Precalculus : Trigonometric Identities

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Solving Trigonometric Equations And Inequalities

Use trigonometric identities to solve the equation for the angle value.

Possible Answers:

Correct answer:

Explanation:

The simplest way to solve this problem is using the double angle identity for cosine.

Substituting this value into the original equation gives us:

Example Question #1 : Solve Trigonometric Equations And Inequalities In Quadratic Form

If  exists in the domain from  , solve the following:  

Possible Answers:

Correct answer:

Explanation:

Factorize .

Set both terms equal to zero and solve.

This value is not within the  domain.

This is the only correct value in the  domain.

Example Question #1 : Solving Trigonometric Equations And Inequalities

Solve for  in the equation  on the interval .

Possible Answers:

Correct answer:

Explanation:

If you substitute  you obtain a recognizable quadratic equation which can be solved for

.

Then we can plug  back into our equation and use the unit circle to find that 

.

Example Question #1 : Solving Trigonometric Equations And Inequalities

Given that theta exists from , solve:  

Possible Answers:

Correct answer:

Explanation:

In order to solve  appropriately, do not divide  on both sides.  The effect will eliminate one of the roots of this trig function.

Substract  from both sides.

Factor the left side of the equation.

Set each term equal to zero, and solve for theta with the restriction .

The correct answer is:

Example Question #4 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Solve  for 

Possible Answers:

There is no solution.

Correct answer:

There is no solution.

Explanation:

By subtracting  from both sides of the original equation, we get . We know that the square of a trigonometric identity cannot be negative, regardless of the input, so there can be no solution. 

Example Question #5 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Solve  when 

Possible Answers:

There are no solutions.

Correct answer:

There are no solutions.

Explanation:

Given that, for any input, , we know that, and so the equation  can have no solutions.

Example Question #1 : Solve Trigonometric Equations And Inequalities In Quadratic Form

Solve  when 

Possible Answers:

There are no solutions.

Correct answer:

Explanation:

By adding one to both sides of the original equation, we get , and by taking the square root of both sides of this, we get  From there, we get that, on the given interval, the only solutions are  and .

Example Question #221 : Pre Calculus

Evaluate the following.

Possible Answers:

Correct answer:

Explanation:

We can use the angle sum formula for sine here.

If we recall that, 

,

we can see that the equation presented is equal to 

 because .

We can simplify this to , which is simply

Example Question #2 : Product/Sum Identities

Evaluate the following.

Possible Answers:

Correct answer:

Explanation:

The angle sum formula for cosine is, 

.

First, we see that . We can then rewrite the expression as,

All that is left to do is to recall the unit circle to evaluate,

.

Example Question #1 : Product/Sum Identities

Evaluate the following.

Possible Answers:

Correct answer:

Explanation:

This one is another angle sum/difference problem, except it is using the trickier tangent identity.

The angle sum formula for tangent is

We can see that .

We can then rewrite the expression as , which is .

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