Without a calculator: 9 sin 30° + 7 tan 45° = □

11.5. sin 30° = 1/2, tan 45° = 1: 9 × 1/2 + 7 × 1 = 11.5.

Without a calculator: 8 sin² 60° + 1 cos² 45° = □

6.5. sin² 60° = (√3/2)^2 = 3/4, cos² 45° = (√2/2)^2 = 1/2: 8 × 3/4 + 1 × 1/2 = 6.5.

sin 45° = cos 60°. True or false?

False. sin θ = cos(90° − θ): sin 45° equals cos(90° − 45°). Check with exact values: sin 30° = cos 60° = 1/2, sin 45° = cos 45° = √2/2.

Trigonometry · Ages 14–15

Exact values at 30°, 45° and 60°

sin 30° = 12, sin 45° = 22, sin 60° = 32.

How to do it

Method

sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2

2 sin 30° + tan 45° = 1 + 1 = 2

Another way

√1/2, √2/2, √3/2

cos 60° = sin 30° = 12

In full

At 30°, 45° and 60° the ratios have exact values, no calculator needed. Half an equilateral triangle of side 2 has sides 1, 3 and 2, so sin 30° = 12, cos 30° = 32, tan 30° = 13; sin 60° = 32, cos 60° = 12, tan 60° = 3. A right-angled isosceles triangle has legs 1 and hypotenuse 2, so sin 45° = cos 45° = 12 = 22 and tan 45° = 1. Put the exact values into an expression: 2 sin 30° + tan 45° = 2 × 12 + 1 = 2; sin² 60° = (32)² = 34. Another way: remember sin 30°, sin 45° and sin 60° as 12, 22 and 32, cos the same the other way round, and tan = sin ÷ cos.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
Without a calculator: 9 sin 30° + 7 tan 45° =
Show the answer and working
Answer: 11.5
sin 30° = 12, tan 45° = 1: 9 × 12 + 7 × 1 = 11.5.
Example 2
Without a calculator: 8 sin² 60° + 1 cos² 45° =
Show the answer and working
Answer: 6.5
sin² 60° = (32)² = 34, cos² 45° = (22)² = 12: 8 × 34 + 1 × 12 = 6.5.
Example 3
sin 45° = cos 60°.
True or false?
Show the answer and working
Answer: False
sin θ = cos(90° − θ): sin 45° equals cos(90° − 45°). Check with exact values: sin 30° = cos 60° = 12, sin 45° = cos 45° = 22.

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.Without a calculator: 8 tan 45° − 5 cos 60° =
Answer:
2.Without a calculator: 2 sin 30° + 5 tan 45° =
Answer:
3.sin 45° = cos 30°.
True or false?
Answer:
4.In a right-angled triangle the side beside a 30° angle is 9 cm and the side opposite it is x cm. In exact values: x =
Answer:
5.A right-angled isosceles triangle has a hypotenuse of 162 cm. How long is each shorter side, in cm?
Answer:

All the answers

1. 5.5
2. 6
3. False
4. 27
5. 16 cm

Free worksheet: Exact values at 30°, 45° and 60°Working out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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