College Algebra & Trigonometry, 2018a

10.3. TRIGONOMETRIC EQUATIONS 463
Some trigonometric equations have no real number solutions. The equation
sin x =2has no real number solutions. Recall that the sine ratio was originally
defined as the ratio of the side opposite an angle to the hypotenuse. The hypotenuse
is always the longest side in a right triangle so there is no way the sine
function could be greater than 1 if we’re working with real-valued angles. However,
in the same way that complex numbers are used to solve equations like
x 2 = −7, complex-valued angles can be used to solve equations such as sin x =2.
We won’t go into this here, however, there is a relatively straightforward way to
solve these equations.
If you encounter an equation like cos x =3and are solving for values of x
0 ◦ ≤ x