The Sine and Cosine Ratios

Understand sine and cosine ratios for right triangles. Math 10-C Alberta mathematics.

Lesson 3.2 of Trigonometry in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• The Sine and Cosine Ratios. Meet the two trig ratios that involve the hypotenuse. Learn the definitions of sine and cosine, understand the SOH CAH TOA mnemonic, and see why both values are always between 0 and 1 for acute angles.
• Writing and Evaluating Ratios. Practice choosing the correct ratio for a given pair of sides, write all three trig ratios for a 3-4-5 triangle from two different angles, and discover the complementary angle pattern between sine and cosine.
• Finding Missing Angles. Use inverse sine and inverse cosine to find unknown angles in right triangles. Work through two applied problems involving a train track and a ski course.
• Finding Missing Sides. Solve for unknown side lengths by setting up sine and cosine equations. Learn when to multiply versus divide, and apply the technique to real-world surveyor and pilot problems.

Inside the lesson — a free preview

Why Do We Need More Ratios?

The tangent ratio connects the opposite and adjacent sides, but many problems involve the hypotenuse. That is where sine and cosine come in.

SOH CAH TOA — The Master Mnemonic. Each letter-group tells you which pair of sides a ratio uses:

SOH — A = (Opposite)/(Hypotenuse)

CAH — A = (Adjacent)/(Hypotenuse)

TOA — A = (Opposite)/(Adjacent)

The Definitions

For a given acute angle A in a right triangle, the ratios are defined as follows.

Sine and Cosine Defined. Sine: A = (length of opposite side)/(length of hypotenuse)

Cosine: A = (length of adjacent side)/(length of hypotenuse)

What Do the Values Tell You?

Just like tangent, the decimal value of sine or cosine tells you how two sides compare in length.

Interpreting the Values — an Example at 32°. 32° = 0.5299 means the opposite side is about 0.53 times the hypotenuse — just over half as long.

32° = 0.8480 means the adjacent side is about 0.85 times the hypotenuse.

Key Observation — Sine and Cosine Stay Between 0 and 1. The hypotenuse is always the longest side of a right triangle, so the opposite and adjacent sides are always shorter than it.

This means: for any acute angle, both and values are always strictly between 0 and 1.

Tangent, by contrast, can be any positive number — it compares two legs, and the opposite side can be longer than the adjacent side.

Worked example: Write All Three Ratios for a 3-4-5 Triangle

Given a right triangle with angle θ at vertex A (bottom-left), the right angle at C (bottom-right), and sides AC = 3, BC = 4, AB = 5.

From Angle A's Perspective — opposite = 4, adjacent = 3, hypotenuse = 5. Step 1 — Identify the sides relative to A:

Opposite = BC = 4

Adjacent = AC = 3

Hypotenuse = AB = 5

Step 2 — Write each ratio:

A = (4)/(5)

A = (3)/(5)

A = (4)/(3)

From Angle B's Perspective — opposite = 3, adjacent = 4, hypotenuse = 5. Step 1 — Identify the sides relative to B:

Opposite = AC = 3

Adjacent = BC = 4

Hypotenuse = AB = 5

Step 2 — Write each ratio:

B = (3)/(5)

B = (4)/(5)

B = (3)/(4)

The Complementary Angle Pattern. Notice: A = B and A = B.

This always happens in a right triangle. The side opposite one acute angle is the side adjacent to the other acute angle — so their sine and cosine values swap.

This is why sine and cosine are called co-functions: the sine of any angle equals the cosine of its complement.

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.