A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 10: Trigonometric identities and equations · Lesson 2

Trigonometric identities

70 marks

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Quick questions to start.

A1
Last lesson
Find the exact value of $cos 15 0^{\circ}$ .
[2 marks]
Answer
A2
Needed today
Factorise fully $16 x^{4} - 81$ .
[2 marks]
Answer
A3
From chapter 9
A triangle has sides $5$ cm, $7$ cm and $8$ cm. Find the angle opposite the $7$ cm side.
[2 marks]
Answer
A4
From chapter 7
Write $\frac{1}{x} - x$ as a single fraction.
[2 marks]
Answer

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Given that $cos θ = - \frac{8}{17}$ and that $θ$ is reflex, find the exact values of $sin θ$ and $tan θ$ .

Hint Find

sin^{2} θ

from

sin^{2} θ + cos^{2} θ = 1

. Then decide the sign: a reflex angle with a negative cosine is in the third quadrant.

B1
Given that $sin α = \frac{3}{7}$ and that $α$ is obtuse, find the exact values of $cos α$ and $tan α$ . Give each answer with a rational denominator.
[4 marks]
Answer $cos α = \dots\dots$ $tan α = \dots\dots$

Example 2

Prove that $\frac{cos θ}{1 + sin θ} + tan θ \equiv \frac{1}{cos θ}$ .

Hint Start from the left-hand side. Write

tan θ

\frac{sin θ}{cos θ}

, then put the two terms over the common denominator

cos θ (1 + sin θ)

B2
Prove that $\frac{sin θ}{1 + cos θ} + \frac{1 + cos θ}{sin θ} \equiv \frac{2}{sin θ}$ .
[4 marks]

Example 3

In triangle $A B C$ , $A B = 6$ cm, $B C = 5$ cm and $A C = 4$ cm. Find the exact values of $cos B$ and $sin B$ .

Not drawn accurately

Hint The cosine rule gives

cos B

exactly. Angle

B

is in a triangle, so

sin B

is positive.

B3
Triangle $P QR$ has $P Q = 9$ cm, $QR = 7$ cm, $P R = 5$ cm. Find $cos Q$ and $sin Q$ exactly, then the exact area.
[6 marks]
Not drawn accurately
Answer $sin Q = \dots$ area $= \dots$

A-Level

C1
Simplify (a) $8 - 8 cos^{2} 4 θ$ (b) $\frac{sin ^{2} θ}{cos θ - cos ^{3} θ}$
[3 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

C2
$21 tan θ + 20 = 0$ and $9 0^{\circ} < θ < 18 0^{\circ}$ . Find $sin θ$ and $cos θ$ exactly.
[3 marks]
Answer $sin θ = \dots\dots$ $cos θ = \dots\dots$
C3
$2 sin x cos y - cos x sin y = 4 cos x cos y$ . Find $tan y$ in terms of $tan x$ .
[3 marks]
Answer $tan y = \dots\dots$
C4
Prove that $tan^{2} θ - sin^{2} θ \equiv tan^{2} θ sin^{2} θ$ .
[3 marks]

Edexcel exam style

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use sin squared plus cos squared equals 1 and tan equals sin over cos to simplify.
I can prove a trigonometric identity, starting from one side.
I can find an exact ratio from another one and the quadrant.
I can eliminate an angle from two equations.

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