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In this lesson, the concept of similarity will be used to understand that side ratios in right triangles are properties of the angles. This will lead to the definition of trigonometric ratios for acute angles.

Catch-Up and Review

Here are a few recommended readings before getting started.

Similar triangles
Right angle
Right triangles
Pythagorean Theorem

Try your knowledge on these topics.

a Select all the triangles that are similar to

b Which of the following angles is a right angle?

c Which of the following are right triangles?

d Find the value of

by using the Pythagorean Theorem.

Challenge

Investigating a Side of a Right Triangle

The Leaning Tower of Pisa has a tilt of degrees. Once, a worker maintaining it accidentally dropped a hammer from the top. The hammer landed meters away from the base of the tower. Luckily, it did not hurt anyone!

Error loading image.

What was the vertical distance traveled by the hammer?

Explore

Comparing Ratios of Sides in Similar Right Triangles

Similar triangles have congruent angles and proportional sides. In the following applet, some of the ratios of the side lengths of two similar right triangles are compared.

What conclusion can be made about the ratios of the side lengths of two similar triangles?

Discussion

Analyzing Ratios of Sides in Similar Right Triangles

Because all right angles are congruent, all right triangles have one pair of congruent angles. If they also have one pair of congruent acute angles, then the triangles have two pairs of congruent angles. Therefore, by the Angle-Angle Similarity Theorem, two triangles with one pair of congruent acute angles are similar.

Since corresponding sides of similar polygons are proportional, the ratios between corresponding sides of similar right triangles are the same.

Discussion

Trigonometric Ratios

The ratios between side lengths of right triangles depend on the acute angles of the triangle. Some of these ratios receive a special name.

A trigonometric ratio relates two side lengths of a right triangle. Consider the right triangle One of its acute angles has been named

Since it is opposite to the right angle, is the hypotenuse of the right triangle. The remaining sides — the legs — can be named relative to the marked angle Because is next to it is called the adjacent side. Conversely, because lies across from it is called the opposite side.

The names of the three main ratios between side lengths are stated in the following table.

Name	Definition	Notation
Sine of
Cosine of
Tangent of

Example

Understanding Trigonometric Ratios in Right Triangles

Dominika is helping Tadeo understand trigonometric ratios. She drew three right triangles for him to write trigonometric ratios with respect to the acute angle Help Tadeo grasp this topic by selecting the correct answers!

a

b

c

Hint

a Identify the hypotenuse of the right triangle and the opposite side to

b Identify the hypotenuse of the right triangle and the adjacent side to

c Identify the opposite and adjacent sides to

Solution

a The sine of

is defined as the ratio of the length of the side

to the

of the right triangle. Therefore, Tadeo needs to identify these two sides.

It can be seen above that the hypotenuse of the right triangle is

and that the length of the opposite side to

is

With this information, the sine of

can be written.

b The cosine of

is the ratio of the length of the

side to

to the

of the right triangle. Therefore, Tadeo needs to identify these two sides.

The hypotenuse of the right triangle is

and the length of the adjacent side to

is

With this information, the cosine of

can be written.

c The tangent of

is the ratio of the length of the side

to the length of the side

Therefore, Tadeo needs to identify these two sides.

It can be seen above that the lengths of the opposite and adjacent sides to

are

and

respectively. With this information, the tangent of

can be written.

Example

Explaining Trigonometric Ratios in Right Triangles

Despite the awesome explanations Dominika provided, Tadeo still does not get how to find trigonometric ratios. To help his friend, Dominika thought of one more exercise.

This time, Dominika drew one right triangle and stated its three side lengths. She also labeled one of the triangle's acute angles.

Tadeo has now been asked to find the sine, cosine, and tangent of

Help him find the answers!

Hint

Start by identifying the hypotenuse of the right triangle. Then identify the opposite and adjacent sides to Finally, recall the definitions of sine, cosine, and tangent of an acute angle of a right triangle.

Solution

To find the sine, cosine, and tangent of

the

of the right triangle needs to be identified. The

and

sides to the angle also need to be identified.

It can be seen above that the hypotenuse is

and that the lengths of the opposite and adjacent sides to

are

and

respectively. This information can be substituted into the definitions for sine, cosine, and tangent.

