The domain of a function refers to all possible values of  for which an answer can be obtained. Cosine, as a function, cycles endlessly between  and  (subject to modifiers of the amplitude). Because there is no real number value that can be inserted into  in this case which does not produce a value between  and , the domain of cosine is effectively infinite.

Note that in this case, neither the  nor the  on the outside affect the domain of the function. They do affect the amplitude, which means the value for range will change, but there is no such thing as "three times infinity" nor "three times negative infinity", so the effective domain remains infinite.

If ,

and being that ,

.

From this, you can construct a right triangle where 5 is the opposite side to the reference angle and 12 is the adjacent side to the reference angle. 

After constructing this triangle, you can then proceed to apply the Pythagorean Theorem.

As 5 and 12 are the legs of the constructed right triangle, 5 and 12 are  and  in no particular order. Hence:

We have calculated the hypotenuse of our right triangle. With this knowledge, we can now assign a value to . As , our values of adjacent (12) and our hypotenuse (13) will now be very useful. 

The .

A helpful pneumonic to remember trig functions: . (Sound it out as phonetically as you can)

:

:

:

Finding a missing side in a right triangle generally utliizes one of two concepts:  the Pythagorean Theorem or trigonometry.  When we have two sides given and need the third side, we use the Pythagorean Theorem.  When we only have one side and need a second but do have the help of an angle other than the right angle, we can use trigonometry.  That is the case our problem falls into.

The three sides of every right triangle can be identified in terms of one of the non-right angles.  The hypotenuse is always the side opposite the right angle (and the longest side).  But if we take one of the non-right angles as our starting point, one of the legs is opposite the angle, and another is adjacent to the angle.  For example, in our triangle, from the starting point of the the 18-degree angle, our hypotenuse is the side of length 17, the adjacent leg is the side labeled , and the unlabeled side is the opposite side.

Given these distinctions, right triangle trigonometry hinges on three ratios:  sine, cosine, and tangent.  Sine is the ratio of the opposite leg to the hypotenuse.  Cosine is the ratio of the adjacent leg to the hypotenuse.  Tangent is the ratio of the opposite leg to the adjacent leg.

Our goal is to find , the length of the adjacent leg.  We know the length of the hypotenuse is 17.  Therefore, our ratio of choice is cosine.  Cosine of our angle (18 degrees) is the ratio of  to 17.  We can write this as an equation.

Fortunately for us living in our technological age, most calculators have buttons for these three ratios.  But before you can use your calculator, you have to be sure it is in the proper mode.  The calculator needs to be in degree mode, but the way to check and/or change modes depend on the calculator, so if you don't know how to do this, check the manual or consult a knowledgeable person.

But with a calculator in degree mode we can continue.  For most calculators, simply hit the cosine button (abbreviated "cos"), followed by the angle measure (18), and then hit the equal button.

You should get 0.9510565.  That means we get the following equation.

From there we have an easy equation to solve.  We simply multiply both sides of the equation by 17.

Rounded to the nearest tenth, our answer is 16.2.

Trig functions are only possible to use when the triangle is a right triangle. Although the figure may provide ample information and look like it is a right triangle, the latter cannot be assumed because there is no indication of a right angle. It lacks the notation that may be observed in the following:

Therefore the correct answer to this problem is that it is not possible using a Trigonometric function. 

SOHCAHTOA tells us that . Since this is true, we know that our right triangle has sides  and . With that as the case, we can use the Pythagorean Theorem to figure out the hypotenuse.

Thus, . Now, we can calculate cosine using SOHCAHTOA again.

.

Thus, .

SOHCAHTOA tells us that . Since this is true, we know that our right triangle has one side  and hypotenuse . With that as the case, we can use the Pythagorean Theorem to figure out the hypotenuse.

 ---> 

Thus, . Now, we can calculate cosine using SOHCAHTOA again.

.

Thus, .

In the triangle below, the tangent of , or the opposite side of the angle divided by the adjacent side of the angle. According to the Pythagorean

Theorem, the 

Thus the hypotenuse equals .

The cosine of an angle is the adjacent side of the angle divided by the hypotenuse of the triangle, giving us .

However, since  is , and when  is between ,  is positive while  is negative. Thus,  is negative, giving us the final answer of .

cos = adjacent/hypotenuse = 

To get the radical out of the denominator, we multiply:

You do not even need to compute the line for this question.  All that you need to do is note that you can make a little right triangle with a height of  and a base of , which you get from the slope of the line.  So, to compute the cosine, you will need to find the hypotenuse using the Pythagorean theorem:

So, our little triangle looks like:

The cosine will be the adjacent side, , divided by the hypotenuse, :

, or approximately .

Since the sine of the angle is , that means the opposite side of the triangle is  and the hypotenuse is . Automatically, you know this is a special 3-4-5 right triangle, and that the missing side is 3. If not, you could also find the 3rd side by doing Pythagorean Theorem. This gives you your answer of