Fundamentals of Trigonometric Functions and Special Triangles

Trigonometric Functions and Right Triangles

Basic Trigonometric Ratios

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. These functions are fundamental in precalculus and are used to solve problems involving triangles and periodic phenomena.

• Sine (sin): Ratio of the length of the side opposite the angle to the hypotenuse.

• Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse.

• Tangent (tan): Ratio of the length of the side opposite the angle to the adjacent side.

The basic trigonometric ratios for an angle $\theta$ in a right triangle are:

• $\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$

• $\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$

• $\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}$

Reciprocal functions:

• $\csc \theta = \dfrac{1}{\sin \theta}$

• $\sec \theta = \dfrac{1}{\cos \theta}$

• $\cot \theta = \dfrac{1}{\tan \theta}$

Special Right Triangles

Special right triangles, such as the 45°-45°-90° and 30°-60°-90° triangles, have side ratios that are useful for evaluating trigonometric functions without a calculator.

• 45°-45°-90° Triangle: The sides are in the ratio $1:1:\sqrt{2}$.

• 30°-60°-90° Triangle: The sides are in the ratio $1:\sqrt{3}:2$.

These ratios allow for quick calculation of trigonometric values for common angles.

Evaluating Trigonometric Functions for Special Angles

Trigonometric functions for angles such as $0, \dfrac{\pi}{6}, \dfrac{\pi}{4}, \dfrac{\pi}{3}, \dfrac{\pi}{2}$ (or $0°, 30°, 45°, 60°, 90°$) are frequently used in precalculus. Their values can be summarized as follows:

$\theta$	$0$	$\dfrac{\pi}{6}$	$\dfrac{\pi}{4}$	$\dfrac{\pi}{3}$	$\dfrac{\pi}{2}$
$\sin \theta$	$0$	$\dfrac{1}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{3}}{2}$	$1$
$\cos \theta$	$1$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{1}{2}$	$0$
$\tan \theta$	$0$	$\dfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

Additional info: The table above is inferred from the standard values for trigonometric functions at special angles, as the original images contain incomplete or symbolic representations.

Example: Using a 30°-60°-90° Triangle

Given a triangle with sides $1$,$\sqrt{3}$, and $2$, the trigonometric functions for $30°$ and $60°$ can be found as follows:

• $\sin 30° = \dfrac{1}{2}$

• $\cos 30° = \dfrac{\sqrt{3}}{2}$

• $\tan 30° = \dfrac{1}{\sqrt{3}}$

• $\sin 60° = \dfrac{\sqrt{3}}{2}$

• $\cos 60° = \dfrac{1}{2}$

• $\tan 60° = \sqrt{3}$

Example: Using a 45°-45°-90° Triangle

Given a triangle with sides $1$, $1$, and $\sqrt{2}$, the trigonometric functions for $45°$ are:

• $\sin 45° = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$

• $\cos 45° = \dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$

• $\tan 45° = 1$

Summary Table: Trigonometric Values for Special Angles

Angle	$\sin$	$\cos$	$\tan$
$0$	$0$	$1$	$0$
$30°$	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}}$
$45°$	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$
$60°$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$
$90°$	$1$	$0$	undefined

Additional info: The above table is reconstructed for clarity and completeness based on standard trigonometric values.

Applications

• Solving right triangles given one side and one angle (other than the right angle).

• Evaluating trigonometric expressions without a calculator using special triangles.

• Understanding the unit circle and the relationship between angles and coordinates.