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The ratios of 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°, and evaluating expressions without a calculator
Mathematics Sec 1 Term 1, Unit 4, Lesson 3
GoalKnow the special values and evaluate expressions without a calculator.
New words
special angles · 0°, 30°, 45°, 60° and the quadrantal angles, whose ratios are exactnot defined · tan 90° has no value because x = 0 and division by zero is impossible
Explanation
Learn this table. The values for 30°, 45° and 60° come from a 30-60-90 half equilateral triangle (sides 1, √3, 2) and a 45-45-90 triangle (sides 1, 1, √2). For the axis angles read the point on the unit circle: 0° is (1, 0), 90° is (0, 1), 180° is (−1, 0), 270° is (0, −1). So sin 90° = 1, cos 90° = 0, and tan 90° = 1/0 is not defined; the same for 270°. To evaluate an expression replace each ratio by its value, then simplify with fractions and roots. Check with sin²θ + cos²θ = 1 when you can.
| θ | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 | 0 | −1 | 0 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 | −1 | 0 | 1 |
| tan θ | 0 | √3/3 | 1 | √3 | — | 0 | — | 0 |
Worked examples
Evaluate sin 30° cos 60° + sin 60° cos 30° = ½ × ½ + (√3/2)(√3/2) = ¼ + ¾ = 1.
Evaluate 2 sin 30° + tan 45° − cos 60° = 2 × ½ + 1 − ½ = 1 + 1 − ½ = 3/2.
Evaluate sin²45° + cos²60° = (√2/2)² + (½)² = ½ + ¼ = ¾. And sin 90° + cos 180° + tan 45° = 1 − 1 + 1 = 1.
Class activity
Triangle strips. Cut a 30-60-90 triangle from paper with short side 1 and hypotenuse 2. Measure the third side and compare with 1.73. Use it to check sin 60° = 0.87 and cos 60° = 0.5.
Practice
1
Evaluate tan 60° × tan 30°.
√3 × √3/3 = 3/3 = 1.
2
Evaluate 4 cos²60° − 3 tan 45°.
4 × ¼ − 3 × 1 = 1 − 3 = −2.
3
Evaluate cos 180° + sin 270°.
−1 + (−1) = −2.
4
Evaluate cos 30° × tan 60° − sin 30°.
√3/2 × √3 − ½ = 3/2 − ½ = 1.
5
Why is tan 270° not defined?
The point is (0, −1), x = 0, and tan = y/x needs division by zero.