Post on 27-Mar-2015
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