## Trigonometric Equations Trigonometric equations are equations containing terms such as sin x and cos x.

They can be solved using the trigonometric graphs and, if necessary, a calculator.

Because trigonometric functions are periodic and continue forever, these trigonometric equations often have an infinite number of solutions unless the domain (x-values) is fixed. Usually only values between 0 and 2π or 360° are required.

In the examples below the solutions are given in degrees. If radians are required the mode of the calculator must be changed.

### sin x = ccos x = c

To solve an equation such as sin x = 0.5, consider the functions y = sin x and the line y = 0.5. Where the line and curve meet will be the solutions. A calculator can be used to find the first value. The two solutions of 30° and 150° can be read off the graph, if the graph is clear and big enough

OR

A caclulator can be used for the first solution      30° and the second solution found from the symmetry of the graph (180° − 30°).

### sin ax = ccos ax = c

To solve an equation such as cos 2x = 0, consider the functions y = cos 2x and y = 0. The line y = 0 is also the x-axis and where the x-axis and the curve meet will be the solutions. The four solutions of 45°, 135°, 225° and 315° can be read off the graph, if the graph is clear and big enough.

OR

A caclulator can be used for the first solution    90° but remember this is 2x = 90 therefore x = 45° and the other three solutions worked out from the symmetry of the graph.

### sin (x − b) = ccos (x − b) = c

To solve an equation such as cos(x + 30°) = -0.7, consider the function y = cos(x + 30°) and the line y = -0.7.
Where the line and curve meet will be the solutions. The two solutions of 104.4° and 195.6° can be read off the graph, if the graph is clear and big enough.

OR

a caclulator can be used for the first solution       134.4° but remember this is
x + 30 = 134.4 therefore x = 104.4° and the other solution is 330 -134.4 = 195.6°