Definition	Substitute

Example

Using Trigonometric Ratios to Find Side Lengths

Tadeo finally understands the topic! But wait, Dominika wants to level up and has let him know that trigonometric ratios can also be used to find missing side lengths of a right triangle. "Tell me more," Tadeo responds. An acute angle and the hypotenuse of a right triangle are given. To see whether Tadeo masters this topic, Dominika asked him to find the value of which is the length of the opposite side to the given angle.

Help Tadeo find the value of

If necessary, round the answer to three significant figures.

Hint

Identify the trigonometric ratio that should be used according to the given and desired lengths. Then, with the help of a calculator, set and solve an equation.

Solution

The of the right triangle is given, and the length of the side to the given is to be found.

The trigonometric ratio that relates these two sides and the acute angle is the sine ratio.

Finally, the equation can be solved for

▼

Solve for

MultEqn

FracMultDenomToNumber

Multiply

RearrangeEqn

Rearrange equation

To find the value of

a calculator can be used. First, it must be set to degree mode. This is done by pushing

and selecting Degree in the third row.

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Next the value of can be calculated by pushing followed by the angle measure.

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Now the value of

can be calculated.

▼

UseCalc

Use a calculator

Multiply

RoundSigDig

Round to significant digit(s)

Pop Quiz

Practice Finding Side Lengths Using Trigonometric Ratios

In the right triangles below, one acute angle and one side length are given. By using the corresponding trigonometric ratio, find the length of the side labeled Round the answer to one decimal place.

Discussion

Pythagorean Identities

By using trigonometric ratios, an important property of angles can be derived.

For any angle the following trigonometric identities hold true.

Proof

For Acute Angles

Consider a right triangle with a hypotenuse of

By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to

can be expressed in terms of the angle.

	Definition	Substitute	Simplify

It can be seen that if the hypotenuse of a right triangle is the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.

By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of and is equal to the square of

Therefore, for any acute angle

the sum of the squares of its sine and cosine equals

This property is also valid for any angle. The proof for angles whose measure is greater than or equal to

will be seen later in this course.

Example

Using Trigonometry to Determine the Cosine of an Angle

The property seen before can be used, among other things, to find the sine or cosine ratio of an acute angle in a right triangle.

Kriz and his friends plan to spend Saturday afternoon playing video games. To optimize the space, they decide to tidy up the basement to ensure the console, snacks, and beverages are placed in the form of a right triangle. Kriz decides to set the snacks and the beverages

and

meters away from the console, respectively.

The adjacent side to

connects the snacks and the beverages. To guarantee a good flow between the snacks and the beverages, Kriz wants to find the cosine of

Help the gang have a good time by finding

for them!

Hint

The hypotenuse of the right triangle is and the measure of the opposite side to is With this information, the sine ratio can be found.

Solution

The hypotenuse of the right triangle is

and the measure of the opposite side to

is

With this information, the sine ratio can be found.

This value can be substituted in the equation

Substitute

▼

Solve for

Rewrite as

Subtract fractions

Note that, when solving the equation for

only the principal root was considered. The reason is that the cosine of

is the ratio between two side lengths, and side lengths are always positive. Therefore, the quotient is also positive.

Explore

Using Inverse Trigonometric Ratios

Trigonometric ratios can also be used to find missing angles. Consider a right triangle where the hypotenuse and a leg are given.

Suppose now that the measure of

is desired. Note that, apart from the

the side whose length is known is

to

The trigonometric ratio that relates the

and the

side to an acute angle in a right triangle is the sine ratio.

To find the measure of

the inverse of the sine ratio could be used.

Finally, to find the value of

and therefore the measure of

a calculator will help. In the following example, it will be shown how to use a calculator to find the value of an inverse trigonometric ratio.

Example

Calculating Angles of Right Triangles

Previously, it was said that apart from being useful to find side lengths of a right triangle, trigonometric ratios can also be used to find missing angle measures.

Before playing video games with his friends, Kriz wants to finish his math homework to have a care-free weekend. He wants to find the measure of an acute angle in three different right triangles. By using the corresponding trigonometric ratios, help Kriz find in each triangle. Round the answer to the nearest degree.

a

b

c

Hint

a The lengths of the adjacent and opposite sides to

are

and

respectively.

b The hypotenuse of the right triangle is

and the length of the adjacent side to

is

c The hypotenuse of the right triangle is

and the length of the opposite side to

is

Solution

a In the given diagram, it can be seen that the lengths of the opposite and adjacent sides to

are

and

respectively. The trigonometric ratio that relates these two sides is the tangent ratio.

