Triangles, trigonometry and quadratics
Assorted exercises
Section A: sine and cosine rules
Since the relevant questions on the initial papers were well done, I am giving here a few examples of cases which are not quite straightforward. Any standard introductory ‘A’ level or higher level GCSE textbook will give further examples if required.
1.
(a)Give an argument based on constructions that this is sufficient information to define the triangle uniquely.
(b)Why is it not possible to use the sine rule at this stage?
(c)Use the cosine rule to find the missing side.
(d)Use the cosine rule to find θ. Then use the cosine rule again to find the other angle and check they add up to 180°.
(e)Without using your answer to (d), use the sine rule to find two possible values for θ. Can you explain why there are two possible values given that this triangle is uniquely defined?
2.
(a)Using constructions, show why there is only one possible triangle with these measurements.
(b)Use the cosine rule to find the missing length, explaining why there is only one possible value. How does the second value correspond to the construction?
(c)Without using your answer to (b), use the sine rule to find two possible values of θ. How can we be sure which value it is?
Section B: using trigonometrical identities
3.Using a calculator only to check your answers, evaluate:
(a)cos 80° cos 20° + sin 80° sin 20°
(b)sin 37° cos 7° - cos 37° sin 7°
(c)cos 15°(d)sin 165°
(e)cos 75°(f)tan 75°
4.Define cot θ = 1 / tan θ. Prove that:
5.Prove that:
(a)sin (45° + A) + sin (45° - A) = √2 cos A
(b)(sin A + cos A)(sin B + cos B) = sin(A + B) + cos (A – B)
(c)
(d)sin(θ + 60°) = sin(120° - θ)
(e)
6.Solve the following equations, giving angles from 0° to 360°:
(a)cos (45° - θ) = sin (30° + θ)
(b)3 sin θ = cos (θ + 60°)
(c)tan (A – θ) = 2/3 and tan A = 3
(d)sin (θ + 60°) = cos θ
Section C: quadratics
7.Find the roots of each of these equations by factorisation. Making use of any other information you need, draw a quick sketch diagram:
(a)y = x2 + 5x – 6(b)y = 3x2 – 7x
(c)y = 4x2 – 8x + 4
8.Using factorisation where appropriate, find the value(s) of x which satisfy:
(a)x(1-x) = x(2x-1)(b)
(c)
9.(a)Find the value(s) of t which satisfy:
t (t – 3) = t2 – 4
(b)If m = 1/t, rewrite the above equation in terms of m, and solve.
(c)How do you reconcile the solutions to (a) and (b)?
10.Solve the following quadratic equations by completing the square:
(a)x2 + 4x – 8 = 0(b)2x2 – 6x + 4 = 0
(c)x2 – 2x + a = 0
11.For what values of k is 9x2 + kx + 16 a perfect square?
12.The roots of 3x2 + kx + 12 = 0 are equal. Find k.
13.Prove that kx2 + 2x – (k-2) = 0 has real roots for any value of k.
14.Find a relationship between p and q if the roots of px2 + qx + 1 = 0 are equal.