How To Find The Hypotenuse Of A Right Triangle

Correct Triangles and the Pythagorean Theorem

The Pythagorean Theorem, [latex]{\displaystyle a^{2}+b^{2}=c^{2},}[/latex] tin exist used to detect the length of whatever side of a right triangle.

Learning Objectives

Use the Pythagorean Theorem to find the length of a side of a right triangle

Key Takeaways

Key Points

• The Pythagorean Theorem, [latex]{\displaystyle a^{two}+b^{2}=c^{2},}[/latex] is used to find the length of whatever side of a right triangle.
• In a correct 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 contrary the 90 degree bending.
• If the length of the hypotenuse is labeled [latex]c[/latex], and the lengths of the other sides are labeled [latex]a[/latex] and [latex]b[/latex], the Pythagorean Theorem states that [latex]{\displaystyle a^{2}+b^{ii}=c^{2}}[/latex].

Cardinal Terms

• legs: The sides next to the right angle in a correct triangle.
• right triangle: A [latex]3[/latex]-sided shape where one bending has a value of [latex]90[/latex] degrees
• hypotenuse: The side opposite the right bending of a triangle, and the longest side of a correct triangle.
• Pythagorean theorem: The sum of the areas of the 2 squares on the legs ([latex]a[/latex] and [latex]b[/latex]) is equal to the area of the square on the hypotenuse ([latex]c[/latex]). The formula is [latex]a^two+b^2=c^2[/latex].

Right Triangle

A right angle has a value of xc degrees ([latex]ninety^\circ[/latex]). A right triangle is a triangle in which one bending is a right angle. The relation betwixt the sides and angles of a right triangle is the basis for trigonometry.

The side opposite the correct bending is called the hypotenuse (side [latex]c[/latex] in the figure). The sides adjacent to the right bending are chosen legs (sides [latex]a[/latex] and [latex]b[/latex]). Side [latex]a[/latex] may be identified equally the side side by side to angle [latex]B[/latex] and opposed to (or opposite) angle [latex]A[/latex]. Side [latex]b[/latex] is the side adjacent to angle [latex]A[/latex] and opposed to angle [latex]B[/latex].

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

If the lengths of all 3 sides of a correct triangle are whole numbers, the triangle is said to exist a Pythagorean triangle and its side lengths are collectively known as a Pythagorean triple.

The Pythagorean Theorem

The Pythagorean Theorem, likewise known as Pythagoras' Theorem, is a key 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 contrary 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 [latex]a[/latex], [latex]b[/latex] and [latex]c[/latex], oftentimes called the "Pythagorean equation":^[1]

[latex]{\displaystyle a^{2}+b^{2}=c^{2}} [/latex]

In this equation, [latex]c[/latex] represents the length of the hypotenuse and [latex]a[/latex] and [latex]b[/latex] the lengths of the triangle'south other ii sides.

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

The Pythagorean Theorem: The sum of the areas of the two squares on the legs ([latex]a[/latex] and [latex]b[/latex]) is equal to the surface area of the square on the hypotenuse ([latex]c[/latex]).  The formula is [latex]a^two+b^2=c^2[/latex].

Finding a Missing Side Length

Example i:  A right triangle has a side length of [latex]10[/latex] feet, and a hypotenuse length of [latex]20[/latex] feet.  Find the other side length.  (circular to the nearest tenth of a foot)

Substitute [latex]a=10[/latex] and [latex]c=20[/latex] into the Pythagorean Theorem and solve for [latex]b[/latex].

[latex]\displaystyle{ \begin{align} a^{2}+b^{2} &=c^{ii} \\ (x)^two+b^2 &=(xx)^2 \\ 100+b^2 &=400 \\ b^2 &=300 \\ \sqrt{b^2} &=\sqrt{300} \\ b &=17.3 ~\mathrm{feet} \end{align} }[/latex]

Example 2:  A right triangle has side lengths [latex]3[/latex] cm and [latex]4[/latex] cm.  Detect the length of the hypotenuse.

Substitute [latex]a=3[/latex] and [latex]b=iv[/latex] into the Pythagorean Theorem and solve for [latex]c[/latex].

[latex]\displaystyle{ \begin{align} a^{2}+b^{2} &=c^{2} \\ 3^ii+4^2 &=c^two \\ 9+xvi &=c^2 \\ 25 &=c^2\\ c^2 &=25 \\ \sqrt{c^ii} &=\sqrt{25} \\ c &=5~\mathrm{cm} \end{marshal} }[/latex]

How Trigonometric Functions Work

Trigonometric functions tin exist used to solve for missing side lengths in right triangles.

