high school Trigonometric Functions

Trigonometric Functions

Trigonometry begins with a point circling a circle of radius 1: its x-coordinate is the cosine, its y-coordinate is the sine, and their ratio is the tangent. This works for any angle, not just inside a right triangle, and the quadrant the point sits in sets each sign. From radius 1 comes sin²θ + cos²θ = 1. Rotate the point with the angle slider and sine and cosine change, with the special-angle values and quadrant signs.

Points on the Unit Circle — Defining sin, cos

Angle θ45°

💡 The unit circle is the starting point

①Point P(cos θ, sin θ) on the unit circle

②cos θ = x-coordinate (red), sin θ = y-coordinate (blue)

③As the angle changes, the point moves around the circle

Trig Values for Special Angles

30°-60°-90° Ratios

sin 30° = 12, cos 30° = √32, tan 30° = 1√3

Side ratios 1 : √3 : 2

45° Ratios

sin 45° = √22, cos 45° = √22, tan 45° = 1

Right triangle made from a square diagonal

Basic Relationships

Pythagorean Identity

sin²θ + cos²θ = 1

Since the unit circle has radius 1: x² + y² = 1

Tangent Definition

tan θ = sin θcos θ

Slope = y/x

Reciprocal Relations

sec θ = 1cos θ, csc θ = 1sin θ, cot θ = 1tan θ

Reciprocals of the trig functions

Signs by Quadrant

📐 ASTC Rule

①Q1 (0°–90°): sin+, cos+, tan+ (All)

②Q2 (90°–180°): sin+, cos−, tan− (Sin)

③Q3 (180°–270°): sin−, cos−, tan+ (Tan)

④Q4 (270°–360°): sin−, cos+, tan− (Cos)

⑤tan θ = sin θ / cos θ, and it is undefined when cos θ = 0

Wrap-up

Trig Core

sin²θ + cos²θ = 1, tan θ = sin θcos θ

All trig functions defined on the unit circle

🎯 Exam Points

①Unit-circle definition: P(cos θ, sin θ)

②Memorize trig values for 30°, 45°, 60°

③Pythagorean identity: sin²θ + cos²θ = 1

④Quadrant signs (ASTC)

⑤tan θ = sin θ / cos θ (undefined when cos θ = 0)

Worked Examples & Past Exam

Example 1

If sin θ = 3/5 and θ is in the first quadrant, find cos θ.

Use the Pythagorean identity sin²θ + cos²θ = 1.

cos²θ = 1 - sin²θ = 1 - (3/5)² = 16/25

In the first quadrant cos θ > 0, so take the positive root.

cos θ = √(16/25) = 4/5

▸ cos θ = 4/5

Find the other value via the Pythagorean identity, then fix the sign by quadrant.

Example 2

If tan θ = 1 and θ is in the third quadrant, find sin θ.

tan θ = 1 gives reference angle 45°; in the third quadrant θ = 225°.

θ = 180° + 45° = 225°

In the third quadrant sin θ < 0, so sin 225° = −sin 45°.

sin θ = -√22

▸ sin θ = −√2/2

Find the trig value of the reference angle, then attach the quadrant sign (ASTC).

exam-style

If cos θ = −1/2 and 0° < θ < 180°, find θ.

①60°

②120°

③135°

④150°

⑤240°

▸ ② 120°

cos θ = −1/2 gives reference angle 60°.

cos 60° = 12

For 0°<θ<180°, cos θ < 0 means the second quadrant, so θ = 180° − 60°.

θ = 180° - 60° = 120°

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Common Logarithm

Graphs of Trigonometric Functions

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