Ratios of 0°, 30°, 45°, 60° and 90°
Learn one table: sin rises 0, 1/2, 1/√2, √3/2, 1 from 0° to 90°, and cos runs the same values backwards. tan 90° is not defined.
The table to learn
| ∠A | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin A | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos A | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan A | 0 | 1/√3 | 1 | √3 | Not defined |
| cosec A | Not defined | 2 | √2 | 2/√3 | 1 |
| sec A | 1 | 2/√3 | √2 | 2 | Not defined |
| cot A | Not defined | √3 | 1 | 1/√3 | 0 |
How to draw the triangles behind the values
- 45°: draw a right triangle with two equal sides a. The hypotenuse is a√2. So sin 45° = a/(a√2) = 1/√2 and tan 45° = a/a = 1.
- 30° and 60°: draw an equilateral triangle of side 2a. Drop a perpendicular from the top vertex to the base. It cuts the base into two halves of length a.
- Each half is a right triangle with angles 30°, 60°, 90° and sides a, a√3 and 2a (by Pythagoras, √(4a² − a²) = a√3).
- So sin 30° = a/2a = 1/2, cos 30° = a√3/2a = √3/2, sin 60° = √3/2 and cos 60° = 1/2.
- 0° and 90°: as angle A shrinks to 0°, the opposite side shrinks to 0, so sin 0° = 0 and cos 0° = 1. As A grows to 90°, sin 90° = 1 and cos 90° = 0.
A quick way to write the table in the exam
Write 0, 1, 2, 3, 4 under 0°, 30°, 45°, 60°, 90°. Divide each by 4 and take the square root. You get 0, 1/2, 1/√2, √3/2, 1: this is the sin row. Write the same row backwards for cos. Then tan = sin ÷ cos. Flip each value to get cosec, sec and cot. Where you would divide by 0, write not defined.
Evaluating an expression
Evaluate 4 sin²30° + 3 tan²60° − 2 cos²45°.
- From the table: sin 30° = 1/2, tan 60° = √3, cos 45° = 1/√2.
- Square them: sin²30° = 1/4, tan²60° = 3, cos²45° = 1/2.
- 4 sin²30° = 4 × 1/4 = 1.
- 3 tan²60° = 3 × 3 = 9.
- 2 cos²45° = 2 × 1/2 = 1.
- Value = 1 + 9 − 1 = 9.
9
Finding two angles
Given tan (A − B) = √3 and sin (A + B) = 1. Here 0° < A + B ≤ 90° and A > B. Find A and B.
- tan 60° = √3, so A − B = 60° ... (1)
- sin 90° = 1, so A + B = 90° ... (2)
- Add (1) and (2): 2A = 150°, so A = 75°.
- Put A = 75° in (2): B = 90° − 75° = 15°.
A = 75°, B = 15°
The set squares in your geometry box have angles 30°, 60°, 90° and 45°, 45°, 90°. These are exactly the triangles behind this table, so you can check the values with a ruler.
1 marks: “Find the value of sin 30° + cos 60°.” 1/2 + 1/2 = 1.
2 marks: “Evaluate (tan 60° − tan 30°)/(1 + tan 60° tan 30°).” Top = √3 − 1/√3 = 2/√3. Bottom = 1 + √3 × 1/√3 = 2. Value = (2/√3)/2 = 1/√3.
3 marks: “Evaluate (3 sin²60° + 2 cot²45° − cosec²30°)/(sec²45° + cos²0°).” Numerator = 3 × 3/4 + 2 × 1 − 4 = 9/4 − 2 = 1/4. Denominator = 2 + 1 = 3. Value = (1/4)/3 = 1/12.
sin row: √0/2, √1/2, √2/2, √3/2, √4/2. The cos row is the same, read backwards.