Learning Objectives

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

Fundamental Takeaways

Key Points

• A correct triangle has one angle with a value of 90 degrees ([latex]90^{\circ}[/latex])The three trigonometric functions nearly oft used to solve for a missing side of a right triangle are: [latex]\displaystyle{\sin{t}=\frac {opposite}{hypotenuse}}[/latex], [latex]\displaystyle{\cos{t} = \frac {adjacent}{hypotenuse}}[/latex], and [latex]\displaystyle{\tan{t} = \frac {contrary}{adjacent}}[/latex]

Trigonometric Functions

We can define the trigonometric functions in terms an angle [latex]t[/latex] and the lengths of the sides of the triangle. The adjacent side is the side closest to the angle. (Next means "next to.") The reverse 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 [latex]t[/latex].

When solving for a missing side of a correct triangle, just the only given data is an acute angle measurement and a side length, utilize the trigonometric functions listed below:

• Sine           [latex]\displaystyle{\sin{t} = \frac {contrary}{hypotenuse}}[/latex]
• Cosine       [latex]\displaystyle{\cos{t} = \frac {side by side}{hypotenuse}}[/latex]
• Tangent    [latex]\displaystyle{\tan{t} = \frac {opposite}{adjacent}}[/latex]

The trigonometric functions are equal to ratios that chronicle sure side lengths of a  right triangle.  When solving for a missing side, the start stride is to place what sides and what angle are given, and then select the appropriate function to employ to solve the problem.

Evaluating a Trigonometric Function of a Correct Triangle

Sometimes you know the length of 1 side of a triangle and an angle, and need to detect other measurements.  Employ i of the trigonometric functions ([latex]\sin{}[/latex], [latex]\cos{}[/latex], [latex]\tan{}[/latex]), place the sides and angle given, set upwards the equation and use the reckoner and algebra to find the missing side length.

Example 1:
Given a right triangle with astute angle of [latex]34^{\circ}[/latex] and a hypotenuse length of [latex]25[/latex] feet, discover the length of the side reverse the acute angle (round to the nearest 10th):

Right triangle: Given a right triangle with acute angle of [latex]34[/latex] degrees and a hypotenuse length of [latex]25[/latex] feet, find the opposite side length.

Looking at the figure, solve for the side opposite the astute angle of [latex]34[/latex] 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.

[latex]\displaystyle{ \begin{align} \sin{t} &=\frac {reverse}{hypotenuse} \\ \sin{\left(34^{\circ}\right)} &=\frac{x}{25} \\ 25\cdot \sin{ \left(34^{\circ}\right)} &=10\\ x &= 25\cdot \sin{ \left(34^{\circ}\right)}\\ x &= 25 \cdot \left(0.559\dots\right)\\ x &=14.0 \end{align} }[/latex]

The side opposite the acute angle is [latex]fourteen.0[/latex] feet.

Example 2:
Given a right triangle with an acute angle of [latex]83^{\circ}[/latex] and a hypotenuse length of [latex]300[/latex] anxiety, notice the hypotenuse length (round to the nearest 10th):

Right Triangle: Given a right triangle with an acute angle of [latex]83[/latex] degrees and a hypotenuse length of [latex]300[/latex] feet, find the hypotenuse length.

Looking at the figure, solve for the hypotenuse to the astute angle of [latex]83[/latex] 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.

[latex]\displaystyle{ \brainstorm{marshal} \cos{t} &= \frac {adjacent}{hypotenuse} \\ \cos{ \left( 83 ^{\circ}\correct)} &= \frac {300}{ten} \\ ten \cdot \cos{\left(83^{\circ}\correct)} &=300 \\ x &=\frac{300}{\cos{\left(83^{\circ}\right)}} \\ ten &= \frac{300}{\left(0.1218\dots\right)} \\ 10 &=2461.7~\mathrm{feet} \stop{align} }[/latex]

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

Cardinal Points

• A mutual mnemonic for remembering the relationships betwixt the Sine, Cosine, and Tangent functions is SohCahToa.
• SohCahToa is formed from the commencement messages of "Southwardine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over side by side (Toa)."

Definitions of Trigonometric Functions

Given a right triangle with an acute angle of [latex]t[/latex], the outset iii trigonometric functions are:

• Sine             [latex]\displaystyle{ \sin{t} = \frac {reverse}{hypotenuse} }[/latex]
• Cosine        [latex]\displaystyle{ \cos{t} = \frac {adjacent}{hypotenuse} }[/latex]
• Tangent      [latex]\displaystyle{ \tan{t} = \frac {reverse}{adjacent} }[/latex]

A mutual mnemonic for remembering these relationships is SohCahToa, formed from the kickoff messages of "Sine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over adjacent (Toa)."

Correct triangle: The sides of a right triangle in relation to bending [latex]t[/latex]. The hypotenuse is the long side, the opposite side is beyond from angle [latex]t[/latex], and the side by side side is next to angle [latex]t[/latex].

