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In this lesson, we will discuss how to solve trigonometry problems using angle and side relationships for special right triangles and the basic trigonometric ratios. You will also learn two different standard measurements for angles, degrees and radians, and how to convert between the two.
Trigonometry is a collection of techniques that can be used to calculate the length of sides and the measure of angles in triangles. When working with trigonometry, it may be helpful to think of yourself as a detective. This module will teach you how to use the pieces of information given to you to determine everything about certain types of triangles.
Trigonometry is also sometimes referred to as pre-calculus because many of the concepts in trigonometry serve as precursors to the derivatives, integrals, and rates of change that you will learn about in the upcoming calculus module.
What are the components of a right triangle?
In order to fully illustrate the trigonometric ratios that will be covered later in this module, it is important that we first understand the parts that make up a right triangle.
Side c, the hypotenuse, is the longest side of the right triangle and does not have a right angle at either end. The other two sides of the triangle (a and b) are referred to as legs. The sum of the angles in a right triangle, as with all triangles, is 180°. So, every right triangle contains one 90° angle and two acute angles. (It is impossible for a right triangle to contain an obtuse angle as it would force the sum of angles to be greater than 180°.) You will find that the application of trigonometric ratios often requires the parts of a triangle to be referred to with respect to a given point of reference (usually an angle). For instance, the diagram below shows how the sides of a right triangle would be referred to using a given angle, in this case 
, as a point of reference.
With respect to the angle 
, side b is the adjacent side. It is immediately between the angle 
and the 90° angle. The adjacent side and the hypotenuse form the angle 
. Side a is the opposite side. It is located across the triangle from the angle 
. The opposite side and adjacent side of a right triangle form the right angle.
Trigonometric ratios are ratios that allow you to associate sides and angles in right triangles. A common mnemonic device for remembering the three basic trigonometric ratios is the term soh cah toa. The “S-O-H” in this term stands for “sine, opposite, hypotenuse,” and is interpreted as, “the sine of 
is equal to the length of the opposite leg over the length of the hypotenuse.” This can be written as a formula as 
. The “C-A-H” in this term stands for “cosine, adjacent, hypotenuse,” and is interpreted as, “the cosine of 
is equal to the length of the adjacent leg over the length of the hypotenuse.” This can be written as a formula as 
. The “T-O-A” stands for “tangent, opposite, adjacent”, and is interpreted as, “the tangent of 
is equal to the length of the opposite leg over the length of the adjacent leg.” This can be written as a formula as 
. A practical application of these trigonometric ratios is shown in the example below.
On the figure below, which of the following is the length of the hypotenuse?
The correct choice is B. The solution can be found by using the trigonometric ratio that we just learned for finding cosine. We know to use the cosine ratio because we are trying to solve for the length of the hypotenuse, and we know the adjacent measurement and the angle measurement at the intersection point between the hypotenuse and the adjacent angle.
We know from trigonometric ratios (and the C-A-H in soh cah toa) that
.
Using the principle of substitution, we can say
.
Multiply both sides of the equation by c (our unknown hypotenuse).
Divide both sides of the equation by 
to find length of hypotenuse.
It is important that you find a way that is easy for you to remember these three trigonometric ratios. If you have these memorized, trigonometry will be a great deal more fun and, as the material becomes more complex, knowing these basic building blocks will make the process much easier. If you are having trouble spelling soh cah toa, try making up a phrase that will help you remember these three basic trigonometric ratios.
There are actually six trigonometric ratios. The three trigonometric ratios that we have not yet discussed are the secant, cotangent, and cosecant. These ratios can be derived from the three basic ratios we have already learned. We will discuss these remaining ratios in more detail later in the lesson.
The two common special right triangles that we will discuss in this module are the isosceles right triangle, also known as the 45-45-90 triangle, and the 30-60-90 triangle. These triangles have “short cuts” to help us solve for the lengths of their sides.
The isosceles right triangle is composed of two 45° acute angles and one 90° right angle. (Remember that the sum of angles in a triangle must always equal 180 degrees.) The lengths of the legs of an isosceles right triangle are always equal. For instance, let the legs be 
and 
, and the hypotenuse be c. Can you derive the hypotenuse in terms of 
? Let’s try it.
Using the Pythagorean Theorem, we know that 
.
We know that an isosceles right triangle has two legs that are of equal length (in this case, 
and 
), so we can say that
Now we can take the square root of both sides.
So the answer is yes, we can derive the hypotenuse in terms of 
.
An easy way of remembering the short cut to the isosceles right triangle is to remember an isosceles triangle always has sides with a ratio 
.
Using the figure below and the short cut we just learned for an isosceles right triangle, solve for side c.
meters
metersThe correct choice is C. If you use the short cut for an isosceles right triangle, you know that a = 10 meters because the legs of an isosceles triangle are equal. You also know that the hypotenuse of an isosceles right triangle is 
.
Another type of special right triangle is the 30-60-90 triangle. The 30-60-90 triangle has a 30° angle, a 60° angle, and a 90° angle, where the hypotenuse is c, the side opposite the 60° angle is a, and the side opposite the 30° angle is b. So in this case, using the trigonometric ratios:
 |
 |
 |
 |
 |
 |
An easy way of remembering the short cut to the 30-60-90 triangle is to remember that a 30-60-90 triangle always has sides with a ratio 
You may wonder how we know that this ratio is valid for every 30-60-90 triangle. By manipulating the information we have available, we will prove that the ratio of a 30-60-90 triangle always has sides with a ratio 
Since we have already proven that 
, c = c, and 
, we have the ratio 
Now divide the ratio by c, and we have the ratio 
. Lastly, you multiply the answer by 2 and…Bingo! You have
the ratio 1:2:
.
Using the triangle below, what is the length of side b?
mThe correct choice is A. Since this is a 30-60-90 triangle, you can apply the aforementioned 
ratio in order to solve for the length of side b. We know that side c = 20, and we know that side c is twice as long as side b (2:1 ratio). Arithmetically, we can now solve for side b as b = 
m.
The special right triangles we have discussed are shown below with each side in terms of the hypotenuse.
Does the word radian sound familiar? You may have heard this term used before in a math class or by an engineer. We all know how to measure angles in degrees, but scientists and engineers commonly measure angles in radians. A radian is the length of the radius of the circle placed along the circumference. For every 360°, there are 2
radians, and for every 180°, there are 
radians. In other words, if the length of the radius r is equal to the length of the arc a, then 
is one radian. Therefore, the formulae to convert from radians to degrees and degrees to radians are:
An easy way of converting from radians to degrees is to remember that 
. So if you need to convert 
radians into degrees, you could just substitute the 
. So 
radians = 
The unit circle below shows degrees and radians at different points around the circle.
Which of the following shows 
radians converted into degrees?
The correct choice is A. 
Convert 115° into radians.
The correct choice is C. Use the formula radians = degrees 
.
.
.
radians, and for every 180°, there are 
radians.degrees = radians 
.
radians = degrees 
.