Trigonometry Exact Values (AQA A-Level Mathematics): Revision Notes

5.4.2 Trigonometry Exact Values

Exact values in trigonometry refer to the precise values of trigonometric functions for specific angles, typically those that are commonly used in mathematics, such as $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$. These values are often memorised or derived using geometrical methods, like the unit circle or special triangles.

1. Exact Values for Sine, Cosine, and Tangent:

Angle $(\theta)$	$\sin \theta$	$\cos \theta$	$\tan \theta$
$0^\circ \ or \ 0 \ radians$	$0$	$\ 1$	$\ 0$
$\ 30^\circ \ or \ \frac{\pi}{6} \ radians$	$\ \frac{1}{2}$	$\ \frac{\sqrt{3}}{2}$	$\ \frac{1}{\sqrt{3}} \ or \ \frac{\sqrt{3}}{3}$
$\ 45^\circ \ or \ \frac{\pi}{4} \ radians$	$\ \frac{\sqrt{2}}{2}$	$\ \frac{\sqrt{2}}{2}$	$\ 1$
$\ 60^\circ \ or \ \frac{\pi}{3} \ radians$	$\ \frac{\sqrt{3}}{2}$	$\ \frac{1}{2}$	$\ \sqrt{3}$
$\ 90^\circ \ or \ \frac{\pi}{2} \ radians$	$\ 1$	$\ 0$	Undefined

2. Exact Values for Cosecant, Secant, and Cotangent:

Angle $(\theta)$	$\ \csc \theta$	$\ \sec \theta$	$\ \cot \theta$
$\ 0^\circ \ or \ 0 \ radians$	Undefined	$\ 1$	Undefined
$\ 30^\circ \ or \ \frac{\pi}{6} \ radians$	$\ 2$	$\ \frac{2}{\sqrt{3}} \ or \ \frac{2\sqrt{3}}{3}$	$\ \sqrt{3}$
$\ 45^\circ \ or \ \frac{\pi}{4} \ radians$	$\ \sqrt{2}$	$\ \sqrt{2}$	$\ 1$
$\ 60^\circ \ or \ \frac{\pi}{3} \ radians$	$\ \frac{2}{\sqrt{3}} \ or \ \frac{2\sqrt{3}}{3}$	$\ 2$	$\ \frac{1}{\sqrt{3}} \ or \ \frac{\sqrt{3}}{3}$
$\ 90^\circ \ or \ \frac{\pi}{2} \ radians$	$\ 1$	Undefined	$\ 0$

3. Deriving Exact Values Using Special Triangles:

1. 30°-60°-90° Triangle:

Consider an equilateral triangle with each side of length 2. If you draw an altitude, it splits the triangle into two $30°-60°-90°$ triangles.

• Hypotenuse = $2$

• Shorter leg (opposite 30°) = $1$

• Longer leg (opposite 60°) =$\sqrt{3}$ From this, you get:

• $\sin 30^\circ = \frac{1}{2}$

• $\cos 30^\circ = \frac{\sqrt{3}}{2}$

• $\tan 30^\circ = \frac{1}{\sqrt{3}}$

• $\sin 60^\circ = \frac{\sqrt{3}}{2}$

• $\cos 60^\circ = \frac{1}{2}$

• $\tan 60^\circ = \sqrt{3}$

2. 45°-45°-90° Triangle:

Consider a right-angled isosceles triangle, where each leg is of length 1.

• Hypotenuse = $\sqrt{2}$ From this, you get:

• $\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

• $\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

• $\tan 45^\circ = 1$

3. Unit Circle Approach:

The unit circle is a circle with a radius of 1 centred at the origin of the coordinate plane. The coordinates of any point on the unit circle correspond to $\ (\cos \theta, \sin \theta) .$

• At $\ \theta = 0^\circ \ or \ 0$ radians, the coordinates are $\ (1, 0) ,$ so $\ \cos 0^\circ = 1$ and $\sin 0^\circ = 0 .$
• At $\ \theta = 90^\circ \ or \ \frac{\pi}{2}$ radians, the coordinates are $\ (0, 1)$, so $\ \cos 90^\circ = 0$ and $\sin 90^\circ = 1$ .
• At $\ \theta = 180^\circ \ or \ \pi$ radians, the coordinates are$\ (-1, 0)$, so $\ \cos 180^\circ = -1$ and $\ \sin 180^\circ = 0 .$
• At $\ \theta = 270^\circ \ or \ \frac{3\pi}{2}$ radians, the coordinates are $\ (0, -1)$, so $\ \cos 270^\circ = 0$ and $\ \sin 270^\circ = -1 .$

Summary:

• Exact values of trigonometric functions are crucial in solving problems involving trigonometric equations, identities, and geometric applications.
• Special triangles like the $30°-60°-90°$ and $45°-45°-90°$ triangles are useful tools for deriving these exact values.
• The unit circle provides a visual and conceptual way to understand and recall the exact values of trigonometric functions at key angles.

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