How To Find Hypotenuse With 2 Sides

Right Triangles and the Pythagorean Theorem

The Pythagorean Theorem, [latex]{\displaystyle a^{ii}+b^{2}=c^{2},}[/latex] can be used to find the length of any side of a right triangle.

Learning Objectives

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

Cardinal Takeaways

Cardinal Points

• The Pythagorean Theorem, [latex]{\displaystyle a^{2}+b^{ii}=c^{2},}[/latex] is used to find the length of any side of a right triangle.
• In a right triangle, i 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 reverse the xc degree angle.
• 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^{2}=c^{two}}[/latex].

Primal Terms

• legs: The sides adjacent to the right angle in a right triangle.
• right triangle: A [latex]iii[/latex]-sided shape where one angle has a value of [latex]xc[/latex] degrees
• hypotenuse: The side reverse the right angle of a triangle, and the longest side of a right triangle.
• Pythagorean theorem: The sum of the areas of the ii 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^2+b^2=c^ii[/latex].

Right Triangle

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

The side opposite the right angle is called the hypotenuse (side [latex]c[/latex] in the figure). The sides adjacent to the right bending are called legs (sides [latex]a[/latex] and [latex]b[/latex]). Side [latex]a[/latex] may be identified as the side adjacent to angle [latex]B[/latex] and opposed to (or reverse) 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 be used to observe the value of a missing side length in a right triangle.

If the lengths of all three sides of a right triangle are whole numbers, the triangle is said to be a Pythagorean triangle and its side lengths are collectively known equally a Pythagorean triple.

The Pythagorean Theorem

The Pythagorean Theorem, also known equally Pythagoras' Theorem, is a fundamental relation in Euclidean geometry. It defines the relationship amidst the iii sides of a right triangle. Information technology states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other ii sides. The theorem can be written every bit an equation relating the lengths of the sides [latex]a[/latex], [latex]b[/latex] and [latex]c[/latex], often called the "Pythagorean equation":^[1]

[latex]{\displaystyle a^{two}+b^{ii}=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's other 2 sides.

Although information technology is often said that the cognition of the theorem predates him,^[2] the theorem is named afterward 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 ([latex]a[/latex] and [latex]b[/latex]) is equal to the area of the foursquare on the hypotenuse ([latex]c[/latex]).  The formula is [latex]a^2+b^2=c^2[/latex].

Finding a Missing Side Length

Example 1:  A right triangle has a side length of [latex]10[/latex] anxiety, and a hypotenuse length of [latex]20[/latex] feet.  Discover the other side length.  (round 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{marshal} a^{2}+b^{2} &=c^{2} \\ (10)^ii+b^ii &=(20)^ii \\ 100+b^two &=400 \\ b^two &=300 \\ \sqrt{b^two} &=\sqrt{300} \\ b &=17.3 ~\mathrm{feet} \end{marshal} }[/latex]

Case 2:  A correct triangle has side lengths [latex]3[/latex] cm and [latex]four[/latex] cm.  Find the length of the hypotenuse.

Substitute [latex]a=3[/latex] and [latex]b=4[/latex] into the Pythagorean Theorem and solve for [latex]c[/latex].

[latex]\displaystyle{ \begin{marshal} a^{2}+b^{ii} &=c^{2} \\ 3^2+4^2 &=c^2 \\ nine+16 &=c^ii \\ 25 &=c^2\\ c^two &=25 \\ \sqrt{c^2} &=\sqrt{25} \\ c &=5~\mathrm{cm} \stop{align} }[/latex]

How Trigonometric Functions Work

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

Learning Objectives

Recognize how trigonometric functions are used for solving issues most correct triangles, and identify their inputs and outputs

Key Takeaways

Fundamental Points

• A correct triangle has i angle with a value of xc degrees ([latex]90^{\circ}[/latex])The three trigonometric functions well-nigh oftentimes 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 {opposite}{adjacent}}[/latex]

