Mathematics
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Graphs of Sine, Cosine and Tangent
The combined graphs
Summary
Solving trigonometric equations
Graphs
x
Sin x
Cos x
Tan x
0 30 60 90 120 150 180 210 240 270 300 330 360
Graphs
x
0 30 60 90 120 150 180 210 240 270 300 330 360
Sin 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 0.0
Cos x
Tan x
Graphs
x
0 30 60 90 120 150 180 210 240 270 300 330 360
Sin 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 0.0
Cos 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 1.0
Tan x
Graphs
x
0 30 60
Sin x 0.0 0.5 0.9
Cos x 1.0 0.9 0.5
Tan x 0.0 0.6 1.7
90
1.0
0.0
???
120
0.9
-0.5
-1.7
150
0.5
-0.9
-0.6
180
0.0
-1.0
0.0
210
-0.5
-0.9
0.6
240
-0.9
-0.5
1.7
270
-1.0
0.0
???
What about tan 70°?
tan 80°?
tan 85°?
Can you explain what’s happening?
300
-0.9
0.5
-1.7
330
-0.5
0.9
-0.6
360
0.0
1.0
0.0
Graph of Sin x°
1
0
90
180
270
360
Sin xº
-1
xº
Graph of Cos x°
Cos xº
1
0
90
-1
180
270
360
xº
Graph of Tan x°
Tan xº
1
0
90
-1
180
270
360
xº
This isn’t drawn
to scale- but it
looks something
like this!
Combined Graphs
Tan xº
Cos xº
1
0
0 - 90°
90
180
270
360
Sin xº
-1
Sin x ° +ve
Cos x ° +ve
Tan x ° +ve
xº
Combined Graphs
Tan xº
Cos xº
1
0
90
90°-180°
180
270
360
Sin xº
-1
Sin x ° +ve
Cos x ° -ve
Tan x ° -ve
xº
Combined Graphs
Tan xº
Cos xº
1
0
90
180
180°-270°
270
360
Sin xº
-1
Sin x ° -ve
Cos x ° -ve
Tan x ° +ve
xº
Combined Graphs
Tan xº
Cos xº
1
0
90
180
270
270°-360°
360
Sin xº
-1
Sin x ° -ve
Cos x ° +ve
Tan x ° -ve
xº
Summary
90°
180°
0°
270°
Summary
90°
Sin x ° +ve
CosSin
x ° -ve
Tan x ° -ve
180°
Sin x ° -ve
CosTan
x ° -ve
Tan x ° +ve
Sin x ° +ve
Cos All
x ° +ve
Tan x ° +ve
Sin x ° -ve
Cos Cos
x ° +ve
Tan x ° -ve
270°
Which are positive?
0°
Summary
90°
Sin x ° +ve
Sinners
Cos
x ° -ve
Tan x ° -ve
180°
Sin x ° -ve
Take
Cos
x ° -ve
Tan x ° +ve
Sin x ° +ve
Cos All
x ° +ve
Tan x ° +ve
Sin x ° -ve
CosCare!
x ° +ve
Tan x ° -ve
270°
Which are positive?
0°
Example 1
Cos x° = 0.5
0 ≤x⁰≤360
So x = 60°, 300°
Cos xº
1
0.5
0
60°
-1
90
180
270
300°
360
xº
Example 2
Cos x° = 0.5
0≤x⁰≤360
(Cos⁻¹ 0.5 = 60°)
x = 60°, 300°
90°
S
180°
T
Cos A
+ve
60°
300° 60°
Cos
+ve
C
270°
0°
Example 3
Sin x° = -0.5
0≤x⁰≤360
(Sin⁻¹ 0.5 = 30°)
x = 210°, 330°
90°
S
180°
Sin
T-ve
A
30°
30°
Sin
-ve
270°
0°
C
Example 4
2Sin x° = 1 0≤x⁰≤360
Sin x° = ½
(Sin⁻¹ ½ = 30°)
x = 30°,150°
90°
SSin
+ve
Sin A
+ve
30º
30º
180°
T
C
270°
0°
Example 5
3 cos x°+1 = 0 0≤x⁰≤360
3 cos x° = -1
cos x° = -⅓ (cos⁻¹ ⅓ = 70.5°)
x = 109.5°, 250.5°
90°
Scos
-ve
180°
A
70.5°
70.5°
0°
cos
T-ve
C
270°
Mathematics
Department
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Cos xº