ATM - Trigonometry Lesson
Trigonometry
Trigonometry is the study of triangles and the relationships of their sides and angles. This topic is important because it allows us to determine distances and angles. Did you ever want to know the distance across a river at a certain point? Trigonometry would help you solve the problem.
We are going to study what is known as right triangle trigonometry because all of our triangles are going to be right triangles. First, some basic knowledge review with the right triangle below.
Pythagorean Theorem
Now that we have covered the basics of right triangles, let's look at the right triangle in terms of the Pythagorean Theorem. Using the right triangle above, any of the side lengths of the triangle can be found if two of the lengths are known. The Pythagorean Theorem indicates the square of each of the legs of the right triangle added together equals the hypotenuse squared.
hypotenuse2 = leg2 + leg2
c2 = a2 + b2
Example 1: Find the length of missing leg if the hypotenuse of the right triangle is 5 feet and one leg is 3 feet
Draw a picture to help you visualize the solution and what to do. The drawing does not have to be fancy, but it will allow you transfer the words to a visual. Note that the 3 could have been placed on either leg, it does not matter.
c2 = a2 + b2 Write the formula in use.
52 = a2 + 32 Substitute.
25 = a2 + 9 Simplify.
16 = a2 Subtract 9.
±4 = a Take the square root of both sides.
Note that a = 4 and a = −4. We were solving for the variable. Can length be negative? No, so the final answer is a = 4. The answer a = −4 is an extraneous answer that does not make sense in the context of the problem so it is eliminated from the answer solution.
The right triangle in the example above is called a Pythagorean Triple. The triangle above is called a 3-4-5 right triangle as it is one of few that will provide a whole number answer as a solution making it a Pythagorean Triple. More of these may be created by multiplying each side of this one by the same number. Try it! Does a 6-8-10 work?
Solution:
62 + 82 = 102 36 + 64 = 100
Are there other Pythagorean Triples besides the 3-4-5 right triangle? Yes, the 5-12-13 right triangle is another and there are more with larger numbers. These are not common. Most solutions require fractions, decimals, or square roots for exact answers.
Special Right Triangles
There are two triangles that are called special right triangles. The first is a 45-45-90 triangle and the second is a 30-60-90 triangle. They have standard formulas for finding their sides and may be seen in the SAT Math Reference Sheet Links to an external site.. These two triangles are also shown below. What you will need to know is how to use them and find any of the sides by without a calculator. In other words, a special right triangle may have square root answers, do not change them to decimals.
The 45-45-90 triangle is also called an isosceles right triangle as one angle is 90 degrees and the other two are 45 degrees as they must match since the sides labeled s will match in length. Sides opposite equal angles are congruent.
Congruent means that the sides are exactly the same length. We studied similarity earlier and this means that the two objects are similar in proportion. If we indicate that two objects are congruent, then they have the same shape and all of the dimensions (lengths and angles) are the same.
Example 2: Given a 45-45-90 triangle, what is the length of the remaining sides if a side is 5 inches?
Use the formula to your advantage and you will know what to do.
45 - 45 - 90
s s
5 5 Substitute s = 5.
The hypotenuse is inches and the other side is 5 inches.
Notice in the example I laid out the degrees of the triangle and then substituted what I knew and solved for the remaining sides. Use this technique no matter what you are solving for in the special right triangles.
Let's look at the 30-60-90 triangle.
Example 3: Given a 30-60-90 triangle, what is the length of the remaining sides if the side labeled x is 7 inches?
Use the formula to your advantage and you will know what to do.
30 - 60 - 90
x 2x Substitute x = 7.
7 2(7) = 14
The hypotenuse is 14 inches and the longer leg is .
It is important to realize that the relationships are the formulas. If you are given side of a special right triangle that is not the variable x or s, then set the number given equal to the formula associated with it and solve for the variable. Then complete finding the rest of the sides. For example, in the 30-60-90 triangle, if you are given the hypotenuse to be 16, then 2x = 16 and x = 8. Then find the other missing side using the x-value.
Right Triangle Trigonometry
How we will look at right triangles that could have the two non-right angles add to 90 degrees using any combination of angle measure, for example 52 degrees + 38 degrees equals 90 degrees. How do we determine side lengths with these triangles?
Let's start with the basic idea of relabeling the triangle to indicate an opposite side and an adjacent side.
The hypotenuse remains, it is always located directly across from the right angle. Opposite and adjacent are new words. Here is the difference in the representation of the blue triangle above and the green triangles below.
The blue triangle has an angle measure marked with the Greek letter theta and a curve indicating the angle goes from one side to the hypotenuse. Theta refers to the measure of the angle that is marked with the curve line. Theta is also the reference point for the blue triangle. Looking from the perspective of theta, directly across the triangle is the opposite side, and forming one side of the angle theta is the adjacent side.
The curve line is optional as if there is either a letter inside the angle or the angle measure inside the angle, the angle is the created by the two segments meeting in the point by the label.
The green triangles have angles labeled with capital letters, indicating that this is the name of the angle. The perspective of this Triangle 1 is from angle B and the perspective of Triangle 2 is angle A. The perspective is determined based on where the opposite and adjacent sides are located. The opposite side is always opposite the angle being referenced and the adjacent side is always attached to the referenced angle. Thus every right triangle has two perspectives, the two non-right angles.
So now we are ready to put the trigonometry formulas to work for us. First I want you to learn a new word, SOH CAH TOA .
The new word, SOH CAH TOA , will help us remember the relationships between the sides of a right triangle that go with the trig functions.
SOH means the formula sin = opposite / hypotenuse
CAH means the formula cos = adjacent / hypotenuse
TOA means the formula tan = opposite / adjacent
What does sin, cos, and tan stand for? Well the full spelling on the trig words is sine, cosine, and tangent respectively. These are relationships between the sides of a triangle based on the angle measure. In other words, if we know the angle and one side one side of the triangle we can find the other sides of the triangle. A calculator or graphing utility is needed to calculate these answers.
To find the perspective that you are supposed to look at the triangle with, the trig function is always followed by the letter of the angle. The angle provides the perspective. For examples, sin A = opp/hyp and yes, you will see all of these shortened to 3 letters.
Do this activity to verify your understanding of the basic trig rules of SOH CAH TOA which creates a ratio of side lengths for a specific angle.
Did you notice the exchange of sin and cos depending on the angle reference in the activity above? If not, return and make note of the similarities.
Now that we understand how to set up the ratios for the trig functions, make sure that you have found on your calculator where the trig functions are located. Watch the video to learn how to find all the parts of a right triangle if you are given only two, an angle and a side.
Solving a right triangle means to find all of the missing parts to the triangle. In the video above, you were given an angle and a side with the right angle. In the next video, only the right angle is given with two sides. Solving for the angles requires using the inverse trig functions, sin−1,cos−1,ortan−1. This is the only method to disconnect the angle from the trig function to solve for the angle.
The video below will show you how to find the side angles of a right triangle if you are given only two legs for the triangle.
Applications of trigonometry are all around us. Watch this video to find the distance across the river.
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