Presents Mathematics Department Graphs of Sine, Cosine and Tangent The combined graphs Summary...
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Transcript of Presents Mathematics Department Graphs of Sine, Cosine and Tangent The combined graphs Summary...
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
