GCSE · Maths · AQA · Spec 8300 · Foundation

Exact values of sine, cosine and tangent

Your calculator says sin 60° = 0.8660254…, a decimal that never ends. The exact answer is √3/2, and it's hiding in a triangle you can draw in ten seconds.

Maths · Trigonometry

Slide the angle and watch the sides

The hypotenuse stays at 10, so sin and cos are just a side ÷ 10. Stop at 30°, 45° and 60°, then push the slider to 0° and 90° and watch the triangle flatten.

adjacent8.7opposite5.0hypotenuse10.0θ = 30°
Angle θ30°
0°90°

SIN θ

sin θ =oppositehypotenuse

= 5.0 / 10.0 = 0.50

For sin, the renderer highlights the side opposite θ and the hypotenuse. Adjacent fades.

Exam line: The panel rounds to two decimal places. 0.87 is really √3/2 = 0.8660…, and 0.71 is √2/2 = 0.7071…. When a question asks for an exact value, give the surd, not the decimal.
Watch out: Pick TAN and slide to 90°. The panel tries to divide by an adjacent side of 0.0 and prints a giant, meaningless number. You can't divide by zero, so tan 90° has no value. That's why it's missing from the list.

Build the 30° and 60° values

Problem

Find the exact values of sin, cos and tan for 30° and 60°, using nothing but an equilateral triangle.

Your turn

Rebuild the 45° values yourself

Use the same method on the second special triangle to find the exact values of sin 45°, cos 45° and tan 45°.

  1. Draw a right-angled triangle whose two shorter sides are both 1.Equal shorter sides mean the other two angles are equal: (180° − 90°) ÷ 2 = 45° each.
  2. missing step
Which line is step 2?

Predict, then check

Where do the values really sit?

Drag each value to where it belongs between 0 and 2, then check. Before you do: 30°, 60° and 90° are equally spaced angles. Are their sines equally spaced too?

Spot the slip

One line loses the mark

A ramp makes an angle of 30° with the flat ground. It covers 6 m horizontally. Without a calculator, find the exact height the ramp rises.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Two small triangles hold every value you need, so there's no table to cram.

What you need to know

  • sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent. Every exact value is one of these ratios in a special triangle.
  • Half an equilateral triangle of side 2 has sides 1, √3 and 2 and angles 30°, 60° and 90°. It gives all the 30° and 60° values.
  • The right-angled isosceles triangle with shorter sides 1 has hypotenuse √2. It gives the 45° values.
  • 0° and 90° are the limiting cases, where the triangle flattens into a line. tan 90° has no value, because the adjacent side is 0 and you can't divide by 0.

The big picture

Sin and cos have exact values at 0°, 30°, 45°, 60° and 90°, and tan has them at 0°, 30°, 45° and 60°. You don't need to learn them as a list: each one is a side divided by a side in one of two triangles. Half an equilateral triangle of side 2 (sides 1, √3, 2) gives 30° and 60°. The right-angled isosceles triangle (sides 1, 1, √2) gives 45°. At 0° and 90° the triangle flattens into a line, which gives the 0s and 1s, and no value at all for tan 90°.

Key points

1sin 0° = 0, sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
2cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = ½, cos 90° = 0. These are the sine values in reverse.
3tan 0° = 0, tan 30° = √3/3 (the same as 1/√3), tan 45° = 1, tan 60° = √3.
4Sine is not proportional to the angle: sin 60° is √3/2, not double sin 30°.
5The corner you stand at decides which side is opposite and which is adjacent. Getting that right is what stops sin and cos, or tan 30° and tan 60°, being swapped.

Worked example

Problem

A right-angled triangle has a hypotenuse of 8 cm. The side next to angle x (not the hypotenuse) is 4 cm. Without a calculator, find angle x.

⚠ Watch out

Swapping sin and cos at 30° and 60°, or swapping tan 30° and tan 60°. Stand at the right corner of the half-equilateral triangle. From 30°, the side opposite you is the short one, 1, so sin 30° = ½ and tan 30° is the small one, √3/3.

🧠

Memory hook

Two triangles, ten seconds: 1, 1, √2 for 45°, and 1, √3, 2 for 30° and 60°. To check your sines, count up under the root: sin 0°, 30°, 45°, 60°, 90° = √0/2, √1/2, √2/2, √3/2, √4/2. Cosine reads the same list backwards.

✓

Check yourself

From memory, sketch both special triangles and label all three sides of each. Then use them, not a list, to write down cos 30°, tan 60° and sin 45°.

Flashcards

(13)
Which triangle gives the exact values for 30° and 60°?
Half an equilateral triangle of side 2: hypotenuse 2, shortest side 1, height √3.
Which triangle gives the exact values for 45°?
The right-angled isosceles triangle: shorter sides 1 and 1, hypotenuse √2.
Why is the height of the half-equilateral triangle √3?
Pythagoras: h² + 1² = 2², so h² = 3 and h = √3.
Exact values of sin 0°, sin 30°, sin 45°, sin 60°, sin 90°?
0, ½, √2/2, √3/2, 1.
Exact values of cos 0°, cos 30°, cos 45°, cos 60°, cos 90°?
1, √3/2, √2/2, ½, 0: the sine values in reverse order.
Exact values of tan 0°, tan 30°, tan 45°, tan 60°?
0, √3/3 (= 1/√3), 1, √3.
Why does tan 90° have no value?
At 90° the adjacent side shrinks to 0, and tan divides by the adjacent. You can't divide by zero.
Why is tan 45° exactly 1?
In the 45° triangle, opposite and adjacent are both 1, so opposite ÷ adjacent = 1 ÷ 1 = 1.
Why is 1/√3 the same as √3/3?
Multiply the top and bottom by √3: (1 × √3)/(√3 × √3) = √3/3. Multiplying by √3/√3 is multiplying by 1.
Is sin 60° double sin 30°?
No. sin 30° = ½, but sin 60° = √3/2 ≈ 0.87, not 1. Sine is not proportional to the angle.
What happens to the triangle at 0° and 90°?
It flattens into a line. These are the limiting cases: at 0° the opposite side is 0 (sin 0° = 0, cos 0° = 1), and at 90° the adjacent side is 0 (sin 90° = 1, cos 90° = 0).
tan 30° or tan 60°: which one is bigger than 1?
tan 60° = √3 ≈ 1.73. tan 30° = √3/3 ≈ 0.58 is less than 1, because 30° is less than 45°.
Why are sin 30° and cos 60° both ½?
They're the same side of the same triangle: the short side 1 over the hypotenuse 2. From the 30° corner that side is opposite; from the 60° corner it's adjacent.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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