Lesson 8.2.5

The Sine & Cosine Ratios

Sine and cosine involve the hypotenuse. Use them when you know (or need) the hypotenuse and one leg of a right triangle.

Introduction

Tangent relates the two legs. But if the hypotenuse is involved — either as a known or unknown — you need sine (opposite / hypotenuse) or cosine (adjacent / hypotenuse). Together with tangent, these three ratios let you solve any right triangle given one side and one acute angle.

Past Knowledge

SOH CAH TOA (8.2.3). Tangent ratio (8.2.4). Calculator skills.

Today's Goal

Use sine and cosine to find missing sides involving the hypotenuse.

Future Success

Complementary angles (8.2.6), inverse trig (8.3.1), Law of Sines (8.3.3).

Key Concepts

Sine and Cosine

Choosing the Right Ratio

You Know	You Want	Use
Angle + hypotenuse	Opposite	sin
Angle + hypotenuse	Adjacent	cos
Angle + opposite	Hypotenuse	sin
Angle + adjacent	Hypotenuse	cos

Worked Examples

Basic

Finding the Opposite with Sine

Angle = 40°, hypotenuse = 20. Find the opposite side.

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Intermediate

Finding the Hypotenuse with Cosine

Angle = 28°, adjacent = 15. Find the hypotenuse.

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Advanced

Solving the Complete Triangle

Angle , hypotenuse = 30. Find both legs and angle .

Opposite of A:

Adjacent to A:

Common Pitfalls

Choosing Sin vs Cos Incorrectly

Both involve the hypotenuse, so you must identify whether the known/unknown leg is opposite (use sin) or adjacent (use cos) to the angle.

Multiplying When You Should Divide

If the hypotenuse is unknown, you divide: . Don't multiply — the hypotenuse is always the largest side, so the answer must be bigger than the leg.

Real-Life Applications

Aviation — Glide Path

A plane descending at 3° from 30,000 feet uses to find the ground distance: about 573,000 ft (≈ 108 miles). That's why planes begin descent so far from the airport.

Rock Climbing — Route Length

If a cliff is 200 ft tall and the climbing route goes at an average 70° angle, the route length is ft — longer than the height because the route isn't perfectly vertical.