Trigonometry Flashcards
1
Q
Prove that ( 1 - cos x ) ( 1 + sec x ) = tanxsinx.
A
- ( 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
2
Q
*State the period of y = cosec x
A
3
Q
Deduce the number of solutions of cosec x = 3 for - pi <= x <= pi
( Graph has a equation of y = cosec x )
A
- Solutions are where y = 3 intersects with y = cosec x
- So there are two solutions in the given interval
4
Q
Sketch the graph of y = pi / 3 + arctan x, indicating the y-intercept
A
- 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 )
5
Q
State the range of y = pi / 3 + arctan x in radians
Graph plateaus at - pi / 6 and 5pi / 6
A
- Range = - pi / 6 < pi / 3 + arctan x < 5pi / 6
- ( Range is in respect of the y-axis )
6
Q
Given that p = sin x and q = 2sec x, show that 4 + p^2q^2 = q^2
A
- 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
7
Q
Given that tan A = - 45 / 28 and the angle A is obtuse, find the exact values of:
a ) sec A
b ) cosec A
A
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
