Presents

Mathematics

Department

Graphs of Sine, Cosine and Tangent

The combined graphs

Summary

Solving trigonometric equations

Menu

x 0 30 60 90 120 150 180 210 240 270 300 330 360Sin xCos xTan x

Graphs

x 0 30 60 90 120 150 180 210 240 270 300 330 360Sin x 0.0 0.5 0.9 1.0 0.9 0.5 0.0 -0.5 -0.9 -1.0 -0.9 -0.5 0.0Cos xTan x

Graphs

x 0 30 60 90 120 150 180 210 240 270 300 330 360Sin x 0.0 0.5 0.9 1.0 0.9 0.5 0.0 -0.5 -0.9 -1.0 -0.9 -0.5 0.0Cos x 1.0 0.9 0.5 0.0 -0.5 -0.9 -1.0 -0.9 -0.5 0.0 0.5 0.9 1.0Tan x

Graphs

x 0 30 60 90 120 150 180 210 240 270 300 330 360Sin x 0.0 0.5 0.9 1.0 0.9 0.5 0.0 -0.5 -0.9 -1.0 -0.9 -0.5 0.0Cos x 1.0 0.9 0.5 0.0 -0.5 -0.9 -1.0 -0.9 -0.5 0.0 0.5 0.9 1.0Tan x 0.0 0.6 1.7 ??? -1.7 -0.6 0.0 0.6 1.7 ??? -1.7 -0.6 0.0

Graphs

What about tan 70°?

tan 80°?

tan 85°?

Can you explain what’s happening?

Sin xº

0

-1

1

90 360270180xº

Graph of Sin x°

Cos xº

Graph of Cos x°

0

-1

1

90 360270180xº

Tan xº

Graph of Tan x°

0

-1

1

90 360270180xº

This isn’t drawn to scale- but it looks something like this!

0 - 90°

Sin x ° +ve

Cos x ° +ve

Tan x ° +ve

Combined Graphs

0

-1

1

90 360270180xº

Sin xº

Cos xºTan xº

Sin x ° +ve

Cos x ° -ve

Tan x ° -ve

Combined Graphs

0

-1

1

90 360270180xº

Sin xº

Cos xºTan xº

90°-180°

Sin x ° -ve

Cos x ° -ve

Tan x ° +ve

Combined Graphs

0

-1

1

90 360270180xº

Sin xº

Cos xºTan xº

180°-270°

Sin x ° -ve

Cos x ° +ve

Tan x ° -ve

Combined Graphs

0

-1

1

90 360270180xº

Sin xº

Cos xºTan xº

270°-360°

270°

180°

90°

0°

Summary

270°

180°

90°

0°

Sin x ° +ve Cos x ° +ve Tan x ° +ve

Sin x ° +ve Cos x ° -ve Tan x ° -ve

Sin x ° -ve Cos x ° -ve Tan x ° +ve

Sin x ° -ve Cos x ° +ve Tan x ° -ve

Sin

Tan Cos

All

Which are positive?

Summary

270°

180°

90°

0°

Sin x ° +ve Cos x ° +ve Tan x ° +ve

Sin x ° +ve Cos x ° -ve Tan x ° -ve

Sin x ° -ve Cos x ° -ve Tan x ° +ve

Sin x ° -ve Cos x ° +ve Tan x ° -ve

Sinners

Take

Care!

All

Which are positive?

Summary

Cos x° = 0.5

0 ≤x⁰≤360

Cos xº

0

-1

1

90 360270180 xº

0.5

60° 300°

Example 1

So x = 60°

, 300°

270°

180°

90°

0°

Cos x° = 0.5

0≤x⁰≤360

A

T

S

C

(Cos⁻¹ 0.5 = 60°)

300°

x = 60°

, 300°

Example 2

60°60°

Cos +ve

Cos +ve

270°

180°

90°

0°

Sin x° = -0.5

0≤x⁰≤360

A

T

S

C

30°Sin -ve

(Sin⁻¹ 0.5 = 30°)

Sin -ve

, 330°

x = 210°

30°

Example 3

270°

180°

90°

0°

2Sin x° = 1

0≤x⁰≤360

A

T

S

C

(Sin⁻¹ ½ = 30°)x = 30°

Sin x° = ½

,150°

30º 30º

Example 4

Sin +ve

Sin +ve

270°

180°

90°

0°

3 cos x° = -10≤x⁰≤360

A

T

S

C

cos -ve

(cos⁻¹ ⅓ = 70.5°)

cos -ve

, 250.5°

x = 109.5°

3 cos x°+1 = 0

cos x° = -⅓

70.5°

70.5°

Example 5

Mathematics

Department