Solution of Simple Trigonometric Equations for a Given Interval

Understanding Simple Trigonometric Equations

A trigonometric equation uses trigonometric functions such as sine, cosine, or tangent.
Simple trigonometric equations look like sin(x) = a or cos(x) = a, where ‘a’ is a known number.
A key part of solving these equations is to know the values of sine, cosine, and tangent at key points (e.g., 0°, 30°, 45°, 60°, and 90°).
You should also be familiar with the graphs of y = sin(x), y = cos(x), and y = tan(x) as this can help you visualize the solutions.

Solving Simple Trigonometric Equations

Start by isolating the trigonometric function, i.e., put it on one side of the equation by itself.
Use the inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to solve for lower case ‘x’. This gives you the angle.
Be aware that these functions will give you one solution, but there could be more than one angle with the same sine, cosine, or tangent value, depending on the interval you’re looking at.

Understanding Intervals for Solutions

An interval typically refers to the range of values for which you are trying to find possible solutions.
For example, the interval [0, 360°] means you are looking for all possible angles between 0 and 360° that satisfy your equation.
For sine and cosine functions, which have a period of 360°, this interval will yield at most two solutions.
For the tangent function, which has a period of 180°, this interval will yield at most one solution.

Checking Your Solutions

When you find potential solutions, you should check them by substituting them back into the original equation to ensure they satisfy it.
Be aware that if an equation is multiplied by a constant or if there is a phase shift, the period may change.
Consult your formula sheet and remember the co-function identities and Pythagorean identities, as these can also be used to check and find solutions.

Using Scientific Calculator for Solutions

A scientific calculator is useful for solving trigonometric equations, particularly when the exact angle cannot be easily determined.
Remember to make sure your calculator is set to the correct mode (degrees or radians) before you start. If you’re finding an angle and you’re given an interval in degrees, make sure your calculator is in the degree mode.