Example 2
What are the tangents of and in the triangle below?
To find these ratios, first identify the sides opposite and adjacent to each angle.
So, the tangent of is about and the tangent of is .
It is common to write instead of . In this text we will use both notations.
Complementary Angles in Right Triangles
Recall that in all triangles, the sum of the measures of all angles must be . Since a right angle has a measure of , the remaining two angles in a right triangle must be complementary. Complementary angles have a sum of . This means that if you know the measure of one of the smaller angles in a right triangle, you can easily find the measure of the other. Subtract the known angle from and you’ll have the measure of the other angle.
Example 3
What is the measure of in the triangle below?
To find , you can subtract the measure of from .
So, the measure of is since and are complementary.
Tangents of Special Right Triangles
It may help you to learn some of the most common values for tangent ratios. The table below shows you values for angles in special right triangles.
Tangent
Notice that you can derive these ratios from the special right triangle. You can use these ratios to identify angles in a triangle. Work backwards from the ratio. If the ratio equals one of these values, you can identify the measurement of the angle.
Example 4
What is in the triangle below?
Find the tangent of and compare it to the values in the table above.
So, the tangent of is . If you look in the table, you can see that an angle that measures has a tangent of . So, .
Example 5
What is in the triangle below?
Find the tangent of and compare it to the values in the table above.
So, the tangent of is about . If you look in the table, you can see that an angle that measures has a tangent of . So, .
Notice in this example that is a triangle. You can use this fact to see that .
Lesson Summary
In this lesson, we explored how to work with different radical expressions both in theory and in practical situations. Specifically, we have learned:
How to identify the different parts of right triangles.
How to identify and use the tangent ratio in a right triangle.
How to identify complementary angles in right triangles.
How to understand tangent ratios in special right triangles.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Review Qusetions
Use the following diagram for exercises 1-5.
How long is the side opposite angle
How long is the side adjacent to angle
How long is the hypotenuse?
What is the tangent of
What is the tangent of
What is the measure of in the diagram below?
What is the measure of in the diagram below?
Use the following diagram for exercises 8-9.
What is the tangent of
What is the tangent of
What is the measure of in the triangle below?
Review Answers
Sine and Cosine Ratios
Learning Objectives
Review the different parts of right triangles.
Identify and use the sine ratio in a right triangle.
Identify and use the cosine ratio in a right triangle.
Understand sine and cosine ratios in special right triangles.
Introduction
Now that you have some experience with tangent ratios in right triangles, there are two other basic types of trigonometric ratios to explore. The sine and cosine ratios relate opposite and adjacent sides of a triangle to the hypotenuse. Using these three ratios and a calculator or a table of trigonometric ratios you can solve a wide variety of problems!
Review: Parts of a Triangle
The sine and cosine ratios relate opposite and adjacent sides to the hypotenuse. You already learned these terms in the previous lesson, but they are important to review and commit to memory. The hypotenuse of a triangle is always opposite the right angle, but the terms adjacent and opposite depend on which angle you are referencing. A side adjacent to an angle is the leg of the triangle that helps form the angle. A side opposite to an angle is the leg of the triangle that does not help form the angle.
Example 1
Examine the triangle in the diagram below.
Identify which leg is adjacent to angle , which leg is opposite to angle , and which segment is the hypotenuse.
The first part of the question asks you to identify the leg adjacent to . Since an adjacent leg is the one that helps to form the angle and is not the hypotenuse, it must be . The next part of the question asks you to identify the leg opposite . Since an opposite leg is the leg that does not help to form the angle, it must be . The hypotenuse is always opposite the right angle, so in this triangle it is segment .
The Sine Ratio
Another important trigonometric ratio is sine. A sine ratio must always refer to a particular angle in a right triangle. The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. Remember that in a ratio, you list the first item on top of the fraction and the second item on the bottom.
So, the ratio of the sine will be
Example 2
What are and in the triangle below?
All you have to do to find the solution is build the ratio carefully.
So, and .
