# Trigonometry and Right Triangles

## Right Triangles and the Pythagorean Theorem

The Pythagorean Theorem,

a2+b2=c2,{\displaystyle a^{2}+b^{2}=c^{2},}

a

2

+

b

2

=

c

2

,

can be used to find the length of any side of a right triangle.

### Learning Objectives

Use the Pythagorean Theorem to find the distance of a side of a correct triangle

### Key Takeaways

#### Key Points

• The Pythagorean Theorem,

a2+b2=c2,{\displaystyle a^{2}+b^{2}=c^{2},}

a

2

+

b

2

=

c

2

,

is used to find the length of any side of a right triangle.

• In a right triangle, one of the angles has a value of 90 degrees.
• The longest side of a right triangle is called the hypotenuse, and it is the side that is opposite the 90 degree angle.
• If the length of the hypotenuse is labeled

cc

c

, and the lengths of the other sides are labeled

aa

a

and

bb

b

, the Pythagorean Theorem states that

a2+b2=c2{\displaystyle a^{2}+b^{2}=c^{2}}

a

2

+

b

2

=

c

2

.

#### Key Terms

• legs: The sides adjacent to the right angle in a right triangle.
• right triangle: A

33

3

-sided shape where one angle has a value of

9090

90

degrees

• hypotenuse: The side opposite the right angle of a triangle, and the longest side of a right triangle.
• Pythagorean theorem: The sum of the areas of the two squares on the legs (

aa

a

and

bb

b

) is equal to the area of the square on the hypotenuse (

cc

c

). The formula is

a2+b2=c2a^2+b^2=c^2

a

2

+

b

2

=

c

2

.

### Right Triangle

A right angle has a value of 90 degrees (

90∘90^\circ

9

0

). A right triangle is a triangle in which one angle is a right angle. The relation between the sides and angles of a right triangle is the basis for trigonometry.

The side opposite the right angle is called the hypotenuse (side

cc

c

in the figure). The sides adjacent to the right angle are called legs (sides

aa

a

and

bb

b

). Side

aa

a

may be identified as the side adjacent to angle

BB

B

and opposed to (or opposite) angle

AA

A

. Side

bb

b

is the side adjacent to angle

AA

A

and opposed to angle

BB

B

.

Right triangle: The Pythagorean Theorem can be used to find the value of a missing side length in a right triangle. The Pythagorean Theorem can be used to find the value of a miss side distance in a right triangle .

### The Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras’ Theorem, is a fundamental relation in Euclidean geometry. It defines the relationship among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides

aa

a

,

bb

b

and

cc

c

, often called the “Pythagorean equation”:[1]

a2+b2=c2{\displaystyle a^{2}+b^{2}=c^{2}}

a

2

+

b

2

=

c

2

In this equation,

cc

c

represents the length of the hypotenuse and

aa

a

and

bb

b

the lengths of the triangle’s other two sides.

Although it is often said that the knowledge of the theorem predates him,[2] the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC). He is credited with its first recorded proof.

The Pythagorean Theorem: The sum of the areas of the two squares on the legs (

aa

a

and

bb

b

) is equal to the area of the square on the hypotenuse (

cc

c

).  The formula is

a2+b2=c2a^2+b^2=c^2

a

2

+

b

2

=

c

2

. The sum of the areas of the two squares on the leg ( and ) is equal to the area of the square on the hypotenuse ( ). The formula is

Example 1:  A right triangle has a side length of

1010

10

feet, and a hypotenuse length of

2020

20

feet.  Find the other side length.  (round to the nearest tenth of a foot)

Substitute

a=10a=10

a

=

10

and

c=20c=20

c

=

20

into the Pythagorean Theorem and solve for

bb

b

.

a2+b2=c2(10)2+b2=(20)2100+b2=400b2=300b2=300b=17.3 feet\displaystyle{
\begin{align}
a^{2}+b^{2} &=c^{2} \\
(10)^2+b^2 &=(20)^2 \\
100+b^2 &=400 \\
b^2 &=300 \\
\sqrt{b^2} &=\sqrt{300} \\
b &=17.3 ~\mathrm{feet}
\end{align}
}

a

2

+

b

2

(

10

)

2

+

b

2

100

+

b

2

b

2

b

2

b

=

c

2

=

(

20

)

2

=

400

=

300

=

300

=

17.3

feet

Example 2:  A right triangle has side lengths

33

3

cm and

44

4

cm.  Find the length of the hypotenuse.

