Trigonometry (5)

Question 1

Endpoints for arcsine

Answer 1

(1,pi/2) and (-1,-pi/2)

Question 2

(1,pi/2) and (-1,-pi/2)

Answer 2

Endpoints for arcsine

Question 3

Endpoints for arccosine

Answer 3

(-1, pi) and (1,0). Crosses the y-axis at (0,pi/2)

Question 4

(-1, pi) and (1,0). Crosses the y-axis at (0,pi/2)

Answer 4

Endpoints for arccosine

Question 5

Asymptotes for arctangent

Answer 5

y = pi/2 and y=-pi/2

Question 6

y = pi/2 and y=-pi/2

Answer 6

Asymptotes for arctangent

Question 7

Working out cos x = -(1/2)

Answer 7

(cos^-1)(-1/2)

Question 8

(cos^-1)(-1/2)

Answer 8

Working out cos x = -(1/2)

Question 9

Interval of sin 3x for 0<=x<=2pi

Answer 9

0<=3x<=6pi

Question 10

0<=3x<=6pi

Answer 10

Interval of sin 3x for 0<=x<=2pi

Question 11

Range for 2cos(x - pi/4) = ¬3 for 0<=x<=2pi

Answer 11

-pi/4 <= x - pi/4 <= 2pi - pi/4

Question 12

Approximate cos(pi/12)

Answer 12

1 - (1/2)(pi/12)^2 = 0.966

Question 13

1 - (1/2)(pi/12)^2 = 0.966

Answer 13

Approximate cos(pi/12)

Question 14

Find the small angle approximation for cosec^2pheta

Answer 14

1/pheta^2

Question 15

1/pheta^2

Answer 15

Find the small angle approximation for cosec^2pheta

Question 16

Find the small angle approximation for sin^2pheta * cospheta

Answer 16

pheta^2 * (1-(1/2)pheta^2) = pheta^2 -(1/2)pheta^4

Question 17

Write 9sinx+12cosx in the form R sin (x+a)

Answer 17

9sinxcosa + 12cosxsina
R cos a = 9, R sin a = 12
tan^-1(12/9) = 0.9272...
¬(12^2 + 9^2) = 15
9sinx+12cosx = 15 sin (x + 0.927)

Question 18

Solve 9sinx + 12cosx = 3 for ranges 0<=x<=2pi

Answer 18

As 9sinx+12cosx = 15 sin (x + 0.9272...)
15 sin (x + 0.9272...) = 3 => sin (x + 0.9272...) = 0.2
0.9272...<= x + 0.9272... <= 7.2104...

draw sinx

x + 0.9272… = sin^-1(0.2)
x + 0.9272… = 0.2013…
(This is outside the range)

pi - 0.2013… = 2.940…
2pi + 0.2013… = 6.484…

(x + 0.9272…) = 2.940… and 6.484
x = 2.01 and 5.56

Question 19

Find the maximum and minimum value of f(x) = 10 - 9 sin x - 12 cos x

Answer 19

9sinx + 12cosx = 15 sin (x + 0.927)

max: 10 + 15 = 25
min: 10 - 15 = -5

Question 20

Solve sin2x = -(1/2) for 0<=x<=360

Answer 20

The new range is 0<=2x<=720

Draw sinx to get the values 210, 330, 570, 690
Divide these values by 2 to get the solutions for sinx. Which are 105, 165, 285, 345