MULUNGUSHI UNIVERSITY
SCHOOL OF NATURAL AND APPLIED SCIENCES
DEPARTMENT OF MATHEMATICS AND STATISTICS
MSM 111 - Mathematical Methods I
Tutorial Sheet 8 - 2022/2023 - Trigonometric Functions
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1. Find the quadrant that contains the the terminal side of θ if the given condition is true:
(a) sin θ > 0 and cos θ < 0
(b) tan θ < 0 and cos θ < 0
(c) tan θ > 0 and cot θ < 0.
(d) sec θ > 0 and tan θ > 0
(e) csc θ > 0 and cot θ > 0
(f) cos θ > 0 and csc θ < 0.
2. Without using a calculator, change each of the following angles in degree to radian measure:
(a) 420o
(b) 570o
(c) −45o
(d) −150o
3. Without using a calculator, change each of the following angles to degrees measure:
(a)
4π
3
π
12
(b)
(c) − 11π
4
(d) − 5π
6
4. Find the exact values (do not use a calculate or table) for the following
(a) sin 2250
(b) cos 150o
(f) sin 2π
3
(c) tan 3300
(d) sec 4200
(e) csc (−135o )
(h) cos − 5π
(i) cot 13π
(j) sec − 7π
3
6
6
(g) tan 4π
3
5. Use the basic trigonometric identities to find the remaining trigonometric functions:
(a) sin θ =
4
5
and the terminal side of θ lies in the first quadrant.
5
(b) cos θ = − 13
and the terminal side of θ lies in the second quadrant.
(c) tan θ =
12
5
and the terminal side of θ lies in the third quadrant.
(d) csc θ = − 45 and sec θ < 0
6. Simplify the given trigonometric expressions to a single trigonometric function or a constant:
(b) (cos2 x − 1)(tan2 x + 1)
(a) sin x − sin x tan x
(d)
sec y−cos y
tan y
(e)
tan θ sin θ
sec2 θ−1
(f)
cot x
cot2 x+1
(g)
(c) cos z + sin z tan z
1+sec θ
sin θ+tan θ
7. Prove each of the following identities
(a) (tan x − sec x)2 =
(d)
sin x
1+cos x
=
1+sin x
1−sin x
1−cos x
sin x
8. (a) Given that sin α =
and tan (α + β)
(b)
tan x(cot2 x+1)
tan2 x+1
= cotx
(c) (sin x − cos x)2 = 1 − 2sinx cos x
(e) (1 − cos z)(1 + sec z) = sin z tan z.
3
5
with α in the first quadrant, and sin β =
(f)
15
17
1
sec y(1−sin y)
= sec y + tan y
with β in the second quadrant, find tan (α − β)
(a) Given that tan α = − 23 with α in the second quadrant, and tan β =
tan (α − β) and tan (α + β)
3
5
with β in the third quadrant, find
9. Find the exact values without using a calculator or a table:
7
24
(a) cos tan−1 24
− sin−1 45
(b) sin tan−1 34 − cos−1 25
10. Prove each of the following identities
(a) sin(x + 90o ) = cos x
(d) tan(α − π) = tan α
(b) cos(y + 90o ) = − sin x
1+tan α
(e) tan α + π4 = 1−tan
α
(c) cos(z − π) = cos z
α−1
(f) tan α − π4 = tan
tan α+1
11. Solve each of the following equations for 0o ≤ x ≤ 360o . Do not use a calculator or a table:
√
(a) 2 sin x + 3 = 0
(b) tan2 x = 3
(c) 2 cos2 x = cos x
(d) 2 cos3 x = cos x
1
(e) 2 cos2 x − sin x − 1 = 0
(g) cos 2x + 3 sin x − 2 = 0
(f) tan x = cot x
12. Solve each of the following equations for −π ≤ θ ≤ π. Do not use a calculator or a table:
(a) 2 tan θ sec θ − tan θ = 0
(b) 2 sin2 θ + 3 sin θ + 1 = 0
(e) sin θ cos θ − cos θ = 1 − sin θ
(f) sin θ = 1 − cos θ
(c) sec2 θ − sec θ − 2 = 0
(g) tan θ + 1 = sec θ
13. Solve each of the following equations for 0 ≤ θ ≤ 2π. Do not use a calculator or a table:
(a) cos θ − sin 2θ = 0
2
(e) 2 − sin θ =
(b) cos 2θ − 3 sin θ − 2 = 0
2 cos2 θ2
(f) sin
θ
2
(c) tan 2θ + sec 2θ = 12−
+ cos θ = 1
14. Find the period, amplitude and phase shift of the given function and draw it’s graph:
(a) f (x) = 2 sin 2(x − π)
(b) f (x) = 12 cos 3(x + π)
(c) f (x) = cos 2x + π2
(e) f (x) = 2 − 3 sin x + π2
(f) f (x) = −2 + 2 sin 2x − π2
(d) f (x) = 3 sin 21 x − π2
15. Graph each of the following functions in the indicated interval:
(a) f (x) = 1 − cos x, 0 ≤ x ≤ 2π
(b) f (x) = −2 + sin x − π2 , π2 ≤ x ≤ 5π
2
(d) f (x) = 1 + sin 2x + 2π
,
0
≤
x ≤ 2π.
(c) f (x) = 2 cos 21 x − 32 , −π ≤ x ≤ 5π
3
16. Prove each of the following identities
(a)
sin 2θ sin θ
2 cos θ
(d) sec 2x =
+ cos2 θ = 1
2
sec x
2−sec2 x
(b)
1−tan2 α
1+tan2 α
(e) cot 2α =
= cos 2α
2
cot α−1
2 cot α
(f)
(c) cot θ sin 2θ = 1 + cos 2θ
cos 2θ
cos θ+sin θ
= cosθ − sin θ
17. Express:
(a) cos 3θ in terms of cos θ
(b) sin 3θ in terms of sin θ
(b) cos 4θ in terms of cos θ
18. Find the exact values without using using a calculator or a table:
π
(b) cos 12
(c) sin 7π
12
θ
θ
19. Find the exact values of sin 2 , cos 2 and tan
(a) sin 15o
(a) sin θ =
(a) tan θ =
3
5
3
5
o
and 0 ≤ θ ≤ 270
o
and 90o ≤ θ ≤ 180o
(d) tan 11π
12
θ
2
. Do not use a calculator or a table.
(b) cos θ = − 35 and 180o ≤ θ ≤ 90o
(b) sec θ =
3
2
2
and 270o ≤ θ ≤ 360o