Special Angles (Grade 10 NSC Matric Mathematics): Revision Notes

Special Angles

What are special angles?

Special angles are specific angles in trigonometry that have exact trigonometric ratios which can be calculated without using a calculator. The three most important special angles are 30°, 45°, and 60°.

These angles are considered "special" because:

• Their trigonometric values are exact (not decimal approximations)
• They come from well-known right-angled triangles
• They appear frequently in mathematics exams
• They help solve trigonometric problems quickly and accurately

Special right triangles

Special angles come from two fundamental right-angled triangles. Understanding these triangles is essential for remembering the exact trigonometric values.

The 30°-60°-90° triangle

This triangle has angles of 30°, 60°, and 90°. The sides are in the ratio 1 : √3 : 2.

• Shortest side (opposite 30°): 1 unit
• Medium side (opposite 60°): √3 units
• Hypotenuse (opposite 90°): 2 units

The 45°-45°-90° triangle

This triangle has angles of 45°, 45°, and 90°. It's an isosceles right triangle where the two legs are equal.

• Both legs: 1 unit each
• Hypotenuse: √2 units

Exact trigonometric values

Using the special right triangles above, we can determine the exact values for sine, cosine, and tangent of the special angles.

For 30°:

• $\cos 30° = \frac{\sqrt{3}}{2}$
• $\sin 30° = \frac{1}{2}$
• $\tan 30° = \frac{1}{\sqrt{3}}$

For 45°:

• $\cos 45° = \frac{1}{\sqrt{2}}$
• $\sin 45° = \frac{1}{\sqrt{2}}$
• $\tan 45° = 1$

For 60°:

• $\cos 60° = \frac{1}{2}$
• $\sin 60° = \frac{\sqrt{3}}{2}$
• $\tan 60° = \sqrt{3}$

Memory techniques

Pattern for sine values

Notice the pattern for $\sin θ$ as the angle increases from 30° to 60°:

• $\sin 30° = \frac{1}{2}$
• $\sin 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\sin 60° = \frac{\sqrt{3}}{2}$

Pattern for cosine values

The cosine values are the reverse of the sine values:

• $\cos 30° = \frac{\sqrt{3}}{2}$ (same as $\sin 60°$)
• $\cos 45° = \frac{1}{\sqrt{2}}$ (same as $\sin 45°$)
• $\cos 60° = \frac{1}{2}$ (same as $\sin 30°$)

Tangent values

Remember: $\tan θ = \frac{\sin θ}{\cos θ}$

• $\tan 30° = \frac{1}{\sqrt{3}}$ (rationalised form: $\frac{\sqrt{3}}{3}$)
• $\tan 45° = 1$ (because $\sin 45° = \cos 45°$)
• $\tan 60° = \sqrt{3}$

Worked examples

Exam tips

Here are some practical strategies for working with special angles in examinations:

• Learn the exact values: Memorise the table of values for 30°, 45°, and 60°
• Draw the triangles: If you forget a value, quickly sketch the appropriate special triangle
• Check your work: Use the fact that $\sin^2 θ + \cos^2 θ = 1$ to verify your answers
• Rationalize denominators: Express $\frac{1}{\sqrt{3}}$ as $\frac{\sqrt{3}}{3}$ when required
• Look for special angles: In exam questions, watch for angles that are multiples or combinations of 30°, 45°, and 60°

Common exam traps

Remember!