Factoring Trigonometric Equations - Trigonometry

Therefore,

and that only happens once in the given interval, at , or 45 degrees.

The fastest way to solve this equation is to simply try the three answers. Plugging in gives

Our first choice is valid.

Plugging in gives

However, since is undefined, this cannot be a valid answer.

Finally, plugging in gives

Therefore, our third answer choice is not correct, meaning only 1 is correct.

Note first that:

and :

.

Now taking . We have

.

Since and .

We therefore have :

We have .

Now since

This last expression can be written as :

.

This shows the required result.

Letting , we have the equivalent expression:

.

We cant factor since .

This shows that we cannot factor the above expression.

We first note that we have:

Then taking , we have the result.

First of all we know that :

and this gives:

.

Now we need to see that: can be written as

and since

we have then:

.

We know that we can write

in the following form

.

Now taking ,

we have:

.

This is the result that we need.

From the expression :

we have:

Now since we know that :

. This expression becomes:

.

This is what we need to show.

First we see that :

.

Now letting

we have

We know that :

and we are given that

, this gives

Step 1: Recall the difference of squares (or powers of four) formula:

Step 2: Factor the question:

Factor more:

Step 3: Recall a trigonometric identity:

.. Replace this

Final Answer:

Don't get scared off by the fact we're doing trig functions! Factor as you normally would. Because our middle term is negative (), we know that the signs inside of our parentheses will be negative.

This means that can be factored to or .

Therefore,

and that only happens once in the given interval, at , or 45 degrees.

The fastest way to solve this equation is to simply try the three answers. Plugging in gives

Our first choice is valid.

Plugging in gives

However, since is undefined, this cannot be a valid answer.

Finally, plugging in gives

Therefore, our third answer choice is not correct, meaning only 1 is correct.

Note first that:

and :

.

Now taking . We have

.

Since and .

We therefore have :

We have .

Now since

This last expression can be written as :

.

This shows the required result.

Letting , we have the equivalent expression:

.

We cant factor since .

This shows that we cannot factor the above expression.

We first note that we have:

Then taking , we have the result.

First of all we know that :

and this gives:

.

Now we need to see that: can be written as

and since

we have then:

.