To solve this equation, the inverse of the tangent function could be used.

To find the value of

a calculator should to be used. First, the calculator must be set in degree mode. This is done by pushing

and selecting Degree in the third row.

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Next the value of can be calculated by pushing followed by and

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Thereofre,

b In the given diagram, it is shown that the length of the adjacent side to

is

and that the hypotenuse of the right triangle is

The trigonometric ratio that relates these two sides is the cosine ratio.

To solve this equation, the inverse of the cosine function is needed.

To find the value of

a calculator should be used. Just like before, the calculator must be set in degree mode. This is done by pushing

and selecting Degree in the third row.

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Next the value of is calculated by pushing followed by and

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It was found that

c In the diagram, it can be seen that the hypotenuse of the right triangle is

and that the length of the opposite side to

is

The trigonometric ratio that relates these two sides is the sine ratio.

To solve this equation, the inverse of the sine function can be used.

To find the value of

a calculator should be used. Just like in Parts A and B, the calculator must be set in degree mode by pushing

and selecting Degree in the third row.

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Next the value of can be calculated by pushing followed by and

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Thereofre,

Pop Quiz

Practice Finding Angles Using Trigonometric Ratios

In the following right triangles, two side lengths are given. By using the corresponding trigonometric ratio, find Round the answer to nearest degree.

Discussion

Reciprocal Trigonometric Ratios

Apart from the sine, cosine, and tangent ratios, there are three other trigonometric ratios that are worth mentioning.

Consider the right triangle

The so called reciprocal ratios are written in the next table.

Name	Definition	Notation
Cosecant of
Secant of
Cotangent of

These ratios can be defined in terms of sine, cosine, and tangent.

Rule

Reciprocal Identities

The trigonometric ratios cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.

Proof

Consider a right triangle with the three sides labeled with respect to an acute angle

Next, the sine, cosine, tangent, cosecant, secant, and cotangent ratios are written.

The reciprocal of the sine ratio will now be calculated.

▼

Solve for

MultEqn

RearrangeEqn

Rearrange equation

It has been found that

which is the reciprocal of

is equal to

By the definition, the cosecant of

is also the ratio of the lengths of the hypotenuse and the opposite side to

Therefore, by the Transitive Property of Equality,

is equal to

By following a similar procedure, the other two identities for secant and cotangent can be proven.

Explore

Finding Reciprocal Identities

If the sine, cosine, and tangent ratios are known, then their reciprocals cosecant, secant, and cotangent can be calculated without too much effort.

LaShay is really good at her favorite subject, Geometry. She has been appointed by Jefferson High's principal to do some tutoring for some of her classmates after school. To do so, she drew a right triangle. She then asked her peers to find all six trigonometric ratios with respect to the marked angle

Help LaShay's classmates find the trigonometric ratios!

Hint

Identify the hypotenuse of the right triangle and the opposite and adjacent sides to

Solution

The of the right triangle and the and sides to will be identified.

It can be seen that the hypotenuse is

and the lengths of the opposite and adjacent sides to

are

and

respectively. With this information, the sine, cosine, and tangent ratios can be found.

The reciprocals of the above ratios are the cosecant, secant, and cotangent of

Closure

Calculating a Side of a Right Triangle

With the topics learned in this lesson, the challenge presented at the onset can now be solved. Previously, it was learned that the Leaning Tower of Pisa has a tilt of degrees. The hammer dropped by the worker landed meters away from the base of the tower.

Error loading image.

With the given information, the vertical distance traveled by the hammer can be calculated. Write the answer to the nearest tenth.

Hint

Draw a right triangle and identify the given information.

Solution

If the tower has a tilt of

then the acute angle formed by the ground and the tower itself is the difference between

and

A right triangle can be drawn with an acute angle whose measure is

Furthermore, the length of the

side to this angle is

meters, and the length of its

side is unknown. The trigonometric ratio that relates an angle of a right triangle with its

and

sides is the tangent ratio.

This equation can be solved for

which is the vertical distance traveled by the hammer.

▼

Solve for

Rearrange equation

Use a calculator

Round to decimal place(s)

The distance traveled by the hammer is about

meters.

Edit Lesson

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution

Proof

Hint

Solution

Hint

Solution

Proof

Hint

Solution

Hint

Solution