Trigonometric Equations - Trigonometry

We can write the system in the equivalent form:

The solution to the first equation is

means that

This means that there is no x that satisfies the system.

Therefore there is no x that solves the 3 inequalities simultaneously.

There are multiple solution paths. We could subtract 1 from both sides and use the quadratic formula with and . Or we could solve using inverse opperations:

divide both sides by 2

take the square root of both sides

The unit circle tells us that potential solutions for are .

To get our final solution set, divide each by 3, giving:

.

This problem has multiple solution paths, including subtracting 5 from both sides and using the quadratic formula with . We can also solve using inverse opperations:

subtract 2 from both sides

divide both sides by 4

take the square root of both sides

If the sine of an angle is , that angle must be one of . Since the angle is , we can get theta by subtracting :

Therefore,

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

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 .

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.

First we see that :

.

Now letting

we have

We know that :

and we are given that

, this gives

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.

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:

This is the figure being described in the problem, and as sine is opposite over hypotenuse and cosine adjacent over hypotenuse, the sine of and the cosine of will yield the correct answer.

Rearrange algebraically so that,

.

Over the interval 0 to 360 degrees, sinx = 1 when x equals 90 degrees.

First, think of the angle values for which . (This is the equivalent of taking the arctan.)

.

The angles for which this is true are 45 degrees and 225 degrees.

We set x-15 equal to those two angles and solve for x, giving us 60 and 240.

Rearrange the equation so that,

.

Recall the angles over the interval 0 to 360 degrees for which sec is equal to 2.

These are 60 and 300 degrees.

Set x+7 equal to these angle measures and then find that x equals 53 and 293.