The Cosine Ratio
The next ratio to examine is called the cosine. The cosine is the ratio of the adjacent side of an angle to the hypotenuse. Use the same techniques you used to find sines to find cosines.
Example 3
What are the cosines of and in the triangle below?
To find these ratios, identify the sides adjacent to each angle and the hypotenuse. Remember that an adjacent side is the one that does create the angle and is not the hypotenuse.
So, the cosine of is about and the cosine of is about .
Note that is NOT one of the special right triangles, but it is a right triangle whose sides are a Pythagorean triple.
Sines and Cosines of Special Right Triangles
It may help you to learn some of the most common values for sine and cosine ratios. The table below shows you values for angles in special right triangles.
Sine
Cosine
You can use these ratios to identify angles in a triangle. Work backwards from the ratio. If the ratio equals one of these values, you can identify the measurement of the angle.
Example 4
What is the measure of in the triangle below?
Note: Figure is not to scale.
Find the sine of and compare it to the values in the table above.
So, the sine of is . If you look in the table, you can see that an angle that measures has a sine of . So, .
Example 5
What is the measure of in the triangle below?
Find the cosine of and compare it to the values in the previous table.
So, the cosine of is about . If you look in the table, you can see that an angle that measures has a cosine of . So, measures about . This is a right triangle.
Lesson Summary
In this lesson, we explored how to work with different trigonometric ratios both in theory and in practical situations. Specifically, we have learned:
The different parts of right triangles.
How to identify and use the sine ratio in a right triangle.
How to identify and use the cosine ratio in a right triangle.
How to apply sine and cosine ratios in special right triangles.
These skills will help you solve many different types of problems. Always be
on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Points to Consider
Before you begin the next lesson, think about strategies you could use to simplify an equation that contains a trigonometric function.
Note, you can only use the , and ratios on the acute angles of a right triangle. For now it only makes sense to talk about the , or ratio of an acute angle. Later in your mathematics studies you will redefine these ratios in a way that you can talk about , and of acute, obtuse, and even negative angles.
Review Questions
Use the following diagram for exercises 1-3.
What is the sine of
What is the cosine of
What is the cosine of Use the following diagram for exercises 4-6.
What is the sine of
What is the cosine of
What is the sine of
What is the measure of in the diagram below?
Use the following diagram for exercises 8-9.
What is the sine of
What is the cosine of
What is the measure of in the triangle below?
Review Answers
Inverse Trigonometric Ratios
Learning Objectives
Identify and use the arctangent ratio in a right triangle.
Identify and use the arcsine ratio in a right triangle.
Identify and use the arccosine ratio in a right triangle.
Understand the general trends of trigonometric ratios.
Introduction
The word inverse is probably familiar to you often in mathematics, after you learn to do an operation, you also learn how to “undo” it. Doing the inverse of an operation is a way to undo the original operation. For example, you may remember that addition and subtraction are considered inverse operations. Multiplication and division are also inverse operations. In algebra you used inverse operations to solve equations and inequalities. You may also remember the term additive inverse, or a number that can be added to the original to yield a sum of . For example, and are additive inverses because .
In this lesson you will learn to use the inverse operations of the trigonometric functions you have studied thus far. You can use inverse trigonometric functions to find the measures of angles when you know the lengths of the sides in a right triangle.
Inverse Tangent
When you find the inverse of a trigonometric function, you put the word arc in front of it. So, the inverse of a tangent is called the arctangent (or arctan for short). Think of the arctangent as a tool you can use like any other inverse operation when solving a problem. If tangent tells you the ratio of the lengths of the sides opposite and adjacent to an angle, then arctan tells you the measure of an angle with a given ratio.
Suppose . The arctangent can be used to find the measure of on the left side of the equation.
Where did that come from? There are two basic ways to find an arctangent. Sometimes you will be given a table of trigonometric values and the angles to which they correspond. In this scenario, find the value that is closest to the one provided, and identify the corresponding angle.
