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 memorized 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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