Trigonometric Ratios

A trigonometric ratio compares two sides of a right triangle, so once the angle is fixed, sin, cos, and tan are fully determined. Thanks to this, you can measure the height of a building, a tree, or a mountain without climbing it. Measure the distance and the angle of elevation from afar, and the height is simply distance × tan(angle). Here you can drag the angle slider to watch the ratios change and compare the special angles 30°, 45°, and 60°.

What is a Trig Ratio? — Sides of a Right Triangle

Angle θ30°

📐 Definitions

①sin θ = opposite / hypotenuse

②cos θ = adjacent / hypotenuse

③tan θ = opposite / adjacent

④If hypotenuse = 1: sin = opposite, cos = adjacent

Special Angles (30°, 45°, 60°)

Special angle45°

⭐ How to Remember

①30°: sin=1/2, cos=√3/2, tan=√3/3

②45°: sin=cos=√2/2, tan=1

③60°: sin=√3/2, cos=1/2, tan=√3

④30°↔60°: sin and cos swap!

Core Relationships

Trig Identity

tan θ = sin θcos θ

tan equals sin/cos

Pythagorean Identity

sin²θ + cos²θ = 1

Pythagorean theorem on a unit-hypotenuse right triangle

💡 Using the Identities

①Given sin θ → cos θ = √(1−sin²θ)

②Knowing one ratio gives all the others

③For 0° < θ < 90°, all three ratios are positive

Application — Heights & Distances

Finding Height

h = d × tan θ

From distance d and angle of elevation θ

🏢 Real-life Examples

①Building height: distance 50m, elevation 30° → h = 50×tan30° ≈ 28.9m

②Same idea for trees, mountains, towers

③Angles of depression work the same way

Work It Out

Example 1

Find the value of sin 30°.

Recall the special-angle table for 30°.

30° → sin=1/2

So sin30° = 1/2.

sin30° = 1/2

▸ 1/2

Memorize the special-angle values (30°, 45°, 60°).

Example 2

In a right triangle, sin A = 3/5 and the hypotenuse is 10. Find the side opposite angle A.

Since sin A = opposite/hypotenuse, opposite = hypotenuse × sin A.

sin A = opp/hyp

opposite = 10 × (3/5) = 6.

opp = 10 × 3/5 = 6

▸ 6

From sin = opp/hyp, opp = hyp × sin.

Exam Wrap-up

Trig Ratio Recap

sin = opphyp, cos = adjhyp, tan = oppadj

Remember SOH-CAH-TOA

G9 school-exam type

What is the value of tan 60°?

①√3/3

②1/2

③√2

④√3

⑤2

▸ ④ √3

Read the special-angle table for 60°.

60° → tan=√3

So tan60° = √3.

tan60° = √3

🎯 Exam Points

①Memorize the special-angle table

②Use sin²θ+cos²θ=1 to find missing ratios

③Use tan = sin/cos

④Height/distance problems: choose the right ratio

⑤Only acute angles (0°~90°) — obtuse comes in high school

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Applications of Quadratic Functions

Circle & Line

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