Right Triangle Trigonometry: Functions, Identities, and Applications

Right Triangle Trigonometry

Introduction to Trigonometric Functions

Trigonometric functions are mathematical relationships between the angles and sides of right triangles. They are fundamental in precalculus and are used to solve problems involving triangles and modeling periodic phenomena.

• Trigonometric functions relate an angle of a right triangle to ratios of two of its sides.

• The three main trigonometric functions are sine, cosine, and tangent.

• The mnemonic SOH-CAH-TOA helps remember the definitions:

Function	Definition	Formula
Sine (sin)	Opposite side / Hypotenuse	$\sin \theta = \frac{\text{Opp}}{\text{Hyp}}$
Cosine (cos)	Adjacent side / Hypotenuse	$\cos \theta = \frac{\text{Adj}}{\text{Hyp}}$
Tangent (tan)	Opposite side / Adjacent side	$\tan \theta = \frac{\text{Opp}}{\text{Adj}}$

• Example: For a right triangle with sides 5 (opposite), 12 (adjacent), and 13 (hypotenuse):

  • $\sin \theta = \frac{5}{13}$

  • $\cos \theta = \frac{12}{13}$

  • $\tan \theta = \frac{5}{12}$

Reciprocal Trigonometric Functions

In addition to sine, cosine, and tangent, there are three reciprocal trigonometric functions: cosecant, secant, and cotangent.

Function	Definition	Formula
Cosecant (csc)	Reciprocal of sine	$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hyp}}{\text{Opp}}$
Secant (sec)	Reciprocal of cosine	$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hyp}}{\text{Adj}}$
Cotangent (cot)	Reciprocal of tangent	$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adj}}{\text{Opp}}$

• Example: For the same triangle as above:

  • $\csc \theta = \frac{13}{5}$

  • $\sec \theta = \frac{13}{12}$

  • $\cot \theta = \frac{12}{5}$

Evaluating Trigonometric Functions in Right Triangles

To evaluate trigonometric functions for a given angle in a right triangle, identify the sides relative to the angle and apply the appropriate ratio.

• Label the sides as opposite, adjacent, and hypotenuse with respect to the angle.

• Substitute the side lengths into the trigonometric ratio.

• Example: For a triangle with sides 8 (opposite), 15 (adjacent), and 17 (hypotenuse):

  • $\cos \theta = \frac{15}{17}$

  • $\sin \theta = \frac{8}{17}$

  • $\tan \theta = \frac{8}{15}$

Inverse Trigonometric Functions

Inverse trigonometric functions are used to find the angle when given a trigonometric ratio. They are denoted as $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$.

• To find an angle $\theta$ given $\sin \theta = x$, use $\theta = \sin^{-1}(x)$.

• Similarly, $\theta = \cos^{-1}(x)$ or $\theta = \tan^{-1}(x)$ as appropriate.

• Example: If $\sin \theta = \frac{5}{13}$, then $\theta = \sin^{-1}\left(\frac{5}{13}\right)$.

Using Calculators for Trigonometric Functions

For non-special angles, calculators are used to evaluate trigonometric functions and their inverses.

• Ensure the calculator is in the correct mode (degrees or radians) as required by the problem.

• For inverse functions, use the $2^{nd}$ or $INV$ button before the trigonometric function key.

• Example: To find $\sin^{-1}(0.5)$ in degrees, enter $\sin^{-1}(0.5)$ to get $30^\circ$.

Summary Table: Trigonometric Functions and Their Reciprocals

Function	Ratio	Reciprocal	Reciprocal Ratio
$\sin \theta$	$\frac{\text{Opp}}{\text{Hyp}}$	$\csc \theta$	$\frac{\text{Hyp}}{\text{Opp}}$
$\cos \theta$	$\frac{\text{Adj}}{\text{Hyp}}$	$\sec \theta$	$\frac{\text{Hyp}}{\text{Adj}}$
$\tan \theta$	$\frac{\text{Opp}}{\text{Adj}}$	$\cot \theta$	$\frac{\text{Adj}}{\text{Opp}}$

How to Use Inverse Trig Functions to Find Angles

1. Choose a trig function that includes the known sides.

2. Write an equation with the chosen trig function.

3. Isolate the sine, cosine, or tangent ratio.

4. Press the $2^{nd}$ (or INV) key, then the trig function key to use the inverse function.

5. Approximate the angle using a calculator.

• Example: For a triangle with sides 5 (opposite) and 13 (hypotenuse): $\theta = \sin^{-1}\left(\frac{5}{13}\right) \approx 22.62^\circ$

Practice Problems (Selected)

• Given a right triangle with sides 8, 15, and 17, evaluate $\cos \theta$.

• Given $\tan \theta = \frac{12}{16}$, find $\theta$.

• If $\sin \theta = \frac{5}{13}$, find the values of the other five trigonometric functions.

• Use a calculator to find $\sin^{-1}(0.4251)$ and round to four decimal places.

Additional info: These notes cover the foundational aspects of right triangle trigonometry, including definitions, reciprocal identities, inverse functions, and calculator usage, as typically required in a Precalculus course.