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 = c

cos 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 = c

cos 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) = c

cos (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°**