Trigonometry

Question 1

Prove that ( 1 - cos x ) ( 1 + sec x ) = tanxsinx.

Answer 1

• ( 1 - cos x ) ( 1 + sec x )
• = 1 + sec x - cos x - cosxsecx
• • cosxsecx = - cos x / cos x = - 1
• ( sec x = 1 / cos x )
• 1 + sec x - cos x - 1
• = sec x - cos x
• = 1 / cos x - cos x
• = 1 - cos^2 x / cos x
• ( 1 - cos^2 x = sin^2 x )
• = sin^2 x / cos^2 x
• = sin x / cos x ( sin x )
• = tanxsinx

Question 2

*State the period of y = cosec x

Question 3

Deduce the number of solutions of cosec x = 3 for - pi <= x <= pi

( Graph has a equation of y = cosec x )

Answer 3

• Solutions are where y = 3 intersects with y = cosec x

- So there are two solutions in the given interval

Question 4

Sketch the graph of y = pi / 3 + arctan x, indicating the y-intercept

Answer 4

• Translation by ( 0 pi / 3 )
• ( Line graph which plateaus at - pi / 6 and 5pi / 6 )
• ( The curve from left to right, is flat, then starts to rise, passing through the y-intercept, and plateaus again at the end )
• ( Has a y-intercept at ( 0, pi / 3 )
• ( Original graph goes through ( 0, 0 ) )
• ( plateaus at pi / 2 and - pi / 2 )

Question 5

State the range of y = pi / 3 + arctan x in radians

Graph plateaus at - pi / 6 and 5pi / 6

Answer 5

• Range = - pi / 6 < pi / 3 + arctan x < 5pi / 6

- ( Range is in respect of the y-axis )

Question 6

Given that p = sin x and q = 2sec x, show that 4 + p^2q^2 = q^2

Answer 6

• 4 + p^2q^2 = 4 + ( sin x )^2 ( 2sec x )^2
• = 4 + 4sin^2 x sec^2 x
• = 4 + 4sin^2 x / cos^2 x
• ( sec^2 x = 1 / cos^2 x )
• = 4cos^2 x + 4sin^2 x / cos^2 x
• = 4 ( cos^2 x + sin^2 x ) / cos^2 x
• ( cos^2 x + sin^2 x = 1 )
• = 4 / cos^2 x
• = 4sec^2 x = q^2

Question 7

Given that tan A = - 45 / 28 and the angle A is obtuse, find the exact values of:

a ) sec A

b ) cosec A

Answer 7

a )

• 1 + tan^2 A = sec^2 A
• 1 + ( - 45 / 28 )^2 = sec^2 A
• sec A = +- Root ( 2809 / 784 )
• = +- 53 / 28
• Sec A = - 53 / 28 when A is obtuse

b )

• tan A = - 45 / 28
• cot A = - 28 / 45
• 1 + cot^2 A = cosec^2 A
• 1 + ( - 28 / 45 )^2 = cosec^2 A
• cosec A = +- Root ( 2809 / 2025 )
• = +- 53 / 45
• cosec A = 53 / 45 when A is obtuse