Introduction to Trigonometry · Lesson 3 of 5
Trigonometric Ratios of Some Specific Angles
“A handful of special angles unlock exact trigonometric values without a calculator.”
• By the end of this lesson, you should be able to. • Derive the trigonometric ratios of 45°. • Derive the ratios of 30° and 60° using an equilateral triangle. • Understand the values at 0° and 90°. • Use the standard-angle table confidently. • Solve problems involving standard trigonometric values.
You will use the angles 0°, 30°, 45°, 60° and 90° so often that their trigonometric values become basic tools. Instead of memorising them blindly, we will first derive them from simple geometry.
Trigonometric Ratios of 45°
Take a right triangle with one acute angle equal to 45°. The other acute angle must also be 45°, so the two legs opposite equal angles are equal. Let each leg be a. Then the hypotenuse is found using Pythagoras theorem.
| Ratio | 45° value |
|---|---|
| sin 45° | 1/√2 = √2/2 |
| cos 45° | 1/√2 = √2/2 |
| tan 45° | 1 |
| cosec 45° | √2 |
| sec 45° | √2 |
| cot 45° | 1 |
Trigonometric Ratios of 30° and 60°
Start with an equilateral triangle of side 2a. Every angle is 60°. Draw a perpendicular from one vertex to the opposite side. It bisects the base and the top angle, creating two congruent 30°–60°–90° right triangles.
| Ratio | 30° | 60° |
|---|---|---|
| sin | 1/2 | √3/2 |
| cos | √3/2 | 1/2 |
| tan | 1/√3 | √3 |
| cosec | 2 | 2/√3 |
| sec | 2/√3 | 2 |
| cot | √3 | 1/√3 |
What Happens at 0° and 90°?
As an acute angle gets closer to 0°, the opposite side becomes extremely small compared with the hypotenuse, while the adjacent side becomes almost the same as the hypotenuse. This leads to sin 0° = 0 and cos 0° = 1.
As the angle gets closer to 90°, the adjacent side becomes extremely small, while the opposite side becomes almost the same as the hypotenuse. This leads to sin 90° = 1 and cos 90° = 0.
tan 90° = sin 90° / cos 90° would require division by 0, so tan 90° is not defined. sec 90° is also not defined. Similarly, cot 0° and cosec 0° are not defined because sin 0° = 0.
The Standard-Angle Table
| Ratio | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | Not defined |
| cosec | Not defined | 2 | √2 | 2/√3 | 1 |
| sec | 1 | 2/√3 | √2 | 2 | Not defined |
| cot | Not defined | √3 | 1 | 1/√3 | 0 |
As the angle increases from 0° to 90°, sin θ increases from 0 to 1, while cos θ decreases from 1 to 0. Also, sin 30° = cos 60°, sin 60° = cos 30°, and sin 45° = cos 45°.
Problem
Evaluate sin 60° cos 30° + sin 30° cos 60°.
- 1.Substitute the standard values.
- 2.sin 60° cos 30° + sin 30° cos 60°
- 3.= (√3/2)(√3/2) + (1/2)(1/2)
- 4.= 3/4 + 1/4
- 5.= 1.
Problem
In right triangle ABC, right-angled at B, angle C = 30° and AB = 6 cm. Find AC.
- 1.Relative to angle C, AB is opposite and AC is the hypotenuse.
- 2.sin 30° = AB/AC.
- 3.1/2 = 6/AC.
- 4.Therefore AC = 12 cm.
Problem
In right triangle PQR, right-angled at Q, PQ = 5 cm and PR = 10 cm. Find angles P and R.
- 1.PR is the hypotenuse.
- 2.For angle R, opposite side = PQ.
- 3.sin R = PQ/PR = 5/10 = 1/2.
- 4.Therefore R = 30°.
- 5.Since P + R = 90°, P = 60°.
Quiz
What is the value of sin 45°?
What is the value of tan 60°?
Which trigonometric value is not defined?
How does cos θ change as θ increases from 0° to 90°?
If θ is a standard acute angle and sin θ = 1/2, what is θ?
Practice Problems
- Evaluate: (i) sin 60° cos 30° + sin 30° cos 60° (ii) 2 tan²45° + cos²30° - sin²60°.
- Evaluate: cos 45° / (sec 30° + cosec 30°).
- If tan(A + B) = √3 and tan(A - B) = 1/√3, where 0° < A + B ≤ 90° and A > B, find A and B.
- State true or false with reason: (i) sin(A + B) = sin A + sin B, (ii) sin θ increases from 0° to 90°, (iii) cos θ increases from 0° to 90°, (iv) cot 0° is defined.
Key Takeaways
• The 45° values come from a right isosceles triangle with side ratio 1 : 1 : √2. • The 30° and 60° values come from splitting an equilateral triangle, giving side ratio 1 : √3 : 2. • sin 0° = 0, cos 0° = 1, sin 90° = 1 and cos 90° = 0. • tan 90°, sec 90°, cot 0° and cosec 0° are not defined. • The standard-angle table should be understood first and then memorised through patterns.
Next, we will connect the trigonometric ratios through identities such as sin²A + cos²A = 1, and learn how to prove more complicated identities step by step.