Tutorial 1
Instructions: No Calculators; Show all workings
π
1. Calculate the area of a sector in a circle of radius 3m, with a central angle of 3.
2. Evaluate
π
(a) cos( 12 ) (b) sin( 5π
3 ), (c) csc(225◦ ) (d) csc( 7π
4 ) (e) cot(225).
(f ) cot( 13π
2 ) (g) tan(− 11π
6 ) (h) csc(− 7π
4 ) (i) cot( 3π
4 ) (j) sin( 19π
12 )
3. If θ is a Quadrant II angle with sin(θ) = 53 , find cos(θ)
√
3π 5
4. If π < θ < 2 with cos(θ) = − 5 , find sin(θ).
2
5. If cos θ = 3 and 0 < θ < π2 , find
(a) sin 2θ (b) cos 2θ
6. If tan α = 34 and tan β = 2, where 0 < α < π
2 and 0<β< π
2, find
(a) sin (α − β) (b) cos(α + β)
7. Find all of the angles which satisfy the given equation
(a) cos(θ) = 21 (b) sin(θ) = − 12 (c) cos(θ) = 0
8. Find all values of t for which sin 2t − 1 = 0.
9. (a) Express 3 sin α + 5 cos α in the form C sin(α + φ)
(b) Show that a sum of the form A sin α + B cos α can be written in the form C sin(α + φ)
10. Verify the following identities
1 3 3
(a) = sin(θ) (b) 6 sec(θ) tan(θ) = −
csc(θ) 1 − sin(θ) 1 + sin(θ)
(c) tan(θ) = sin(θ) sec(θ) (d) sec(θ) − tan(θ) sec(θ) + tan(θ) = 1
2 tan(θ)
(e) sin(2θ) =
1 + tan2 (θ)
11. Find all values of x for which sin2 x = 2 sin x − 1.
12. Find all values of x for which
√
(1 − 3) tan x
√ =1
3 − tan2 x
13. Determine the values of θ ∈ [0, 2π] for which cos(2θ) = 3 cos(θ) − 2.
14. Determine the values of θ ∈ [0, 2π] for which cos(2θ) = 2 sin θ cos θ
15. Determine the values of θ ∈ [0, 2π] for which sin θ = cos( θ2 )
16. Determine the values of θ ∈ [0, 2π] for which 3 cos θ + 3 = 2 sin2 θ
17. Determine the values of θ ∈ [0, 2π] for which sin2 θ = 2 cos θ + 2
√
3
18. Determine the values of θ ∈ [0, 2π] for which cos θ cos(2θ) + sin θ sin(2θ) = 2
19. Find the general solution for the equation cos(π + x) = sin(π − x) .
20. Find the general solution for the equation cos(2θ) = cos θ.
21. By factoring the polynomial x5 + x3 + 8x2 + 8 completely, find all real zeroes of the polynomial.
22. Find all real solutions of the equations
(a) x4 − 6x3 + 22x + 15 = 0 (b) x4 − 7x3 + 3x2 + 31x + 20 = 0
(c) 3x4 + 14x3 + 14x2 − 8x − 8 = 0 (d) x5 − 2x4 − 6x3 + 5x2 + 8x + 12 = 0
(e) 2x4 − x3 − 14x2 − 5x + 6 = 0 (f ) x5 + 4x4 − 4x3 − 34x2 − 45x − 18 = 0
23. Let p(x) = x3 + 4x2 + x − 6. Find q(x) and r such that
(a) p(x) = (x + 1)q(x) + r (b) p(x) = (x − 1)q(x) + r
24. Use synthetic division to find the quotient q(x) and the remainder r(x) that result when p(x) is
divided by s(x)
(a) p(x) = 3x2 − 4x − 1; s(x) = x − 2
4 3 2
(b) p(x) = 2x + 3x − 17x − 27x − 9; s(x) = x − 1
7
(c) p(x) = x + 1; s(x) = x − 1
25. Given that two of the roots of x4 + 3x3 − 7x2 − 27x − 18 = 0 have the same modulus but different
sign, solve the equation. (Hint - let two of the roots be a and −a and use the technique of equating
coefficients).
26. Find all integer zeros of
p(x) = x6 + 5x5 − 16x4 − 15x3 − 12x2 − 38x − 21
27. Show that the cubic equation of the form x3 + bx2 + cx + d = 0 can be reduced to the form
y 3 + py + q = 0 [Hint: let x = y − 3b ].
28. Show that the general quartic equation x4 + ax3 + bx2 + cx + d = 0 can be reduced to the form
y 4 + py 2 + qy + r = 0 [Hint:let x = y − a4 ]
29. Solve the following quartic equations by reduction.
(a) x4 − x2 − 3x + 2 = 0 (b) x4 + x3 − 7x2 − x + 6 = 0
(c) x4 − 4x3 + 3x2 − 5x + 2 = 0
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