Substitute

a=3a=3

a

=

3

and

b=4b=4

b

=

4

into the Pythagorean Theorem and solve for

cc

c

.

a2+b2=c232+42=c29+16=c225=c2c2=25c2=25c=5 cm\displaystyle{
\begin{align}
a^{2}+b^{2} &=c^{2} \\
3^2+4^2 &=c^2 \\
9+16 &=c^2 \\
25 &=c^2\\
c^2 &=25 \\
\sqrt{c^2} &=\sqrt{25} \\
c &=5~\mathrm{cm}
\end{align}
}

a

2

+

b

2

3

2

+

4

2

9

+

16

25

c

2

c

2

c

=

c

2

=

c

2

=

c

2

=

c

2

=

25

=

25

=

5

cm

## How Trigonometric Functions Work

Trigonometric functions can be used to solve for missing side lengths in right triangles.

### Learning Objectives

Recognize how trigonometric functions are used for solving problems about right triangles, and identify their inputs and outputs

### Key Takeaways

#### Key Points

• A right triangle has one angle with a value of 90 degrees (

90∘90^{\circ}

9

0

)The three trigonometric functions most often used to solve for a missing side of a right triangle are:

sin⁡t=oppositehypotenuse\displaystyle{\sin{t}=\frac {opposite}{hypotenuse}}

sin

t

=

h

y

p

o

t

e

n

u

se

o

pp

os

i

t

e

,

cos

t

=

h

y

p

o

t

e

n

u

se

a

d

ja

ce

n

t

, and

tan

t

=

a

d

ja

ce

n

t

o

pp

os

i

t

e

### Trigonometric Functions

We can define the trigonometric functions in terms an angle

tt

t

and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle. (Adjacent means “next to.”) The opposite side is the side across from the angle. The hypotenuse  is the side of the triangle opposite the right angle, and it is the longest.

Right triangle: The sides of a right triangle in relation to angle

tt

t

. The sides of a correct triangle in relation to angle

• Sine

sin⁡t=oppositehypotenuse\displaystyle{\sin{t} = \frac {opposite}{hypotenuse}}

sin

t

=

h

y

p

o

t

e

n

u

se

o

pp

os

i

t

e

• Cosine

cos

t

=

h

y

p

o

t

e

n

u

se

a

d

ja

ce

n

t

• Tangent

tan

t

=

a

d

ja

ce

n

t

o

pp

os

i

t

e

The trigonometric functions are equal to ratios that relate certain side lengths of a  right triangle.  When solving for a missing side, the first step is to identify what sides and what angle are given, and then select the appropriate function to use to solve the problem.

### Evaluating a Trigonometric Function of a Right Triangle

Sometimes you know the length of one side of a triangle and an angle, and need to find other measurements.  Use one of the trigonometric functions (

sin⁡\sin{}

sin

,

cos⁡\cos{}

cos

,

tan⁡\tan{}

tan

), identify the sides and angle given, set up the equation and use the calculator and algebra to find the missing side length.

Example 1:

Given a right triangle with acute angle of

34∘34^{\circ}

3

4

and a hypotenuse length of

2525

25

feet, find the length of the side opposite the acute angle (round to the nearest tenth):

Right triangle: Given a right triangle with acute angle of

3434

34

degrees and a hypotenuse length of

2525

25

feet, find the opposite side length. Given a right triangle with acute lean ofdegrees and a hypotenuse length offeet, find the reverse side duration .

3434

34

degrees.  The ratio of the sides would be the opposite side and the hypotenuse.  The ratio that relates those two sides is the sine function.

sin⁡t=oppositehypotenusesin⁡(34∘)=x2525⋅sin⁡(34∘)=xx=25⋅sin⁡(34∘)x=25⋅(0.559… )x=14.0\displaystyle{
\begin{align}
\sin{t} &=\frac {opposite}{hypotenuse} \\
\sin{\left(34^{\circ}\right)} &=\frac{x}{25} \\
25\cdot \sin{ \left(34^{\circ}\right)} &=x\\
x &= 25\cdot \sin{ \left(34^{\circ}\right)}\\
x &= 25 \cdot \left(0.559\dots\right)\\
x &=14.0
\end{align}
}

sin

t

sin

(

3

4

)

25

sin

(

3

4

)

x

x

x

=

h

y

p

o

t

e

n

u

se

o

pp

os

i

t

e

=

25

x

=

x

=

25

sin

(

3

4

)

=

25

(

0.559

)

=

14.0

The side opposite the acute angle is

14.014.0

14.0

feet.

