Graphical Solutions of Trigonometric Equations

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?

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

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.

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, 132, 228, 252 or 348.