Applications of Trigonometric Functions

Application in Equations

An equation that includes a trigonometric function with an unknown angle, such as , can be solved using the graph of the trigonometric function as follows:

Steps:

1. Rewrite the given equation in the form .
2. In the given range, graph and the line , and find the -coordinates of the points of intersection between the two graphs to determine the solution.

Example:
Find the solution of the equation when .

The -coordinates of the points of intersection between the graph of and the line are the solutions.

From the graph, the solutions are:

Note:
Using the unit circle:
If points and are where the unit circle and the line intersect, the solutions to the equation are the angles formed by the radii and , where is the origin.

Application in Inequalities

An inequality that includes a trigonometric function with an unknown angle, such as , can also be solved using the graph of the trigonometric function as follows:

Steps:

1. Rewrite the given inequality in the form .
2. In the given range, find the values of where the graph of is above the line , by drawing both graphs and identifying the -coordinates of the points of intersection.

Example:
Find the solution of the inequality when .

For the equation , the solutions are and .

For the inequality, the solution is the range of -values where the graph of is above the line , which is:

Note:

1. Inequalities involving trigonometric functions can also be solved using the unit circle, similar to equations.
2. If an equation or inequality involves multiple trigonometric functions, it is convenient to use identities like to express everything in terms of one trigonometric function.