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. Rotate the point with the angle slider to watch sine and cosine change and meet the 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)

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).

2023 CSAT Math type, adapted

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

Trig Graphs

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