Foundation and Higher · exact trig values

Exact trig values.
No calculator.
Just two triangles.

draw them, then
read the answers off

first, what the question wants —

Exact

no decimals · no rounding

leave the √ in your answer

triangle 1 of 2

Half a square: 45°

1. Halve a square of side 1: 45°, 45°, 90°.

2. Pythagoras: 1² + 1² = 2, diagonal √2.

3. tan 45° = 1 ÷ 1 = 1.

✓ sin 45° = cos 45° = 1√2 = √22

triangle 2 of 2

Half an equilateral

1. Start with an equilateral triangle, side 2.

2. Halve it: 30°, 60°, 90°, and the base is 1.

3. Pythagoras: 2² − 1² = 3, height √3.

✓ Sides 1, √3 and 2

now use SOH CAH TOA

Read off 30° and 60°

1. From 30°: opposite 1, adjacent √3,
hypotenuse 2.

2. From 60°: opposite and adjacent swap.

✓ sin 30° = cos 60° = 12

add 0° and 90°

The full table

1. sin: √0, √1, √2, √3, √4, all over 2.

2. cos: the same list backwards.

3. tan = sin ÷ cos.

✓ tan 90° does not exist:
you cannot divide by 0

what to remember —

Two triangles

1, 1, √2 and 1, √3, 2

sin goes up

√0, √1, √2, √3, √4 over 2

cos goes down

the same list backwards

Exact means leave the √ in.
No decimals.

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