UNIVERSITY OF STRATHCLYDE
DEPARTMENT OF MATHEMATICS
SUMMER SCHOOL
Exercises on Circular Functions
1. Sketch the graphs of the following functions for −π ≤ x ≤ π :
π
π
(i) y = sin x +
;
(ii) y = cos x −
;
(iii) y = sin(x + π) + 1.
6
3
2. Sketch in the same diagram the graphs of y = cos 2x and y = 2 cos x for 0 ≤ x ≤ 2π.
3. Show that
√
1 √
(i) sin 15◦ = ( 6 − 2),
4
√
1 √
(ii) cos 15◦ = ( 6 + 2),
4
(iii) tan 15◦ = 2 −
√
3.
4. Show that
√
1 √
(i) sin 105◦ = ( 6 + 2),
4
√
(iii) tan 105◦ = −2 − 3 .
√
1 √
(ii) cos 105◦ = − ( 6 − 2),
4
7
5. Given that α and β are acute angles with sin α = 25
and cos β =
using tables or a calculator, sin(α + β) and tan(α + β).
5
13
find, without
6. Prove that
(i) cos(A + B) cos(A − B) = cos2 A − sin2 B,
(ii) sin(A + B) sin(A − B) = sin2 A − sin2 B.
7. Prove that
sin 2θ
(i)
= tan θ ,
1 + cos 2θ
(iii) cos2 θ + cos 2θ = 2 − 3 sin2 θ ,
(v)
(vii)
8.
cos4 θ − sin4 θ = cos 2θ ,
sin θ + sin 2θ
= tan θ ,
1 + cos θ + cos 2θ
(ii) csc 2θ + cot 2θ = cot θ,
(iv) tan 2θ − 2 tan 2θ sin2 θ = sin 2θ ,
1 − cos 2θ
= tan2 θ ,
1 + cos 2θ
1 + tan2 θ
(viii)
= sec 2θ .
1 − tan2 θ
(vi)
(i) Express sin 75◦ + sin 15◦ in product form and hence evaluate it exactly.
(ii) Express sin 75◦ − sin 15◦ in product form and hence evaluate it exactly.
(iii) Use the results in (i) and (ii) to evaluate sin 15◦ and sin 75◦ exactly.
9. Prove the following identities:
sin 5x + sin 3x
tan 4x
(i)
=
;
sin 5x − sin 3x
tan x
sin 2x − sin x
x
(iii)
= tan ;
cos 2x + cos x
2
◦
◦
cos x + cos 9x
= cot 5x ;
sin x + sin 9x
cos 5x + cos 2x
7x
(iv)
= cot
.
sin 5x + sin 2x
2
(ii)
◦
◦
10. Evaluate 2 sin 52 21 cos 7 21 and 2 sin 52 21 sin 7 12 exactly by first expressing in sum
or difference form.
11. For each of the following equations, find all solutions in the interval [0, 2π] :
(i)
sin x cos x = cos x ;
(ii) 2 sin2 x − sin x − 1 = 0 ;
(iii)
tan2 x + 3 sec x + 3 = 0 ;
(iv) cos2 x + sin x + 1 = 0 ;
(v)
2 sin2 x − 2 cos2 x = 3 ;
(vi) cos 2x − cos x = 0 ;
(vii)
tan2 2x + sec2 2x = 3 ;
(viii) sin 2x = 1 − cos 2x .
12. Without using tables or a calculator, evaluate tan θ given that
12
3
−1
−1
+ cos
.
θ = sin
5
13
13. Without using tables or a calculator, evaluate sin θ given that
1
1
−1
−1
θ = sin
+ cos
.
3
3
Hence determine a value of θ in the range [0, 2π].
14. Without using tables or a calculator show that, if
12
3
−1
−1
− cos
,
φ = 2 sin
5
13
253
.
325
15. By considering the tan of both sides, find a value of x that satisfies the equation
then sin φ =
tan−1 (3x) + tan−1 (2x) =
π
.
4
16. The following are xy-coordinates. Find the corresponding polar coordinates.
(i) (0, 2);
(ii) (−1, 0);
(iii) (−2, 2);
(iv) (2, −1);
(v) (−0.5, −0.5).
17. The following are polar coordinates. Plot the points in the xy-plane and find the
corresponding xy-coordinates.
π
5π
5π
3π
; (ii) 4,
; (iii) 1,
; (iv) 2,
.
(i) 3,
2
3
6
4