Evaluating a Trigonometric Function of a Right Triangle

Example ane:
Given a right triangle with an acute angle of [latex]62^{\circ}[/latex] and an adjacent side of [latex]45[/latex] anxiety, solve for the reverse side length. (round to the nearest tenth)

Correct triangle: Given a right triangle with an astute angle of [latex]62[/latex] degrees and an adjacent side of [latex]45[/latex] feet, solve for the opposite side length.

First, determine which trigonometric function to use when given an adjacent side, and you demand to solve for the opposite side.  Always decide which side is given and which side is unknown from the acute angle ([latex]62[/latex] degrees).  Remembering the mnemonic, "SohCahToa", the sides given are opposite and adjacent or "o" and "a", which would utilize "T", significant the tangent trigonometric part.

[latex]\displaystyle{ \begin{align} \tan{t} &= \frac {opposite}{adjacent} \\ \tan{\left(62^{\circ}\correct)} &=\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 \correct) \\ x &=84.6 \end{align} }[/latex]

Example 2:  A ladder with a length of [latex]30~\mathrm{feet}[/latex] is leaning against a building.  The bending the ladder makes with the ground is [latex]32^{\circ}[/latex].  How high up the building does the ladder accomplish? (circular to the nearest 10th of a foot)

Right triangle: After sketching a picture of the trouble, nosotros have the triangle shown. The angle given is [latex]32^\circ[/latex], the hypotenuse is 30 anxiety, and the missing side length is the opposite leg, [latex]x[/latex] feet.

Determine which trigonometric part to use when given the hypotenuse, and you need to solve for the reverse side.  Remembering the mnemonic, "SohCahToa", the sides given are the hypotenuse and opposite or "h" and "o", which would use "S" or the sine trigonometric role.

[latex]\displaystyle{ \begin{align} \sin{t} &= \frac {opposite}{hypotenuse} \\ \sin{ \left( 32^{\circ} \right) } & =\frac{x}{30} \\ 30\cdot \sin{ \left(32^{\circ}\correct)} &=10 \\ x &= 30\cdot \sin{ \left(32^{\circ}\right)}\\ ten &= 30\cdot \left( 0.5299\dots \right) \\ x &= 15.9 ~\mathrm{feet} \end{align} }[/latex]

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

Utilize changed trigonometric functions in solving bug involving right triangles

Key Takeaways

Key Points

• A missing acute angle value of a correct triangle tin be found when given two side lengths.
• To find a missing angle value, use the trigonometric functions sine, cosine, or tangent, and the inverse fundamental on a calculator to employ the changed function ([latex]\arcsin{}[/latex], [latex]\arccos{}[/latex], [latex]\arctan{}[/latex]), [latex]\sin^{-1}[/latex], [latex]\cos^{-1}[/latex], [latex]\tan^{-1}[/latex].

Using the trigonometric functions to solve for a missing side when given an acute bending is as simple equally identifying the sides in relation to the acute angle, choosing the correct part, setting upwards the equation and solving.  Finding the missing acute bending when given two sides of a right triangle is just equally uncomplicated.

Inverse Trigonometric Functions

In order to solve for the missing astute angle, use the same iii trigonometric functions, but, utilise the inverse fundamental ([latex]^{-i}[/latex]on the calculator) to solve for the bending ([latex]A[/latex]) when given ii sides.

[latex]\displaystyle{ A^{\circ} = \sin^{-1}{ \left( \frac {\text{opposite}}{\text{hypotenuse}} \right) } }[/latex]

[latex]\displaystyle{ A^{\circ} = \cos^{-1}{ \left( \frac {\text{side by side}}{\text{hypotenuse}} \right) } }[/latex]

[latex]\displaystyle{ A^{\circ} = \tan^{-ane}{\left(\frac {\text{opposite}}{\text{adjacent}} \right) }}[/latex]

Example

For a right triangle with hypotenuse length [latex]25~\mathrm{anxiety}[/latex] and acute angle [latex]A^\circ[/latex]with opposite side length [latex]12~\mathrm{feet}[/latex], find the acute angle to the nearest degree:

Right triangle: Detect the measure of angle [latex]A[/latex], when given the opposite side and hypotenuse.

From angle [latex]A[/latex], the sides reverse and hypotenuse are given.  Therefore, use the sine trigonometric function. (Soh from SohCahToa)  Write the equation and solve using the inverse key for sine.

[latex]\displaystyle{ \brainstorm{align} \sin{A^{\circ}} &= \frac {\text{contrary}}{\text{hypotenuse}} \\ \sin{A^{\circ}} &= \frac{12}{25} \\ A^{\circ} &= \sin^{-1}{\left( \frac{12}{25} \correct)} \\ A^{\circ} &= \sin^{-i}{\left( 0.48 \right)} \\ A &=29^{\circ} \end{marshal} }[/latex]

Source: https://courses.lumenlearning.com/boundless-algebra/chapter/trigonometry-and-right-triangles/

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