CIE IGCSE Mathematics (0580) Exact trigonometric values Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Exact trigonometric values Study Notes
LEARNING OBJECTIVE
- Exact Trigonometric Values for sin/cos/tan
Key Concepts:
- Exact Trigonometric Values
Exact Trigonometric Values
Exact Trigonometric Values
You are expected to remember the exact values of \( \sin x \) and \( \cos x \) for the special angles:
\( x = 0^\circ, \; 30^\circ, \; 45^\circ, \; 60^\circ, \; 90^\circ \)
Exact Trigonometric Values Table:
| Angle | \( \sin x \) | \( \cos x \) |
|---|---|---|
| \( 0^\circ \) | 0 | 1 |
| \( 30^\circ \) | \( \frac{1}{2} \) | \( \frac{\sqrt{3}}{2} \) |
| \( 45^\circ \) | \( \frac{\sqrt{2}}{2} \) | \( \frac{\sqrt{2}}{2} \) |
| \( 60^\circ \) | \( \frac{\sqrt{3}}{2} \) | \( \frac{1}{2} \) |
| \( 90^\circ \) | 1 | 0 |
These values should be memorised – they often appear in non-calculator exam questions.
Example:
Without using a calculator, find the exact value of:
- \( \sin 30^\circ \)
- \( \cos 60^\circ \)
▶️ Answer/Explanation
\( \sin 30^\circ = \frac{1}{2} \), from the table
\( \cos 60^\circ = \frac{1}{2} \), from the table
Example:
Find the exact value of \( \cos^2 45^\circ + \sin^2 45^\circ \).
▶️ Answer/Explanation
\( \cos 45^\circ = \frac{\sqrt{2}}{2} \), \( \sin 45^\circ = \frac{\sqrt{2}}{2} \)
\( \cos^2 45^\circ + \sin^2 45^\circ = \left( \frac{\sqrt{2}}{2} \right)^2 + \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{1}{2} + \frac{1}{2} = \boxed{1} \)
Note: This confirms the identity \( \cos^2 x + \sin^2 x = 1 \)
Exact Values of \( \tan x \) for Special Angles
You should know the exact values of \( \tan x \) for:
\( x = 0^\circ, \; 30^\circ, \; 45^\circ, \; 60^\circ \)
Exact Tangent Values Table:
| Angle | \( \tan x \) |
|---|---|
| \( 0^\circ \) | 0 |
| \( 30^\circ \) | \( \frac{1}{\sqrt{3}} \) |
| \( 45^\circ \) | 1 |
| \( 60^\circ \) | \( \sqrt{3} \) |
These values can be remembered using:
\( \tan x = \frac{\sin x}{\cos x} \)
Example:
Find the exact value of \( \tan 45^\circ \).
▶️ Answer/Explanation
From the table: \( \tan 45^\circ = \boxed{1} \)
Example:
Without using a calculator, evaluate \( \frac{\tan 30^\circ}{\tan 60^\circ} \)
▶️ Answer/Explanation
\( \tan 30^\circ = \frac{1}{\sqrt{3}}, \quad \tan 60^\circ = \sqrt{3} \)
\( \frac{\tan 30^\circ}{\tan 60^\circ} = \frac{\frac{1}{\sqrt{3}}}{\sqrt{3}} = \frac{1}{\sqrt{3} \cdot \sqrt{3}} = \boxed{\frac{1}{3}} \)