Another, easier way of finding the arctangent is to use a calculator. The arctangent button may be labeled “arctan,” “atan,” or “ .” Either way, select this button, and input the value in question. In this case, you would press the arctangent button and enter (or on some calculators, enter , then press “arctan”). The output will be the value of measure .
is about .
Example 1
Solve for if
You can use the inverse of tangent, arctangent to find this value.
Then use your calculator to find the arctangent of .
Example 2
What is in the triangle below?
First identify the proper trigonometric ratio related to that can be found using the sides given. The tangent uses the opposite and adjacent sides, so it will be relevant here.
Now use the arctangent to solve for the measure of .
Then use your calculator to find the arctangent of .
Inverse Sine
Just as you used arctangent as the inverse operation for tangent, you can also use arcsine (shortened as arcsin) as the inverse operation for sine. The same rules apply. You can use it to isolate a variable for an angle measurement, but you must perform the operation on both sides of the equation. When you know the arcsine value, use a table or a calculator to find the measure of the angle.
Example 3
Solve for if
You can use the inverse of sine, arcsine to find this value.
Then use your calculator to find the arcsine of .
Example 4
What is in the triangle below?
First identify the proper trigonometric ratio related to angle that can be found using the sides given. The sine uses the opposite side and the hypotenuse, so it will be relevant here.
Now use the arcsine to isolate the value of angle .
Finally, use your calculator to find the arcsine of .
Inverse Cosine
The last inverse trigonometric ratio is arccosine (often shortened to arccos). The same rules apply for arccosine as apply for all other inverse trigonometric functions. You can use it to isolate a variable for an angle measurement, but you must perform the operation on both sides of the equation. When you know the arccosine value, use a table or a calculator to find the measure of the angle.
Example 5
Solve for if .
You can use the inverse of cosine, arccosine, to find this value.
Then use your calculator to find the arccosine of .
Example 6
What is the measure of in the triangle below?
First identify the proper trigonometric ratio related to that can be found using the sides given. The cosine uses the adjacent side and the hypotenuse, so it will be relevant here.
Now use the arccosine to isolate the value of .
Finally use your calculator or a table to find the arccosine of
General Trends in Trigonometric Ratios
Now that you know how to find the trigonometric ratios as well as their inverses, it is helpful to look at trends in the different values. Remember that each ratio will have a constant value for a specific angle. In any right triangle, the sine of a angle will always be —it doesn’t matter how long the sides are. You can use that information to find missing lengths in triangles where you know the angles, or to identify the measure of an angle if you know two of the sides.
Examine the table below for trends. It shows the sine, cosine, and tangent values for eight different angle measures.
Sine
Cosine
Tangent
Example 7
Using the table above, which value would you expect to be greater: the sine of or the cosine of ?
You can use the information in the table to solve this problem. The sine of is and the sine of is . So, the sine of will be between the values and . The cosine of is and the cosine of is So, the cosine of will be between the values of and Since the range for the cosine is greater, than the range for the sine, it can be assumed that the cosine of will be greater than the sine of
Notice that as the angle measures approach , approaches . Similarly, as the value of the angles approach , the approaches . In other words, as the gets greater, the gets smaller for the angles in this table.
The tangent, on the other hand, increases rapidly from a small value to a large value (infinity, in fact) as the angle approaches .
Lesson Summary
In this lesson, we explored how to work with different radicals both in theory and in practical situations. Specifically, we have learned:
How to identify and use the arctangent ratio in a right triangle.
How to identify and use the arcsine ratio in a right triangle.
How to identify and use the arccosine ratio in a right triangle.
How to understand the general trends of trigonometric
ratios.
These skills will help you solve many different types of problems. Always be on the lookout for new and interesting ways to find relationships between sides and angles in triangles.
Points to Consider
To this point, all of the trigonometric ratios you have studied have dealt exclusively with right triangles. Can you think of a way to use trigonometry on triangles that are acute or obtuse?
Review Questions
Solve for
Solve for
What is the measure of in the triangle below?
Solve for
What is the measure of in the triangle below?
Solve for
Solve for
What is the measure of in the triangle below?
CK-12 Geometry Page 32