Right Triangle Trigonometry

1. Finding Approximate Values of Trigonometric Functions

With the exception of multiples of and most other function values are more convenient to use when approximated by using your calculator.

Your calculator should have buttons labeled “SIN,” “COS,” and “TAN.” To evaluate one of these functions at a specific angle, it is important to make sure that your calculator is set to the appropriate angle mode.

Degree mode: the angle input of a sine or cosine function is treated as a degree measure. For example, when you evaluate on your calculator, you will get the value of

Radian mode: the angle input of a sine or cosine function is treated as a radian measure. For example, when you evaluate on your calculator, the angle is measured in radians.

EXAMPLE

Using a calculator, we’ll approximate the following values to four decimal places.

(The angle 1.2 is measured in radians, so be sure to put your calculator in radian mode before evaluating.)

(The angle is measured in degrees, so be sure to put your calculator in degree mode before evaluating.)

(The angle 3.78 is measured in radians, so be sure to put your calculator in radian mode before evaluating.)

(The angle is measured in degrees, so be sure to put your calculator in degree mode before evaluating.)

Some calculators have an angle feature that allows you to add a “degree” symbol at the end of the angle. If your calculator is in radian mode and you type in , for example, it will return the cosine function of Check to see if your calculator has this feature, and use it if this makes things easier for you. It might be easier than switching the mode all the time.

try it

Consider the expression

Approximate the value of the expression to four decimal places.

try it

Consider the expression

Approximate the value of the expression to four decimal places.

Notice that your calculator does not have the reciprocal functions (secant, cosecant, and cotangent) as dedicated buttons. Remember that each of these functions is the reciprocal of one other trigonometric function. This will help to evaluate function values of secant, cosecant, and cotangent.

EXAMPLE

Evaluate and to four decimal places.

First, evaluate

	Use the identity
	Approximate using a calculator.

Thus,

Next, evaluate

	Use the identity
	Approximate using a calculator.

Thus,

Lastly, evaluate

	Use the identity
	Approximate using a calculator.

Thus,

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Consider the expression

Evaluate the expression to four decimal places.

Remember that Using this fact, you should get approximately 1.3054 as your answer.

try it

Consider the expression

Evaluate the expression to four decimal places.

Remember that Using this fact, you should get approximately -1.2349 as your answer.

2. The Trigonometric Functions as They Relate to Sides of a Right Triangle

Now that we know how to approximate trigonometric functions of any angle, we are ready to solve application problems that involve right triangles.

Since a triangle’s angles have a sum of a right triangle has two angles whose sum is These angles are called complements, and are described as complementary. Since the angles must be positive, they must have measures that are less than which makes them acute angles.

Consider the following picture, which shows two circles centered at the origin, with angle that terminates in the first quadrant.

• The inner circle has radius 1, with angle having terminal point
• The outer circle has radius r, where with angle having terminal point

As it turns out, knowing the point and its distance r from the origin, we can find the values of and and as a consequence, the value of each of the other trigonometric functions of angle

Since the triangle is contained in the first quadrant, we know that meaning that is an acute angle. Note now the similar right triangles that are formed by dropping vertical lines from the points and

Here is a closer look at these triangles, with the lengths of their sides labeled:

Since these triangles are similar, we know that the lengths of the corresponding sides are in proportion to one another. This means that the ratio of any two sides in one triangle is equal to the ratio of the same two sides in the other triangle.

Using the base and the hypotenuse of each triangle, which gives

Using the height and hypotenuse from each triangle, which means

Thus, given that angle has a terminal side that contains the point and

Note: if as with the unit circle, we have and

Recall also that is the slope of the line containing the origin and the terminal point of the angle. It follows that

Given a right triangle as shown in the figure, we can find the sine, cosine, and tangent functions of the angle as follows.

Then, we can label the triangle this way.

The values of and are found as follows:

formula to know

Sine of an Angle in Terms of Sides of a Right Triangle

Cosine of an Angle in Terms of Sides of a Right Triangle

Tangent of an Angle in Terms of Sides of a Right Triangle

One way to remember these ratios is by the mnemonic “SohCahToa.” The S, C, and T stand for trig functions, and the lowercase letters after each capital letter are the sides that are used to compute the value of each function.

