Right Triangle Trigonometry and Trigonometric Functions of Acute Angles

Right Triangle Trigonometry

Definition of Trigonometric Functions of Acute Angles

Trigonometric functions relate the angles of a right triangle to the ratios of its sides. For an acute angle $\theta$ in a right triangle, the six trigonometric functions are defined as follows:

• Sine ($\sin \theta$): Ratio of the length of the side opposite $\theta$ to the hypotenuse.

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

• Tangent ($\tan \theta$): Ratio of the length of the side opposite $\theta$ to the adjacent side.

• Cosecant ($\csc \theta$): Reciprocal of sine.

• Secant ($\sec \theta$): Reciprocal of cosine.

• Cotangent ($\cot \theta$): Reciprocal of tangent.

Function Name	Abbreviation	Value
Sine of $\theta$	sin $\theta$	$\dfrac{\text{opposite}}{\text{hypotenuse}}$
Cosine of $\theta$	cos $\theta$	$\dfrac{\text{adjacent}}{\text{hypotenuse}}$
Tangent of $\theta$	tan $\theta$	$\dfrac{\text{opposite}}{\text{adjacent}}$
Cosecant of $\theta$	csc $\theta$	$\dfrac{\text{hypotenuse}}{\text{opposite}}$
Secant of $\theta$	sec $\theta$	$\dfrac{\text{hypotenuse}}{\text{adjacent}}$
Cotangent of $\theta$	cot $\theta$	$\dfrac{\text{adjacent}}{\text{opposite}}$

Example: Finding the Value of Trigonometric Functions

Given a right triangle with sides of length 3, 4, and 5, find the six trigonometric functions for the angle opposite the side of length 3.

• $\sin \theta = \dfrac{3}{5}$

• $\cos \theta = \dfrac{4}{5}$

• $\tan \theta = \dfrac{3}{4}$

• $\csc \theta = \dfrac{5}{3}$

• $\sec \theta = \dfrac{5}{4}$

• $\cot \theta = \dfrac{4}{3}$

Trigonometric Identities

Reciprocal Identities

Each trigonometric function has a reciprocal identity:

Identity
$\csc \theta = \dfrac{1}{\sin \theta}$
$\sec \theta = \dfrac{1}{\cos \theta}$
$\cot \theta = \dfrac{1}{\tan \theta}$
$\sin \theta = \dfrac{1}{\csc \theta}$
$\cos \theta = \dfrac{1}{\sec \theta}$
$\tan \theta = \dfrac{1}{\cot \theta}$

Quotient Identities

Identity
$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$
$\cot \theta = \dfrac{\cos \theta}{\sin \theta}$

Pythagorean Identities

• $\sin^2 \theta + \cos^2 \theta = 1$

• $1 + \tan^2 \theta = \sec^2 \theta$

• $1 + \cot^2 \theta = \csc^2 \theta$

Finding the Values of Trigonometric Functions

Given One Function Value

To find the values of the remaining trigonometric functions when one value is given:

1. Draw a right triangle representing the given information.

2. Use the Pythagorean Theorem to find the missing side.

3. Use the definitions to find the other function values.

Example: If $\sin \theta = \dfrac{3}{5}$ and $\theta$ is acute, then $\cos \theta = \dfrac{4}{5}$, $\tan \theta = \dfrac{3}{4}$, etc.

Using Identities

Identities can be used to find the value of a trigonometric function given another function value.

Example: If $\sin \theta = \dfrac{3}{5}$, then $\cos \theta = \sqrt{1 - (3/5)^2} = \dfrac{4}{5}$.

Complementary Angle Theorem

Cofunctions of Complementary Angles

The trigonometric function of an angle is equal to the cofunction of its complement:

• $\sin(90^\circ - \theta) = \cos \theta$

• $\tan(90^\circ - \theta) = \cot \theta$

• $\sec(90^\circ - \theta) = \csc \theta$

• And vice versa for each pair.

Degrees	Radians	Identity
$\sin 30^\circ$	$\sin \dfrac{\pi}{6}$	$= \cos 60^\circ = \cos \dfrac{\pi}{3}$
$\tan 20^\circ$	$\tan \dfrac{\pi}{9}$	$= \cot 70^\circ = \cot \dfrac{7\pi}{18}$
$\sec 80^\circ$	$\sec \dfrac{4\pi}{9}$	$= \csc 10^\circ = \csc \dfrac{\pi}{18}$

Exact Values of Trigonometric Functions for Special Angles

45°, 30°, and 60° Angles

The trigonometric functions for $45^\circ$, $30^\circ$, and $60^\circ$ (or $\dfrac{\pi}{4}$, $\dfrac{\pi}{6}$, $\dfrac{\pi}{3}$ radians) have exact values:

Angle	$\sin$	$\cos$	$\tan$	$\csc$	$\sec$	$\cot$
$30^\circ$ ($\dfrac{\pi}{6}$)	$\dfrac{1}{2}$	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{\sqrt{3}}$	$2$	$\dfrac{2}{\sqrt{3}}$	$\sqrt{3}$
$45^\circ$ ($\dfrac{\pi}{4}$)	$\dfrac{\sqrt{2}}{2}$	$\dfrac{\sqrt{2}}{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
$60^\circ$ ($\dfrac{\pi}{3}$)	$\dfrac{\sqrt{3}}{2}$	$\dfrac{1}{2}$	$\sqrt{3}$	$\dfrac{2}{\sqrt{3}}$	$2$	$\dfrac{1}{\sqrt{3}}$

Using a Calculator to Approximate Trigonometric Values

To approximate the value of a trigonometric function, use a scientific calculator and round to the required decimal places.

Example: $\sin 50^\circ \approx 0.77$

Applications: Solving Right Triangles

Solving for Sides and Angles

Given some sides and/or angles of a right triangle, use trigonometric functions and identities to solve for unknown values.

Example: If $AC = 140 \sin 15^\circ$, then $AC \approx 36.23$ ft.

Finding the Height of a Cloud (Applied Problem)

Meteorologists can use trigonometry to find the height of a cloud by measuring angles of elevation and distances on the ground. For example, if the base is 300 ft, and the angles are $75^\circ$ and $60^\circ$, the height can be calculated using trigonometric ratios.

Example: $h = 300 \tan 75^\circ + 600 \tan 60^\circ = 1120$ ft

Summary Table: Fundamental Trigonometric Identities

Identity
$\sin^2 \theta + \cos^2 \theta = 1$
$1 + \tan^2 \theta = \sec^2 \theta$
$1 + \cot^2 \theta = \csc^2 \theta$
$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$
$\cot \theta = \dfrac{\cos \theta}{\sin \theta}$
$\csc \theta = \dfrac{1}{\sin \theta}$
$\sec \theta = \dfrac{1}{\cos \theta}$
$\cot \theta = \dfrac{1}{\tan \theta}$

Key Points for Study

• Understand the definitions and relationships of the six trigonometric functions.

• Be able to use identities to find unknown values.

• Know the exact values for $30^\circ$, $45^\circ$, and $60^\circ$.

• Apply trigonometric functions to solve right triangle problems, including real-world applications.