THE UNIVERSITY OF ZAMBIA

Department of Mathematics and Statistics

MAT1110: Foundation Mathematics and Statistics for Social Sciences
Tutorial Sheet 5 (2020/2021)
1. Find the quadrant that contains the terminal (end) side of θ if the given conditions are
satisfied:

(a) i. iii. v.

sin θ > 0 and tan θ < 0 cos θ > 0 and tan θ < 0 sin θ < 0 and cot θ < 0
ii. iv. vi.
sin θ < 0 and cos θ < 0 tan θ > 0 and sec θ < 0 csc θ > 0 and sec θ > 0

(b) Without the use of a calculator, convert each of the following to radians:

i. −150◦ iii. 15◦ v. 570◦

ii. 225◦ iv. −330◦ vi. 135◦

(c) Without the use of a calculator, convert each of the following to degrees:
π
i. 4π
3
iii. 12 v. 7π
6
ii. − 3π
4
iv. 5π
3
vi. − 11π
4

2. (a) Without using a calculator or table, find the exact values of each of the following:

i. sin 225◦ iv. sin(−135)◦ vii. cot(− 5π

3
)
ii. cos 225◦ v. tan 315◦ 7π
viii. sec 6
π
iii. cos 150◦ vi. cos 7π
3
ix. csc( 12 )

(b) Simplify each of the following to a single trigonometric function or a constant:

sec θ−cos θ
i. sec x − sin x tan x iii. cos x + tan x sin x v. tan θ
sec θ+1
ii. csc x − cos x cot x iv. (sin2 x − 1)(tan2 x + 1) vi. tan θ+sin θ

(c) Prove each of the following identities:

i. v.
1 + sin x (tan x)(1 + cot2 x)
tan x + 1 + tan x sec x =
2 = cot x
cos2 x 1 + tan2 x
vi.
1 − sin x
ii.
cos x + sin x = (tan x − sec x)2 .
= 1 + tan x 1 + sin x
cos x
iii. vii.
csc x − 1 cot x
= sec x + tan x.
1
cot x cscx + 1
=
sec x(1 − sin x)
iv.
viii.
= cos x sin x tan x + tan y
1
tan x + cot x = tan y tan x.
cot x + cot y
 (d) Verify each of the following:

i. iii. v.
π
cos(2x) = 2 cos2 x − 1 cos(θ − π) = − cos θ sin(θ + ) = cos θ
2
ii. iv. vi.
π
cos(2x) = 1 − 2 sin2 x cos(θ + ) = − sin θ tan(θ − π) = tan θ
2

3. (a) Without using a calculator or a table, solve each of the following equations for 0 ≤
x ≤ 2π :

i. iii. v.
√ √
2 sin x + 3=0 2 cos2 x − 3 cos x = 0 2 cos2 x + cos x = sin2 x
ii. iv. vi.
sec x = 2 sin x + 2 = 3 tan x = cot x

(b) Without using a calculator or a table, solve each of the following equations for 0 ≤
x ≤ 2π :

i. iii.
2 tan x sec x − tan x = 0 cos 2x + 3 sin x − 2 = 0
ii. iv.
sec2 x − sec x − 2 = 0 sin x + cos x = 1

(c) Sketch each of the following in the indicated interval:

i. iii.
f(x) = − cos x, 0 ≤ x ≤
2π
f(x) = 1 + sin x, 0 ≤ x ≤ 2π 3
iv.
ii.  
f(x) = sin x , 0 ≤ x ≤ 4π
1 1
f(x) = 1 − cos x, 0 ≤ x ≤ 2π 2 2

(d) Find the period, amplitude and phase shift of each of the following functions, and in
each case sketch 1 complete revolution for x ≥ 0.

i. iv. vi.
f(x) = 2 sin 2x
f(x) = −1 − cos(x + 15◦ ) f(x) = sin 2(x − π)
2
ii.
π 3
f(x) = cos(2x + )
2 v. vii.
iii.
π
f(x) = 3 sin(x − 30◦ ) f(x) = −4 cos(3x + 15◦ ) f(x) = 2 + 3 cos(2x − )
2