The sine, cosine, and tangent functions are the most commonly used. The value of each of the other trigonometric functions is computed by calculating the appropriate reciprocal.

watch

Given a right triangle, we’ll find the values of all six trigonometric functions of

EXAMPLE

Consider the right triangle shown below.



Use the triangle to find all six trigonometric functions of

First, notice that the length of the hypotenuse is unknown. Since this is a right triangle, the Pythagorean theorem is used to find the unknown side.

	This is the Pythagorean theorem, where and b are the lengths of the legs and c is the length of the hypotenuse.
	Let and
	Simplify.
	Apply the square root property. Since c is the length of a side of a triangle, only the positive solution is considered.

Thus, the length of the hypotenuse is 13 units.

Now, we can compute the values of all trigonometric functions of

Note: the opposite side has length 5 units, the adjacent side has length 12 units, and the hypotenuse has length 13 units.

         

        

         

try it

Consider the triangle shown below.

Find the values of all six trigonometric functions of the labeled angle.

First, find the unknown side using the Pythagorean Theorem.

Let y = the unknown side.

	Set up the Pythagorean Theorem with sides 5 and 6 known.
	Simplify.
	Subtract 25 from both sides.
	Apply the square root principle. Since a side cannot have a negative value, only the positive solution is considered.

Thus, the unknown side opposite is

Then, we have the following:


Note: While we could remember the right-triangle ratios for csc, sec, and cot, it is usually easier to just remember that csc, sec, and cot are the reciprocals of sin, cos, and tan respectively.

Now we will look at problems in which the measure of an angle is known, and we wish to find the other sides and angles of a triangle.

EXAMPLE

Consider the triangle shown below.



Find the values of x, y, and rounded to the nearest whole number.

First, remember that the sum of the measures of the angles in a triangle is Then, which means

To find the lengths of the unknown sides, we’ll use the given as reference.

This means that and The last equation is not very helpful since it has two variables in it.

Consider the equation

Multiplying both sides by 25, we have Using a calculator in degree mode,

Next, consider the equation

Multiplying both sides by 25, we have Using a calculator in degree mode,

Note: once you have the lengths of two sides of a right triangle, the Pythagorean theorem can be used to find the third side, but be sure to use all the decimal places, since using the rounded value of a side can result in a rounding error.

think about it

Do you feel that using a trigonometric function was easier?

watch

In this video, we will find the unknown sides and angles of a given right triangle by using trigonometric functions.

try it

Consider the right triangle given below.

Find the approximate lengths of the unknown sides, rounded to the nearest whole number.

There are several ways to solve this problem. Here is one strategy:

Relative to angle x is the adjacent side and 8 is the opposite. These can all be related by the tangent function:

	Multiply both sides by x.
	Divide both sides by
	Round to the nearest whole number.

Next, there are several ways to find r, but since we want to practice trigonometric functions, we’ll use the sin function since 8 is opposite the given angle and we wish to find the hypotenuse.

	Multiply both sides by r.
	Divide both sides by
	Round to the nearest whole number.

To the nearest whole number, and

terms to know

Right Triangle

A triangle that contains a right angle.

Complementary Angles, Complements

Two acute angles whose sum is or in radians, their sum is

Acute Angle

Angle whose measure is positive and less than or in radians, less than

3. Cofunction Identities

The two acute angles in a right triangle are complementary. It turns out that trigonometric functions of complementary angles are related. Consider the right triangle shown below.

If is the measure of one acute angle, then the measure of the other is

Here are the six trigonometric functions evaluated for and its complement,

Trigonometric Functions of	Trigonometric Functions of

By comparing values of the trig functions, we have six pairs of expressions that are equal:

formula to know

Cofunction Identities

The first two equations tell us that the sine of one angle is the cosine of its complement, and vice versa. The functions sine and cosine are called “cofunctions” since one has a “co” and the other doesn’t. Notice that secant and cosecant are cofunctions, as well as tangent and cotangent.

The main idea is that cofunctions of complementary angles are equal.

One can also write the cofunction identities in terms of radians. To do so, replace with

In order to use the cofunction identities effectively, we need to know how to find complementary angles, more specifically with radians.

EXAMPLE

Find the complement of

Two acute angles are complements if their sum is

Thus, the complement of is

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Consider the angle

What is its complement?