Example 2:

Given a right triangle with an acute angle of

83∘83^{\circ}

8

3

and a hypotenuse length of

300300

300

feet, find the hypotenuse length (round to the nearest tenth):

Right Triangle: Given a right triangle with an acute angle of

8383

83

degrees and a hypotenuse length of

300300

300

feet, find the hypotenuse length. Given a correct triangle with an acute angle ofdegrees and a hypotenuse distance offeet, find the hypotenuse length .

8383

83

degrees. The ratio of the sides would be the adjacent side and the hypotenuse.  The ratio that relates these two sides is the cosine function.

\begin{align}
\cos{ \left( 83 ^{\circ}\right)} &= \frac {300}{x} \\
x \cdot \cos{\left(83^{\circ}\right)} &=300 \\
x &=\frac{300}{\cos{\left(83^{\circ}\right)}} \\
x &= \frac{300}{\left(0.1218\dots\right)} \\
x &=2461.7~\mathrm{feet}
\end{align}
}

cos

t

cos

(

8

3

)

x

cos

(

8

3

)

x

x

x

=

h

y

p

o

t

e

n

u

se

a

d

ja

ce

n

t

=

x

300

=

300

=

cos

(

8

3

)

300

=

(

0.1218

)

300

=

2461.7

feet

## Sine, Cosine, and Tangent

The mnemonic

SohCahToa can be used to solve for the length of a side of a right triangle.

### Learning Objectives

Use the acronym SohCahToa to define Sine, Cosine, and Tangent in terms of right triangles

### Key Takeaways

#### Key Points

• A common mnemonic for remembering the relationships between the Sine, Cosine, and Tangent functions is SohCahToa.
• SohCahToa is formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”

### Definitions of Trigonometric Functions

Given a right triangle with an acute angle of

tt

t

, the first three trigonometric functions are:

• Sine

sin⁡t=oppositehypotenuse\displaystyle{ \sin{t} = \frac {opposite}{hypotenuse} }

sin

t

=

h

y

p

o

t

e

n

u

se

o

pp

os

i

t

e

• Cosine

cos

t

=

h

y

p

o

t

e

n

u

se

a

d

ja

ce

n

t

• Tangent

tan

t

=

a

d

ja

ce

n

t

o

pp

os

i

t

e

A common mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa).”

Right triangle: The sides of a right triangle in relation to angle

tt

t

. The hypotenuse is the long side, the opposite side is across from angle

tt

t

, and the adjacent side is next to angle

tt

t

. right triangle : The sides of a right triangle in relation back to angle. The hypotenuse is the long side, the opposite side is across from angle, and the adjacent side is next to angle

Example 1:

Given a right triangle with an acute angle of

62∘62^{\circ}

6

2

4545

45

feet, solve for the opposite side length. (round to the nearest tenth)

Right triangle: Given a right triangle with an acute angle of

6262

62

degrees and an adjacent side of

4545

45

feet, solve for the opposite side length. Given a right triangle with an acute angle ofdegrees and an adjacent side offeet, solve for the face-to-face side length .

6262

62

degrees).  Remembering the mnemonic, “SohCahToa”, the sides given are opposite and adjacent or “o” and “a”, which would use “T”, meaning the tangent trigonometric function.

\begin{align}
\tan{\left(62^{\circ}\right)} &=\frac{x}{45} \\
45\cdot \tan{\left(62^{\circ}\right)} &=x \\
x &= 45\cdot \tan{\left(62^{\circ}\right)}\\
x &= 45\cdot \left( 1.8807\dots \right) \\
x &=84.6
\end{align}
}

tan

t

tan

(

6

2

)

45

tan

(

6

2

)

x

x

x

=

a

d

ja

ce

n

t

o

pp

os

i

t

e

=

45

x

=

x

=

45

tan

(

6

2

)

=

45

(

1.8807

)

