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

Angles in all four quadrants and exact values

70 marks

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Quick questions to start.

A1
Last lesson
Write down the period of $y = cos 3 x$ and the maximum value of $y = 4 - 2 cos 3 x$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$
A2
Needed today
An equilateral triangle has sides of $2$ cm. Find the exact height of the triangle.
[2 marks]
Answer
A3
From chapter 1
Write $\frac{3}{6}$ with a rational denominator, in its simplest form.
[2 marks]
Answer
A4
From chapter 5
$O$ is the origin and $P$ is the point $(- 3, 4)$ . Find the gradient of $O P$ and the length of $O P$ .
[2 marks]
Answer $\dots\dots$ and $\dots\dots$

Example, then your turn

A-Level

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

Example 1

Use the unit circle to write down the values of $sin 63 0^{\circ}$ , $cos (- 54 0^{\circ})$ and $tan 90 0^{\circ}$ .

Hint Bring each angle into

0^{\circ}

36 0^{\circ}

, then read the circle.

B1
Without a calculator, write down the values of $cos 99 0^{\circ}$ , $sin (- 63 0^{\circ})$ and $tan 112 5^{\circ}$ .
[3 marks]
Answer $\dots\dots$ , $\dots\dots$ , $\dots\dots$

Example 2

Write $sin 29 0^{\circ}$ , $cos (- 12 5^{\circ})$ and $tan 47 0^{\circ}$ as ratios of acute angles.

Hint Acute angle to the

x

-axis, then the sign.

B2
Express $cos 21 5^{\circ}$ , $tan (- 32 0^{\circ})$ and $sin 68 5^{\circ}$ in terms of trigonometric ratios of acute angles.
[3 marks]
Answer $\dots\dots$ , $\dots\dots$ , $\dots\dots$

Example 3

$θ$ is an acute angle. Express $sin (90 0^{\circ} - θ)$ , $cos (- 18 0^{\circ} - θ)$ and $tan (θ - 54 0^{\circ})$ in terms of $sin θ$ , $cos θ$ or $tan θ$ .

Hint Take away whole turns of

36 0^{\circ}

first. Then draw the angle and decide its quadrant.

B3
$θ$ is an acute angle. Express $cos (72 0^{\circ} - θ)$ , $sin (- θ - 18 0^{\circ})$ and $tan (- θ - 90 0^{\circ})$ in terms of $sin θ$ , $cos θ$ or $tan θ$ .
[3 marks]
Answer $\dots\dots$ , $\dots\dots$ , $\dots\dots$

Example 4

Find the exact values of $cos 31 5^{\circ}$ , $sin (- 24 0^{\circ})$ and $tan 51 0^{\circ}$ .

Hint The two triangles give the exact values for

3 0^{\circ}

4 5^{\circ}

and

6 0^{\circ}

. Find the acute angle, then the sign.

B4
Find the exact values of $cos (- 15 0^{\circ})$ , $sin 67 5^{\circ}$ , $tan 24 0^{\circ}$ and $cos 84 0^{\circ}$ .
[4 marks]

Practice

A-Level

C1
For each angle, state the quadrant that $O P$ lies in, and whether $sin θ$ , $cos θ$ and $tan θ$ are positive or negative. (a) $θ = 50 0^{\circ}$ (b) $θ = - 29 0^{\circ}$
[4 marks]

C2
Show that $cos 21 0^{\circ} sin 24 0^{\circ} + sin 33 0^{\circ} cos (- 30 0^{\circ}) = \frac{1}{2}$ .
[4 marks]
C3
$sin θ < 0$ and $tan θ > 0$ , where $0^{\circ} < θ < 36 0^{\circ}$ . State the quadrant that $θ$ is in. Given also that $O P$ makes an acute angle of $2 5^{\circ}$ with the $x$ -axis, find $θ$ .
[4 marks]
Answer $\dots\dots$ quadrant $θ = \dots\dots$

Exam-style questions

Edexcel exam style

D1
The diagram shows the unit circle. $O P$ makes an acute angle $θ$ with the positive $x$ -axis, and $O Q$ makes an angle of $18 0^{\circ} + θ$ .
(a)
Write down the coordinates of $P$ . Explain why $Q$ has coordinates $(- cos θ, - sin θ)$ .
[3 marks]
(b)
Hence show that $sin (18 0^{\circ} + θ) = - sin θ$ and $tan (18 0^{\circ} + θ) = tan θ$ .
[3 marks]
(c)
Hence find the exact value of $sin 22 5^{\circ} + tan 24 0^{\circ}$ . Give your answer as a single fraction.
[4 marks]
Answer
Total for D1: 10 marks

D2
Sam writes: " $33 0^{\circ}$ and $3 0^{\circ}$ both make an angle of $3 0^{\circ}$ with the $x$ -axis, so $tan 33 0^{\circ} = tan 3 0^{\circ} = \frac{3}{3}$ ."
Sam is wrong.
(a)
Explain Sam's mistake, and give the exact value of $tan 33 0^{\circ}$ .
[2 marks]
(b)
Find the exact value of $\frac{tan 33 0 ^{\circ} cos ( - 42 0 ^{\circ} )}{sin 48 0 ^{\circ}}$ .
[4 marks]
Answer
Total for D2: 6 marks
D3
The points $P$ and $Q$ lie on a circle with centre $O (0, 0)$ and radius $6$ . $O P$ makes an angle of $12 0^{\circ}$ with the positive $x$ -axis and $O Q$ makes an angle of $33 0^{\circ}$ .
Not drawn accurately
(a)
Find the exact coordinates of $P$ and of $Q$ .
[4 marks]
(b)
Hence find the exact area of triangle $O P Q$ .
[3 marks]
Answer
Total for D3: 7 marks
D4
A regular octagon has its eight corners on a circle with centre $O$ and radius $r$ cm.
(a)
Show that the area of the octagon is $22 r^{2}$ cm².
[3 marks]
(b)
The area of the octagon is $982$ cm². Hence find $r$ .
[2 marks]
Answer $r = \dots\dots$
(c)
Find the exact perimeter of the octagon. Give it in the form $a 2 - 2$ , where $a$ is an integer.
[3 marks]
Answer
Total for D4: 8 marks

Extension

Stretch

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

E1
Without a calculator, find the exact value of $tan 22. 5^{\circ}$ in the form $a + b 2$ . Hence find the exact value of $tan 202. 5^{\circ} \times tan 157. 5^{\circ}$ .
[6 marks]
Hint 1 · What to try
Start with a right-angled triangle $A B C$ with $A B = B C = 1$ and the right angle at $B$ , so angle $A C B = 4 5^{\circ}$ . Halve that $4 5^{\circ}$ with an isosceles triangle.
Hint 2 · The first line
Extend $B C$ past $C$ to $D$ so that $C D = C A = 2$ . Triangle $A C D$ is isosceles and angle $A C D = 13 5^{\circ}$ , so angle $A D C = 22. 5^{\circ}$ .
Hint 3 · The full method
$tan 22. 5^{\circ} = \frac{A B}{B D} = \frac{1}{1 + 2} = 2 - 1$ . $tan 202. 5^{\circ} = tan 22. 5^{\circ}$ and $tan 157. 5^{\circ} = - tan 22. 5^{\circ}$ , so the product is $- (2 - 1)^{2} = 22 - 3$
Answer $\dots\dots$ and $\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the unit circle to find sin, cos and tan of any angle.
I can say whether sin, cos and tan are positive or negative in each quadrant.
I can write down the exact values for $3 0^{\circ}$ , $4 5^{\circ}$ and $6 0^{\circ}$ .
I can find exact values for related angles of any size.

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