposite/hypotenuse, so 0.5 = opposite/10.

2.Multiply both sides by 10: opposite = 5.

Opposite side = 5

Guided practice

1.A right triangle has legs 8 (opposite θ) and 6 (adjacent to θ), and hypotenuse 10. Find sin θ.
Show answer
sin θ = opposite/hypotenuse = 8/10 = 0.8.
2.Using the same triangle as problem 1, find cos θ and tan θ.
Show answer
cos θ = 6/10 = 0.6; tan θ = 8/6 = 4/3 ≈ 1.33.
3.In a right triangle, the side adjacent to θ is 5 and the hypotenuse is 13. Find cos θ.
Show answer
cos θ = adjacent/hypotenuse = 5/13 ≈ 0.385.
4.A right triangle has a 45° angle, and both legs equal 7. Find tan 45°.
Show answer
tan 45° = opposite/adjacent = 7/7 = 1.
5.A right triangle has a 60° angle and hypotenuse 8. Using cos 60° = 0.5, find the side adjacent to the 60° angle.
Show answer
cos 60° = adjacent/hypotenuse, so 0.5 = adjacent/8, giving adjacent = 4.

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