Applications of Trigonometric Functions

Solving Right Triangles

Solving right triangles is a fundamental application of trigonometry, allowing us to find unknown side lengths or angles using trigonometric ratios. These techniques are widely used in fields such as engineering, architecture, and navigation.

• Key Trigonometric Ratios: The three main trigonometric ratios for a right triangle are sine, cosine, and tangent.

• Sine:

• Cosine:

• Tangent:

• Solving for Unknowns: Use the known sides and angles to set up equations and solve for the unknown values.

Example: Finding the Height of a Tower

Given a right triangle with an angle of elevation of 85.4° and a horizontal distance of 80 ft, the height of the tower can be found using the tangent function:

• ft

Example: Finding the Angle of a Guy Wire

Given a right triangle with a vertical side of 6.7 yards and a hypotenuse of 13.8 yards, the angle the wire makes with the ground can be found using the sine function:

Example: Finding the Height of a Sculpture

Given two angles of elevation (32° and 35°) from a point 800 ft away, the height of the sculpture is the difference between the heights calculated for each angle:

• ft

• ft

• Height of sculpture: ft

Solving Problems Involving Bearings

Bearing problems involve navigation and direction, often using right triangles formed by changes in direction. Bearings are measured clockwise from north or south.

• Key Concept: Bearings are used to describe direction in navigation, typically in the format N/S θ° E/W.

• Example: If you hike 2.3 miles on a bearing of S 31° W, then turn 90° and hike 3.5 miles on a bearing of N 59° W, you can use the tangent function to find the bearing from the starting point.

• Bearing: S 87.7° W

Modeling Simple Harmonic Motion

Simple harmonic motion describes periodic movement, such as a ball on a spring or a pendulum. The displacement is modeled using sine or cosine functions.

• General Equation: or

• Amplitude (a): Maximum displacement from rest position.

• Angular Frequency (ω):

• Frequency (f):

• Period: Time required for one complete cycle,

Example: Ball on a String

• Maximum displacement: 6 inches

• Period: 4 seconds

• Equation:

Example: Object in Simple Harmonic Motion

• Equation:

• Maximum displacement: 12 cm

• Frequency: oscillation per second

• Period: seconds

Summary Table: Trigonometric Applications