The Sine Law & The Cosine Law

Solve any triangle using sine and cosine laws. Math 20-1 Alberta Grade 11.

Lesson 4.6 of Trigonometry in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Sine Law. Learn how to solve oblique triangles using the Sine Law. Discover when a matching pair unlocks the equation, and work through three detailed examples: finding a side, finding an angle, and fully solving a triangle.
• The Cosine Law. Master the Cosine Law for the two cases the Sine Law cannot handle: SSS and SAS. See how it generalizes the Pythagorean theorem, learn to rearrange it for finding angles, and work through full examples for both cases.
• Choosing the Right Law. Build a decision framework for every oblique-triangle problem. Use the given information to instantly identify whether to reach for the Sine Law or the Cosine Law, and learn sanity-check strategies to verify your answers.

Inside the lesson — a free preview

What Is an Oblique Triangle?

Definition — Oblique Triangle. An oblique triangle is a triangle that does not contain a right angle.

Because no angle equals 90°, you cannot use the Pythagorean theorem or SOH CAH TOA directly. You need a different approach — the Sine Law or the Cosine Law.

The Sine Law

The Sine Law — Two Equivalent Forms. For any triangle ABC with sides a, b, c opposite to angles A, B, C:

Angle form (use when solving for an angle):

( A)/(a) = ( B)/(b) = ( C)/(c)

Side form (use when solving for a side):

(a)/( A) = (b)/( B) = (c)/( C)

These two forms are identical — one is the reciprocal of the other. Choose the form where the unknown is in the numerator for easiest solving.

When to Use the Sine Law

The Matching Pair Requirement. The Sine Law works any time you have a matching pair — an angle paired with the side directly opposite it (e.g. A and a, or B and b). You also need one more piece of information.

Condition 1: A matching pair plus another side → find the angle opposite that other side.

Condition 2: A matching pair plus another angle → find the side opposite that other angle.

Big idea: The Sine Law uses two of the three ratios at a time — never all three. Pick the two ratios where you have the most information.

Worked example: Find the Length of Side p

Example 1 — Find side p in triangle PQR. Given: P = 75°, R = 39°, q = 8.1 cm. Find side p to the nearest tenth.

Step 1: Find the missing angle.

P + Q + R = 180°

75° + Q + 39° = 180°

Q = 180° - 75° - 39° = 66°

Step 2: Identify the matching pair.

We have Q = 66° paired with q = 8.1 cm.

Step 3: Set up the Sine Law for side p.

Since we want a side, use the side form:

(p)/( P) = (q)/( Q)

(p)/( 75°) = (8.1)/( 66°)

Step 4: Solve for p.

p = (8.1 · 75°)/( 66°)

Step 5: Calculate (degree mode).

75° ≈ 0.9659, 66° ≈ 0.9135

p ≈ (8.1 × 0.9659)/(0.9135) ≈ (7.824)/(0.9135) ≈ 8.56

Answer: p ≈ 8.6 cm

Tip — Keep the Unknown in the Numerator. When solving for a side, place the unknown in the numerator of the ratio. This makes the solution a single multiplication rather than requiring an extra step. If the unknown ends up in the denominator, rearrange before solving.

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