In a right triangle, for sides a and b, with c being the hypotenuse, . Thus if cos(A) is , then c = 14, and the side adjacent to A is 11. Therefore, the side opposite of angle A is the square root of , which is  Since sin is , sin(A) is .

As with all trigonometry problems, begin by considering how you could rearrange the question. They often have hidden easy ways out. So begin by noticing:

Now, you can treat  like it is any standard denominator. Therefore:

Combine your fractions and get:

Now, from our trig identities, we know that , so we can say:

Now, for our triangle, the  is . Therefore,

Recall that the standard  triangle, in radians, looks like:

Since , you can tell that .

Therefore, you can say that  must equal :

Solving for , you get:

sin(30º) = ½

sine = opposite / hypotenuse

½ = opposite / 8

Opposite = 8 * ½ = 4

Use SOHCAHTOA. Sin(30) = 12/x, then 12/sin(30) = x = 24.

You can also determine the side with a measure of 12 is the smallest side in a 30:60:90 triangle. The hypotenuse would be twice the length of the smallest leg.

We can solve for the length of the chord by drawing a line the bisects the angle and the chord, shown below as .

In this circle, we can see the triangle  has a hypotenuse equal to the radius of the circle (), an angle  equal to half the angle made by the chord, and a side  that is half the length of the chord.  By using the sine function, we can solve for .

The length of the entire chord is twice the length of , so the entire chord length is .

We can solve for the length of the chord by drawing a line the bisects the angle and the chord, shown below as .

In this circle, we can see the triangle  has a hypotenuse equal to the radius of the circle (), an angle  equal to half the angle made by the chord, and a side  that is half the length of the chord.  By using the sine function, we can solve for .

The length of the entire chord is twice the length of , so the entire chord length is .

Recall that the sine of an angle is the ratio of the opposite side to the hypotenuse of that triangle. Thus, for this triangle, we can say:

Solving for , we get:

 or 

Begin by drawing out this scenario using a little right triangle:

Note importantly: We are looking for  as the the distance to the top of the building. We know that the sine of an angle is equal to the ratio of the side opposite to that angle to the hypotenuse of the triangle. Thus, for our triangle, we know:

Using your calculator, solve for :

This is . Now, take the decimal portion in order to find the number of inches involved.

 Thus, rounded, your answer is  feet and  inches.

To find the sine of an angle, remember the mnemonic SOH-CAH-TOA. 
This means that 

.

We are asked to find the . So at point  we see that side  is opposite, and the hypotenuse never changes, so it is always . Thus we see that