The Tangent Ratio

Learn tangent ratio in right triangle trigonometry. Math 10-C Alberta mathematics curriculum.

Lesson 3.1 of Trigonometry in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Labeling the Sides of a Right Triangle. Learn to identify the hypotenuse, opposite, and adjacent sides of a right triangle relative to a chosen angle, and understand angles of elevation and depression.
• The Tangent Ratio. Define the tangent ratio as opposite over adjacent, interpret what the value tells you about the triangle, and find angles using inverse tangent.
• Using Tangent to Find a Missing Side. Apply the tangent ratio to calculate an unknown opposite or adjacent side when you know one leg and an acute angle.
• Word Problems with Tangent. Solve real-world problems involving lighthouses, ladders, and rivers by drawing right triangles, identifying sides, and applying the tangent ratio.

Inside the lesson — a free preview

The Three Sides

Before you can use any trig ratio, you need to know how to label the three sides of a right triangle relative to a specific angle. This labeling changes depending on which angle you are looking at.

The Hypotenuse. The hypotenuse (hyp) is the longest side of the right triangle. It is always directly across from the 90° angle.

The hypotenuse never changes — no matter which acute angle you choose to work with, the hypotenuse stays the same.

The Opposite Side. The opposite side (opp) is the side directly across from the angle you are working with.

If you switch to a different angle, the opposite side changes.

The Adjacent Side. The adjacent side (adj) is the side next to the angle you are working with — the one that is not the hypotenuse.

Always ask yourself: "Which angle am I working with?" before labeling.

Important — Opposite and Adjacent Swap When You Switch Angles. If you switch from angle A to angle C, the sides swap roles.

From angle A: side BC is opposite, side AB is adjacent.

From angle C: side AB is opposite, side BC is adjacent.

The hypotenuse is the only side that never changes.

Angles of Elevation and Depression

Angle of Elevation. An angle of elevation is the angle measured upward from a horizontal line to a line of sight.

You use this when you are looking up at something above you — for example, looking up at the top of a building or a mountain peak.

Angle of Depression. An angle of depression is the angle measured downward from a horizontal line to a line of sight.

You use this when you are looking down at something below you — for example, looking down from a cliff to a boat.

Key Point — Both Are Just Angles in Right Triangles. Whether a problem says "angle of elevation" or "angle of depression," you are still working with an acute angle inside a right triangle.

These terms only tell you where the angle sits in the real-world scenario — they do not change how you solve the triangle.

Worked example: Find the Tangent Ratios

Given a right triangle with angle A at the bottom-left, right angle at B, with AB = 12 cm and BC = 5 cm.

Part a) — Find A. Step 1 — Identify the sides from angle A:

Opposite = BC = 5 cm

Adjacent = AB = 12 cm

Step 2 — Apply the tangent formula:

A = (opp)/(adj) = (5)/(12)

Step 3 — Find the angle using inverse tangent:

θ = ^(-1)((5)/(12)) ≈ 22.6°

Answer: A = (5)/(12) ≈ 0.417, and A ≈ 22.6°

Part b) — Find C. Step 1 — Identify the sides from angle C:

From angle C's perspective, the opposite side is AB = 12 cm and the adjacent side is BC = 5 cm.

Step 2 — Apply the tangent formula:

C = (12)/(5)

Step 3 — Find the angle:

θ = ^(-1)((12)/(5)) ≈ 67.4°

Answer: C = (12)/(5) = 2.4, and C ≈ 67.4°

Key Observation — Reciprocals and Complementary Angles. A = (5)/(12) and C = (12)/(5) are reciprocals of each other.

The two angles add up to 90°: 22.6° + 67.4° = 90°.

This always happens with the two acute angles in a right triangle.

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