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

.

The hypotenuse c, opposite the right angle, is the longest side. The legs are shorter and may be the same or different lengths, and are perpendicular to each other in a right triangle. 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.

image 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=adjacenthypotenuse\displaystyle{\cos{t} = \frac {adjacent}{hypotenuse}}

    cos

    t

    =

    h

    y

    p

    o

    t

    e

    n

    u

    se

    a

    d

    ja

    ce

    n

    t

    , and

    tan⁡t=oppositeadjacent\displaystyle{\tan{t} = \frac {opposite}{adjacent}}

    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.

image 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=adjacenthypotenuse\displaystyle{\cos{t} = \frac {adjacent}{hypotenuse}}

    cos

    t

    =

    h

    y

    p

    o

    t

    e

    n

    u

    se

    a

    d

    ja

    ce

    n

    t

  • Tangent    

    tan⁡t=oppositeadjacent\displaystyle{\tan{t} = \frac {opposite}{adjacent}}

    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):

image 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):

image 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.

cos⁡t=adjacenthypotenusecos⁡(83∘)=300xx⋅cos⁡(83∘)=300x=300cos⁡(83∘)x=300(0.1218… )x=2461.7 feet\displaystyle{
\begin{align}
\cos{t} &= \frac {adjacent}{hypotenuse} \\
\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=adjacenthypotenuse\displaystyle{ \cos{t} = \frac {adjacent}{hypotenuse} }

    cos

    t

    =

    h

    y

    p

    o

    t

    e

    n

    u

    se

    a

    d

    ja

    ce

    n

    t

  • Tangent      

    tan⁡t=oppositeadjacent\displaystyle{ \tan{t} = \frac {opposite}{adjacent} }

    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).”

image 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

and an adjacent side of

4545

45

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

image 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.

tan⁡t=oppositeadjacenttan⁡(62∘)=x4545⋅tan⁡(62∘)=xx=45⋅tan⁡(62∘)x=45⋅(1.8807… )x=84.6\displaystyle{
\begin{align}
\tan{t} &= \frac {opposite}{adjacent} \\
\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)

image 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(adjacenthypotenuse)\displaystyle{ A^{\circ} = \cos^{-1}{ \left( \frac {\text{adjacent}}{\text{hypotenuse}} \right) }
}

A

=

cos

1

(

hypotenuse

adjacent

)

A∘=tan⁡−1(oppositeadjacent)\displaystyle{ A^{\circ} = \tan^{-1}{\left(\frac {\text{opposite}}{\text{adjacent}} \right) }}

A

=

tan

1

(

adjacent

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:

image 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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Licenses and Attributions

The Pythagorean Theorem, can be used to find the distance of any side of a right triangle.A correct fish has a measure of 90 degrees ( ). A right triangle is a triangle in which one angle is a justly angle. The relation between the sides and angles of a veracious triangulum is the footing for trigonometry.The side opposite the correct angle is called the hypotenuse ( sidein the visualize ). The sides adjacent to the right angle are called legs ( sidesand ). Sidemay be identified as the side adjacent to angleand opposed to ( or opposite ) slant. Sideis the side adjacent to angleand opposed to angleThe Pythagorean Theorem, besides 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 slant ) is equal to the summarize of the squares of the other two sides. The theorem can be written as an equality relating the lengths of the sidesand, much called the “ pythagorean equation ” : In this equation, represents the distance of the hypotenuse andandthe lengths of the triangulum ‘s early two sides.Although it is often said that the cognition of the theorem predates him, the theorem is named after the ancient greek mathematician Pythagoras ( c. 570 – c. 495 BC ). He is credited with its first gear recorded proof.Example 1 : A right triangulum has a side length offeet, and a hypotenuse distance offeet. Find the other side length. ( rung to the nearest tenth of a foot ) Substituteandinto the Pythagorean Theorem and solve forExample 2 : A right field triangulum has side lengthscm andcm. Find the duration of the hypotenuse.Substituteandinto the Pythagorean Theorem and solve forTrigonometric functions can be used to solve for missing side lengths in good triangles.We can define the trigonometric functions in terms an angleand the lengths of the sides of the triangle. The adjacent side is the side close to the angle. ( adjacent means “ adjacent to. ” ) The inverse side is the english across from the angle. The hypotenuse is the side of the triangulum opposite the right fish, and it is the longest.The trigonometric functions are adequate to ratios that refer certain side lengths of a right triangle. When solving for a miss side, the first step is to identify what sides and what slant are given, and then select the appropriate function to use to solve the problem.Sometimes you know the length of one side of a triangle and an lean, and need to find early measurements. Use one of the trigonometric functions ( ), identify the sides and angle given, set up the equation and use the calculator and algebra to find the miss side length.Example 1 : Given a right triangle with acute lean ofand a hypotenuse length offeet, find the length of the side opposite the acute slant ( circle to the nearest tenth ) : degrees. The proportion of the sides would be the opposite side and the hypotenuse. The ratio that relates those two sides is the sine function.The side opposite the acute fish isfeet.Example 2 : Given a right triangle with an acute slant ofand a hypotenuse duration offeet, find the hypotenuse length ( cycle to the nearest tenth ) : 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.The mnemonicSohCahToa can be used to solve for the length of a side of a good triangle.Given a right triangulum with an acuate angle of, the first base three trigonometric functions are : A coarse mnemonic for remembering these relationships is SohCahToa, formed from the first letters of “ Sine is face-to-face over hypotenuse ( Soh ), Cosine is adjacent over hypotenuse ( Cah ), Tangent is face-to-face over adjacent ( Toa ). ” exercise 1 : Given a right triangle with an acute lean ofand an adjacent side offeet, solve for the face-to-face side distance. ( turn to the nearest tenth ) degrees ). Remembering the mnemonic, “ SohCahToa ”, the sides given are opposite and adjacent or “ oxygen ” and “ a ”, which would use “ T ”, meaning the tangent trigonometric function.Example 2 : A ladder with a length ofis leaning against a building. The slant the ladder makes with the land is. How high up the build does the ladder reach ? ( round to the nearest tenth of a foot ) The inverse trigonometric functions can be used to find the acute accent angle measurement of a right triangle.Using the trigonometric functions to solve for a miss side when given an acute angle is deoxyadenosine monophosphate simple as identifying the sides in relation to the acute angle, choosing the right function, setting up the equality and resolution. Finding the missing acuate lean when given two sides of a right triangle is just equally simple.In order to solve for the missing acute fish, use the like three trigonometric functions, but, use the inverse keystone ( on the calculator ) to solve for the angle ( ) when given two sides.For a proper triangle with hypotenuse lengthand acuate anglewith opposition side length, find the acute fish to the nearest academic degree :, 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 .

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