1. Graphical Solutions of Trigonometric Equations

2. An equation involving trigonometric ratios of unknown angles is called a trigonometric equation.
For example,
sin x = k, cos x = k and tan x = k, where k is a constant, are all trigonometric equations.
How can we solve these trigonometric equations graphically?

3. Similar to solving simultaneous equations graphically, we can solve trigonometric equations as follows:
The trigonometric equation can be written as a pair of simultaneous equations.
y = sin x
y = k
sin x = k
The coordinates of the intersections satisfy both equations. y x
Draw the graphs of y = sin x and y = k.
y = sin x
y = k a b
The x-coordinates of the intersections are the solutions.
solutions of
sin x = k

4. Using the given graph, solve sin x = 0.6 for 0  x  360.
y = sin x
sin x = 0.6
y = sin x
y = 0.6
y = 0.6
Draw the line y = 0.6.
The solutions are x = 36 or 144. ∴
Solutions obtained by graphical method are approximations only.

5. Follow-up question
Refer to the graph of y = cos 3x. Solve cos 3x = 0.8 for 0  x  360 graphically.
cos 3x = 0.8
y = cos 3x
y = 0.8
y = cos 3x
y = 0.8
Draw the line y = 0.8.
60
120
180
240
300
360
∴ The solutions are x = 12, 108, , 228, 252 or 348.