Trigonometry

Question 1

sin

Answer 1

180 - x

Question 2

cos

Answer 2

360 - x

Question 3

tan

Answer 3

180 + x

Question 4

Trigonometry
example
8tanx = 9 0<x<360

Answer 4

tanx = 9/8
x = tan-1 (9/8) 180 + 48.37 = 228.37
x = 48. 37, 228.37

Question 5

Trigonometry
example
7cosx + 1 = 0

Answer 5

7cosx = -1
cox = -1/7
x = cos-1(1/7) 180 - 81.8 = 98.2
x = 81.8 180 + 81.8 = 261.8
x = 98.2, 261.8

Question 6

Solving trig equations
example
3sin2x = 1 0 < x < 360

Answer 6

sin2x = 1/3 0 < x < 720
2x = sin-1(1/3)
2x = 19.5, 160.5 (+ 360)
= 379.5, 520.5
x = 9.75, 80.25, 189.75, 260.25

Question 7

Using exact values
example
sin300

Answer 7

-sin60
-square root3/2

Question 8

Using exact values
example
cos(-135)

Answer 8

-cos45
-1/square root 2

Question 9

change 180 into radians

Answer 9

Pi

Question 10

change 360 into radians

Answer 10

2Pi

Question 11

Radians
change 90 into radians

Answer 11

180 = Pi
90 = Pi/2

Question 12

Radians
example
square root 2sin2x - 1 = 0

Answer 12

square root 2sin2x = 1
sin2x = 1/square root 2
2x = sin-1 (1/square root2)
2x = 45
= Pi/4, 3Pi/4 4Pi/4 - Pi/4 = 3Pi/4
.= 9Pi/4, 11Pi/4 (+ 2Pi)
x = Pi/8, 3Pi/8, 9Pi/8, 11Pi/8

Question 13

Addition Formulae
example
expand & simplify
sin(x + 30)

Answer 13

= sinxcos30 + coxsin30
sinxsquare root3/2 + cosx x 1/2
square root 3/2sinx + 1/2cosx

Question 14

Double Angle Formulae
example
show that 2cos2x = 3cos^2x - sin^2x - 1

Answer 14

2(2cos^2x - 1)
4cos^2x - 2
3cos^2x + (cos^2x -1) -1
cos^2x + sin^2x = 1
cos^2x -1 = sin^2x
3cos^2x - sin^2x - 1

Question 15

Solving trig equations using double angle
example
solve sin2x - cosx = 0

Answer 15

replace sin2x with 2sinxcosx
2sinxcosx - cosx = 0
factorise 2sc -c c(2s - 1)
cosx(2sinx - 1) = 0
cosx = 0 2sinx - 1 = 0
sinx = 1/2 180 - 30 = 150
use the x = 30, 150
graph
x = 90, 270
x = 30, 90, 150, 270

Question 16

Wave Function

Answer 16

1. expand using the addition formula
2. equate the co-efficients
3. calculate k k = square root a^2 + b^2
4. Calculate ~ tanx = sinx/cosx
5. Write out equation

Question 17

Wave Function
example
Write 2cosx + 3sinx in the form kcos(x - ~)

Answer 17

1. kcos(x - ~) = k(cosxcos~ + sinxsin~)
   = kcosxcos~ + ksinxsin~
   2cosx + 3sinx
2. kcos~ = 2 ksin~ = 3
3. k = square root (2)^2 + (3)^2
   square root 13
4. tanx = sinx/cosx = 3/2
   x = tan-1 = (3/2)
   = 56.3
5. square root 13cos(x - 56.3)

Question 18

Solving trig equation with wave function
example
a) express 4sinx + 5cosx in the form ksin(x+a)
b) hence solve 4sinx + 5cosx = 5.5

Answer 18

a) ksinxcosa + kcosxsina
4sinx + 5cosx
kcosa = 4 ksina = 5
k = square root (4)^2 + (5)^2
= square root 41

tansx = sinx/cosx = 5/4 = 0.896

b) square root 41sin(x + 0.896) = 5.5
sin(x + 0,896) = 5.5/square root 41
x + 0.896 = sin-1 (5.5/square root 41) Pi - x
x + 0.896 = 1.033, 2.108
x = 0.137, 1.212

Question 19

Sketching ksin(x + a) & kcos(x - a)

Answer 19

roots
- either knowledge of graph
- when y = 0
y-intercept
- when x = 0
turning points
- knowledge of graphs

Question 20

Sketching ksin(x - 45)
example
y = 3sin(x - 45)

• goes to the right

Answer 20

0 = 45 90 = 135 180 = 215
270 = 305

y = 3sin(0-45)
= 2.12

Question 21

Max/Min values
example
state the maximum value of y = 3sin(x-45)

Answer 21

the maximum value of sinx = 1
therefore 3sinx = 3

the corresponding x value
3sin(x - 45) = 3
sin(x - 45) = 1
x - 45 = 90
x = 135