Trigonometric Equations

Sol A Dividing both sides of the equation by gives or . Since and the tangent has period , for any integer .

Sol B Dividing both sides of the equation by gives or . Since and the tangent has period , the solutions are and .

Sol C Rewriting as gives or . Factoring yields , so either or .

If , then or .

If , then or .

Therefore the solutions are .

Sol 1 Factoring gives , so or . If , then or , and if , then or .

Sol 2 Factoring gives , so or . If , then or ; and if , then .

Sol 3 Squaring both sides of gives .

Since , we have that ; so and therefore or . If , then the original equation gives or ; so .

if , then the original equation gives ; so .

Therefore and are the solutions.

Sol 4 gives , so and therefore or .

If , then , where is any integer.

If , then , where is any integer.

Sol 5 Substituting for in the equation gives or , so . Factoring gives , so or .

If , then or ;

and if , then .

Therefore the solutions are ,,and .

Sol 6 Since ,substituting for gives or . Factoring yields , so or .

If , or .

If , then .

Therefore the solutions are , ,and .

Sol 7 gives , so . Therefore or , so either

or

.

Dividing by 3 to find , we obtain the solutions

and

.

Sol 8 Since , we get the equation ; so and therefore .

Then or , so either

or

.

Dividing by 2 to find , we obtain the solutions

and

.

Lawrence Marx 2015-09-15