Special Angles

1Summary 重點摘要

Sin, cos, tan of 30°, 45°, 60° are fixed values. Memorise them.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Why memorise special anglesThe sin, cos and tan of 30°, 45° and 60° are fixed values, used directly in exams without reaching for the calculator — fast, accurate, and exact rather than approximate. Example: for sin 30°, write 12 directly, which is more rigorous than the calculator's 0.5; some proofs can only continue with exact values. A common error is swapping the values between angles, e.g. writing sin 30° as √32; memorise the table and recite it regularly to stay safe.
2. Step 2: Where 45° comes fromThe 45° values come from an isosceles right triangle: the two legs are equal, say 1 each, and by Pythagoras the hypotenuse = √(1² + 1²) = √2. Then sin 45° = opposite ÷ hypotenuse = 1/√2 = √22, cos 45° is also √22, and tan 45° = 1. Knowing the origin removes the need to memorise blindly: the two legs being equal forces sin and cos to be identical. A common error is in rationalising the denominator — 1/√2 multiplied above and below by √2 gives √22, not 2/√2.
3. Step 3: Where 30° and 60° come fromThe 30° and 60° values come from cutting an equilateral triangle in half. Let the equilateral triangle have side 2; cutting along the height makes two right triangles with hypotenuse 2, short leg 1, and by Pythagoras the other leg = √(2² − 1²) = √3. Then sin 30° = 12, cos 30° = √32, tan 30° = 1/√3; and sin 60° = √32, cos 60° = 12, tan 60° = √3. Note how the sin and cos of 30° and 60° swap places. A common error is mixing the two angles' values; a sketch fixes the memory.
4. Step 4: Memorise the tableThe full table: the 30° row is sin = 12, cos = √32, tan = 1/√3; the 45° row is sin = √22, cos = √22, tan = 1; the 60° row is sin = √32, cos = 12, tan = √3. To recite: write the three sin values first, get the cos row as the sin row reversed, then verify each tan cell as sin ÷ cos. Writing the table twice before the exam is enough. A common error is giving tan 30° as √3 — but √3 belongs to tan 60°.
5. Step 5: How to memorise easilyThe memory trick: the three sin values from 30° to 60° are √12, √22, √32 — the numbers under the root climb 1, 2, 3; cos runs the same list in reverse, √32, √22, √12; and tan = sin ÷ cos can be computed on the spot whenever needed. The trick squeezes nine values into three lines and self-checks: for example, tan 45° = (√22) ÷ (√22) = 1, agreeing with the table. A common error is remembering cos in the same order as sin; remember 'cos reversed' instead.
6. Step 6: How to useOn seeing 30°, 45° or 60°, substitute the exact value directly. Example: in a right triangle the side opposite the 30° angle is 5; find the hypotenuse. From sin 30° = opposite ÷ hypotenuse, 12 = 5 ÷ hypotenuse, so the hypotenuse = 5 ÷ (12) = 10. The steps: pick the right ratio (opposite and hypotenuse take sin), write the defining equation, then substitute the exact value and solve. A common error is choosing the wrong ratio, e.g. tan; a small sketch labelling opposite, adjacent and hypotenuse prevents it.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Three special angles: 30°, 45°, 60°
• Memorise sin, cos, tan values
• Sin from small to big: 12, √22, √32
• Cos reversed: √32, √22, 12
• Tan = sin ÷ cos