Here's a few examples of the trigonometric equations you may be required to solve.
Solve the equation #4sin^2x = 1#
Isolate #x#.
#sin^2x = 1/4#
#sinx = +-1/2#
#x = arcsin(+-1/2)#
#x = 30˚, 150˚, 210˚, 330˚#
Solve the equation #cos^2x = 2sin^2x + 2sinx#
Apply the identity #cos^2x + sin^2x = 1 -> cos^2x = 1 - sin^2x#
#1 - sin^2x = 2sin^2x + 2sinx#
#0 = 3sin^2x + 2sinx - 1#
#0 = 3sin^2x + 3sinx - sinx - 1#
#0 = 3sinx(sinx + 1) - 1(sinx + 1)#
#0 = (3sinx - 1)(sinx + 1)#
#sinx = 1/3 and sinx = -1#
#x = arcsin(1/3) and 270˚#
Solve the equation #cscx xx tanx = cotx xxsinx#
Apply the following identities:
•#cscx = 1/sinx#
•#tanx = sinx/cosx#
•#cotx = cosx/sinx#
#1/sinx xx sinx/cosx = cosx/sinx xx sinx#
#1/cosx = cosx#
#1 = cos^2x#
#0 = cos^2x - 1#
#0 = (cosx + 1)(cosx - 1)#
#cosx = -1 and cosx = 1#
#x = 0˚ and 180˚#
However, these solutions are extraneous, since they render the original equation undefined with cotangent.
Solve the equation #sin(45˚ + x) = 1#
#45˚ + x = arcsin1#
#x = 90˚ - 45˚#
#x = 45˚#
The key to solving trigonometric equations is knowing your identities.
Hopefully this helps!
