Trigonometry Graphs

Learn sine, cosine, and tangent graphs

CAPS Grade 10 Mathematics

Trig graphs show how sine, cosine, and tangent values change as the angle increases. Use the interactive graph below to see how changing values affects each curve.

Interactive Graph

a (height)

p (shift °)

q (up/down)

Current: y = sin θ

Period: 360°, Amplitude: 1

Hover: θ = 0°, y = 0

What You'll Learn

Sketch sine, cosine, and tangent graphs

Find period, amplitude, and asymptotes

Apply transformations (stretch, shift)

Solve equations using graphs

Connect to real-world problems

Quiz 1: Basic Graphs

What is the amplitude of y = sin θ?

A) 0

B) 1

C) 2

D) 360

1. Basic Graphs

Sine: y = sin θ

Period: 360°
Amplitude: 1
Range: -1 to 1
Starts at 0, goes up to 1, down to -1

Cosine: y = cos θ

Period: 360°
Amplitude: 1
Range: -1 to 1
Starts at 1, goes down to -1, back to 1

Tangent: y = tan θ

Period: 180°
Amplitude: None (goes to infinity)
Asymptotes: at 90°, 270°
Passes through (0°,0)

Quiz 2: Key Features

What is the period of y = tan θ?

A) 90°

B) 180°

C) 360°

D) 720°

2. Transformations

y = a·sin(θ + p) + q
a = height (amplitude)
p = horizontal shift (left if +)
q = vertical shift (up if +)

a - Height

Changes wave height
y = 2sin θ goes from -2 to 2
Negative a flips the graph

p - Shift

sin(θ + 30°) shifts LEFT 30°
sin(θ - 30°) shifts RIGHT 30°

q - Up/Down

sin θ + 2 shifts UP 2 units
New middle line at y = q

Quiz 3: Transformations

What is the range of y = 2sin θ + 1?

A) -1 to 1

B) -2 to 2

C) -1 to 3

D) 0 to 2

Example: y = 2sin(θ - 30°) + 1

Step-by-Step

Start with y = sin θ (basic wave)
Multiply by 2 → height doubles (range -2 to 2)
θ - 30° shifts RIGHT 30°
+ 1 shifts UP 1 unit
Final range: -1 to 3

3. Quick Reference

Function	Period	Amplitude	Range	y-intercept
y = sin θ	360°	1	[-1, 1]	0
y = cos θ	360°	1	[-1, 1]	1
y = tan θ	180°	None	All real	0

Quiz 4: Period & Amplitude

What is the period of y = sin θ?

A) 90°

B) 180°

C) 360°

D) 720°

4. Solving Using Graphs

Example: sin θ = 0.5

Draw y = sin θ from 0° to 360°
Draw horizontal line y = 0.5
Find where they cross
Solutions: θ = 30° and θ = 150°

Answer: θ = 30°, 150°

Quiz 5: Solving Equations

Using the graph of y = cos θ, what are the solutions to cos θ = 0.5 between 0° and 360°?

A) 30°, 150°

B) 60°, 300°

C) 45°, 315°

D) 90°, 270°

5. Real-World Uses

Tides

h = 2sin(30t) + 3

Max height: 5m
Min height: 1m
Period: 12 hours

Temperature

T = -5cos(15t) + 20

Range: 15°C to 25°C
Coldest at midnight: 15°C
Hottest at noon: 25°C

6. Watch Out For

Mistake:

sin(θ + 30°) shifts RIGHT? NO! It shifts LEFT.

Mistake:

Tangent has amplitude? NO! Tangent has no amplitude.

Mistake:

Forgetting asymptotes for tan θ at 90° and 270°.

Quiz 6: Real-World

A Ferris wheel: h = 15sin(12t) + 20. What is the maximum height?

A) 20m

B) 25m

C) 35m

D) 15m

7. Practice

What is the amplitude of y = 3cos θ?

Answer: Amplitude = 3

What is the period of y = tan θ?

Answer: Period = 180°

Find the range of y = 2sin θ + 1

Answer: Range = [-1, 3]

8. Summary

Sine & Cosine

Period = 360°
Amplitude = |a|
Range = [q - |a|, q + |a|]

Tangent

Period = 180°
No amplitude
Asymptotes at 90° + k·180°

Basic Trig Functions