=

84.6

Example 2:  A ladder with a length of

30 feet30~\mathrm{feet}

30

feet

is leaning against a building.  The angle the ladder makes with the ground is

32∘32^{\circ}

3

2

.  How high up the building does the ladder reach? (round to the nearest tenth of a foot)

Right triangle: After sketching a picture of the problem, we have the triangle shown. The angle given is

32∘32^\circ

3

2

, the hypotenuse is 30 feet, and the missing side length is the opposite leg,

xx

x

feet. right triangle : After sketching a picture of the problem, we have the triangle shown. The angle given is, the hypotenuse is 30 feet, and the miss side length is the opposite stage, feet .

sin⁡t=oppositehypotenusesin⁡(32∘)=x3030⋅sin⁡(32∘)=xx=30⋅sin⁡(32∘)x=30⋅(0.5299… )x=15.9 feet\displaystyle{
\begin{align}
\sin{t} &= \frac {opposite}{hypotenuse} \\
\sin{ \left( 32^{\circ} \right) } & =\frac{x}{30} \\
30\cdot \sin{ \left(32^{\circ}\right)} &=x \\
x &= 30\cdot \sin{ \left(32^{\circ}\right)}\\
x &= 30\cdot \left( 0.5299\dots \right) \\
x &= 15.9 ~\mathrm{feet}
\end{align}
}

sin

t

sin

(

3

2

)

30

sin

(

3

2

)

x

x

x

=

h

y

p

o

t

e

n

u

se

o

pp

os

i

t

e

=

30

x

=

x

=

30

sin

(

3

2

)

=

30

(

0.5299

)

=

15.9

feet

## Finding Angles From Ratios: Inverse Trigonometric Functions

The inverse trigonometric functions can be used to find the acute angle measurement of a right triangle.

### Learning Objectives

Use inverse trigonometric functions in solving problems involving properly triangles

### Key Takeaways

#### Key Points

• A missing acute angle value of a right triangle can be found when given two side lengths.
• To find a missing angle value, use the trigonometric functions sine, cosine, or tangent, and the inverse key on a calculator to apply the inverse function (

arcsin⁡\arcsin{}

arcsin

,

arccos⁡\arccos{}

arccos

,

arctan⁡\arctan{}

arctan

),

sin⁡−1\sin^{-1}

sin

1

,

cos⁡−1\cos^{-1}

cos

1

,

tan⁡−1\tan^{-1}

tan

1

.

### Inverse Trigonometric Functions

In order to solve for the missing acute angle, use the same three trigonometric functions, but, use the inverse key (

−1^{-1}

1

on the calculator) to solve for the angle (

AA

A

) when given two sides.

A∘=sin⁡−1(oppositehypotenuse)\displaystyle{ A^{\circ} = \sin^{-1}{ \left( \frac {\text{opposite}}{\text{hypotenuse}} \right) } }

A

=

sin

1

(

hypotenuse

opposite

)

}

A

=

cos

1

(

hypotenuse

)

A

=

tan

1

(

opposite

)

### Example

For a right triangle with hypotenuse length

25 feet25~\mathrm{feet}

25

feet

and acute angle

A∘A^\circ

A

with opposite side length

12 feet12~\mathrm{feet}

12

feet

, find the acute angle to the nearest degree:

Right triangle: Find the measure of angle

AA

A

, when given the opposite side and hypotenuse. Find the measure of angle, when given the diametric side and hypotenuse .

AA

A

, the sides opposite and hypotenuse are given.  Therefore, use the sine trigonometric function. (Soh from SohCahToa)  Write the equation and solve using the inverse key for sine.

sin⁡A∘=oppositehypotenusesin⁡A∘=1225A∘=sin⁡−1(1225)A∘=sin⁡−1(0.48)A=29∘\displaystyle{
\begin{align}
\sin{A^{\circ}} &= \frac {\text{opposite}}{\text{hypotenuse}} \\
\sin{A^{\circ}} &= \frac{12}{25} \\
A^{\circ} &= \sin^{-1}{\left( \frac{12}{25} \right)} \\
A^{\circ} &= \sin^{-1}{\left( 0.48 \right)} \\
A &=29^{\circ}
\end{align}
}

sin

A

sin

A

A

A

A

=

hypotenuse

opposite

=

25

12

=

sin

1

(

25

12

)

=

sin

1

(

0.48

)

=

2

9

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