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
.