Since complements add up to the complement of is

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Consider the angle

What is its complement?

The complement is

Using complements, we can rewrite trigonometric expressions using cofunction identities.

EXAMPLE

The expressions and are equal by the cofunction identities, since sine and cosine are cofunctions, and and are complements.

The expressions and are equivalent since tangent and cotangent are cofunctions, and and are complements.

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Consider the expression

Using cofunction identities, write an equivalent expression.

The appropriate cofunction identity is

Using we have and thus the identity tells us that

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Consider the expression

Using cofunction identities, write an equivalent expression.

The cofunction identity states that

Therefore, to complete this identity,

4. Solving Real-Life Problems Using Right Triangles

Trigonometric functions are useful in determining the height of very tall objects.

Consider the figure below.

To measure the height of the tower, it would be dangerous to climb the structure with a tape measure in one hand to measure the height. As the picture suggests, one could stand a distance away from the base of the tower and determine the angle of elevation to the top of the building, then use trigonometry to find the height of the building. The angle of depression is made with the horizontal that points downward toward an object

In order to solve problems in this lesson, you’ll need to do the following:

step by step

1. Draw a right triangle that represents the situation.
2. Label all known sides and angles.
3. Label the side you wish to find.
4. Use an appropriate trigonometric function to solve for that side.

EXAMPLE

From a point 30 feet away from the base of a tree, the angle of elevation to the top of the tree is What is the height of the tree, to the nearest foot?

Here is a picture of this situation:



Notice that the hypotenuse is not known, and it is not needed. Let the height of the tree.

Referencing the given angle we want to find its opposite side, and we know the length of its adjacent side. This means we should use the tangent function.

	Multiply both sides of the equation by 30.
	Approximate the solution.

The height of the tree is approximately 27 feet.

try it

A 33-ft ladder leans against a building so that the angle between the ground and the ladder is

To the nearest tenth of a foot, how high does the ladder reach up the side of the building?

From the picture, we can see that the hypotenuse is the length of the ladder, which is 33 ft, and the angle opposite the height is

This means that the sin function is appropriate to solve this problem. Let the height of the ladder against the building:

	Multiply both sides by 33.
	Approximate h.

To the nearest tenth, the ladder reaches a height of 32.5 feet.

Sometimes two right triangles are needed to solve problems.

watch

In this video, we’ll solve the following problem.

A radio tower is located 400 feet from a building. From a window in the building, a person determines that the angle of elevation to the top of the tower is and that the angle of depression to the bottom of the tower is How tall is the tower to the nearest tenth of a foot?

try it

There is an antenna on top of a building. From a point 300 feet from the base of the building, the angle of elevation to the top of the building is measured to be The angle of elevation to the top of the antenna is measured to be

To the nearest foot, how tall is the antenna?

The base of the triangle is 300 feet.

Let the height of the building.

Let the height of the building and the antenna combined.

Then, we want to find

To find y, notice that y is the side opposite the angle. Since the adjacent side is known, use the tangent function:

	Multiply both sides by 300 to solve for y.
	Approximate y.

Now, follow a similar idea to find z:

	Multiply both sides by 300 to solve for z.
	Approximate z.

Next, find the difference, then round to the nearest whole number.

which rounds to 28 feet.

This means that the height of the antenna is approximately 28 feet.

terms to know

Angle of Elevation

An angle made with the horizontal that points upward toward an object.

Angle of Depression

An angle made with the horizontal that points downward toward an object.

summary

In this lesson, you began by understanding that the most convenient way to find approximate values of trigonometric functions is by using your calculator, making sure that your calculator is set to the appropriate angle mode (degree or radian). Next, you learned that in order to solve problems involving the trigonometric functions as they relate to sides of a right triangle, one must know how to approximate trigonometric functions of angles other than the special angles . Then, you apply the trigonometric functions to right triangles, where the value of a trigonometric function is the ratio of two sides. You also learned that the relationship between the two acute angles in a right triangle (as complementary angles) gives way to the cofunction identities, the main idea being that cofunctions of complementary angles are equal. Finally, you learned that you can solve many real-life problems, particularly with angles of depression and elevation, by using a right triangle as a model.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.