Pythagorean Theorem, Primary Trig Ratios & Solving Right Triangles

Master primary trig ratios and right triangle solving. Grade 11 Math 20-1 - exam prep.

Lesson 4.1 of Trigonometry in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• Triangle Basics and the Pythagorean Theorem. Learn how triangles are labelled, explore the four triangle types, and use the Pythagorean theorem to find missing sides.
• Trigonometric Ratios — SOH CAH TOA. Name the sides of a right triangle relative to an angle and write sine, cosine, and tangent ratios using SOH CAH TOA.
• Using a Calculator. Evaluate trig ratios given an angle and use inverse trig functions to find an angle given a ratio.
• Solving Triangles. Apply a systematic strategy to find every unknown side and angle in a right triangle.
• Word Problems — Angles of Elevation and Depression. Translate real-world scenarios involving angles of elevation and depression into right-triangle diagrams and solve them.

Inside the lesson — a free preview

How Triangles Are Labelled

Vertices, Angles, and Sides. Triangles are named by their vertices (the corners). A triangle with corners A, B, and C is called triangle ABC, written as ABC.

Angles are labelled with capital letters (A, B, C) at each vertex.

Sides are labelled with lowercase letters that match the angle directly opposite the side:

The Opposite-Side Convention.

Example 1 — Label the Sides and Angles of ABC with C = 90°. Angle labels go at the vertices A, B, and C. Mark a small square at vertex C to show the right angle.

Side a is opposite A — connects B and C.

Side b is opposite B — connects A and C.

Side c is opposite C — connects A and B. Since C = 90°, side c is the hypotenuse (the longest side).

Two Key Facts to Remember.

Types of Triangles

Different triangles have different properties based on their sides and angles.

This Lesson's Focus. This lesson focuses on right triangles, since trigonometric ratios are defined specifically for them.

The Pythagorean Theorem

The Formula. For any right triangle with legs a and b and hypotenuse c:

a^2 + b^2 = c^2

The hypotenuse is always the longest side and is always opposite the right angle. The letter c in the formula always represents the hypotenuse.

Rearranging to Solve for Any Side.

Example 3a — Find the Missing Side (given hypotenuse and one leg). A right triangle has one leg = 12 and hypotenuse = 13. Find the missing leg.

Step 1 — Identify c.

The hypotenuse is c = 13.

Step 2 — Apply the theorem.

a^2 + b^2 = c^2

12^2 + b^2 = 13^2

Step 3 — Compute the squares.

144 + b^2 = 169

Step 4 — Solve for b.

b^2 = 169 - 144 = 25

Step 5 — Take the square root.

b = √(25) = 5

Answer: The missing side is 5.

Example 3b — Find the Missing Side (given both legs). A right triangle has legs 2 and 4. Find the hypotenuse.

Step 1 — Identify the missing side.

Both legs are given, so we solve for c.

Step 2 — Apply the theorem.

a^2 + b^2 = c^2

2^2 + 4^2 = c^2

Step 3 — Compute the squares.

4 + 16 = c^2

c^2 = 20

Step 4 — Take the square root and simplify.

c = √(20) = √(4 · 5) = √(4) · √(5) = 2√(5)

Answer: The hypotenuse is 2√(5) (exact value).

Exact Values vs. Decimals. When a problem says "exact value", leave the answer in simplified radical form (e.g. 2√(5)) rather than rounding to a decimal. Exact values avoid rounding errors.

Worked example: Write the Trig Ratios

In right ABC, the right angle is at C, side AB = 13, and side BC = 12. Write A and B.

Solution. Step 1 — Find the missing side using the Pythagorean theorem.

12^2 + AC^2 = 13^2

144 + AC^2 = 169

AC^2 = 25, so AC = 5

Step 2 — Write A.

From angle A: the opposite side is BC = 12 and the hypotenuse is AB = 13.

A = (opposite)/(hypotenuse) = (12)/(13)

Step 3 — Write B.

From angle B: the opposite side is AC = 5 and the adjacent side is BC = 12.

B = (opposite)/(adjacent) = (5)/(12)

Answer: A = (12)/(13) and B = (5)/(12)

Sanity Check. Notice: when we switched from angle A to angle B, the "opposite" and "adjacent" sides swapped. This is a useful check whenever you compute ratios for two different angles in the same triangle.

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