Write the area of the sector in radians.

Substitute the givens and solve.

Katelyn is making a semi-circular design to put on one of her quilts. The design is not a perfect half-circle however, she needs to make the central angle  radians. If the radius of the circle is , what is the area of the semi-circular design?

Recall the following formual for area of a sector:

So, we plug in our knowns and solve for area!

Therefore, our answer is...

Lucy is making a solar panel to cover a portion of a satellite dish. If the central angle of the sector the solar panel will cover is , and the satellite dish has a radius of , what area will the solar panel cover?

To calculate area of a sector, use the following formula:

Where the numerator of the fraction is the measure of the desired angle in radians, and r is the radius of the circle.

Now, we know both our variables, so we simply need to plug them in and simplify.

Now, this looks messy, but we can simplify it to get:

Next, use your calculator to find a decimal answer, and then round to get our final answer.

Making the area of the sector

Equation for sector area  is given by

, where  is the angle measure of the sector in radians, and  is the radius of the circle. 

In our case, the sector encompasses  of the circle or

To determine the radius , given diameter 

The sector area therefore is:

Write the Pythagorean Identity.

Reorganize the left side of this equation so that it matches the form:   

Subtract cosine squared theta on both sides.

Multiply both sides by 3.

Of the six trigonometric functions, four are odd, meaning . These four are:

• sin x
• tan x
• cot x
• csc x

That leaves two functions which are even, which means that . These are:

• cos x
• sec x

Of the aforementioned, only  is incorrect, since secant is an even function, which implies that 

Rewrite  by odd and even identities.

Use the difference identity of sine, and choose the special angles 45 and 30, since their difference equals to 15.

Write the even and odd identities for sine and cosine.

Rewrite the expression  and evaluate.

In order to simplify , rewrite the expression after applying the rule of odd-even identities for the secant function.

Which of the following is equivalent to the expression:

Begin by recalling the following identity:

Next, recall the relationship between cotangent and tangent:

As well as the relationship between tangent, sine and cosine

So to put it all together, we can pull out the negative sign from our original expression:

Next, we can rewrite our cotangent as tangent

Finally, we can change our tangent to sine and cosine, but because we are dealing with the reciprocal of tangent, we will need the reciprocal of our identity.

Making our answer:

Beware trap answer:

This may look good on the surface, but recall