Trigonometric Functions

Nosotros 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 ways "next to.") The opposite side is the side beyond 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 right triangle, but the simply given information is an astute bending measurement and a side length, use the trigonometric functions listed below:

• 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}{next}}[/latex]

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

Evaluating a Trigonometric Function of a Right Triangle

Sometimes you know the length of 1 side of a triangle and an angle, and demand to find other measurements.  Use one of the trigonometric functions ([latex]\sin{}[/latex], [latex]\cos{}[/latex], [latex]\tan{}[/latex]), identify the sides and angle given, set up upwardly the equation and use the figurer and algebra to find the missing side length.

Instance 1:
Given a right triangle with astute angle of [latex]34^{\circ}[/latex] and a hypotenuse length of [latex]25[/latex] anxiety, find the length of the side opposite the astute angle (round to the nearest 10th):

Correct triangle: Given a right triangle with acute bending of [latex]34[/latex] degrees and a hypotenuse length of [latex]25[/latex] anxiety, observe the contrary side length.

Looking at the effigy, solve for the side contrary the astute angle of [latex]34[/latex] degrees.  The ratio of the sides would be the reverse side and the hypotenuse.  The ratio that relates those 2 sides is the sine function.

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

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

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

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

Looking at the effigy, solve for the hypotenuse to the acute bending of [latex]83[/latex] degrees. The ratio of the sides would be the side by side side and the hypotenuse.  The ratio that relates these two sides is the cosine function.

[latex]\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{anxiety} \end{align} }[/latex]

Sine, Cosine, and Tangent

The mnemonic
SohCahToa tin be used to solve for the length of a side of a right triangle.

Learning Objectives

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

Central Takeaways

Central Points

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

Definitions of Trigonometric Functions

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

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

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

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

Evaluating a Trigonometric Function of a Right Triangle

Example 1:
Given a right triangle with an astute angle of [latex]62^{\circ}[/latex] and an adjacent side of [latex]45[/latex] feet, solve for the opposite side length. (circular to the nearest tenth)

Correct triangle: Given a right triangle with an acute 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 apply when given an adjacent side, and you need to solve for the opposite side.  Ever 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 employ "T", pregnant the tangent trigonometric office.

[latex]\displaystyle{ \begin{align} \tan{t} &= \frac {contrary}{adjacent} \\ \tan{\left(62^{\circ}\right)} &=\frac{ten}{45} \\ 45\cdot \tan{\left(62^{\circ}\correct)} &=x \\ x &= 45\cdot \tan{\left(62^{\circ}\right)}\\ x &= 45\cdot \left( 1.8807\dots \correct) \\ x &=84.vi \end{marshal} }[/latex]

Example two:  A ladder with a length of [latex]30~\mathrm{feet}[/latex] is leaning against a building.  The angle the ladder makes with the ground is [latex]32^{\circ}[/latex].  How high up the building does the ladder reach? (round to the nearest tenth of a foot)

Right triangle: Subsequently sketching a film of the problem, we take the triangle shown. The angle given is [latex]32^\circ[/latex], the hypotenuse is 30 feet, and the missing side length is the reverse leg, [latex]10[/latex] feet.

Determine which trigonometric function to use when given the hypotenuse, and you need to solve for the contrary 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 office.

[latex]\displaystyle{ \begin{align} \sin{t} &= \frac {opposite}{hypotenuse} \\ \sin{ \left( 32^{\circ} \correct) } & =\frac{10}{30} \\ 30\cdot \sin{ \left(32^{\circ}\right)} &=x \\ 10 &= 30\cdot \sin{ \left(32^{\circ}\right)}\\ 10 &= 30\cdot \left( 0.5299\dots \right) \\ x &= 15.nine ~\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

Use inverse trigonometric functions in solving issues involving right triangles

Cardinal Takeaways

Key Points

• A missing acute angle value of a right triangle tin can be institute when given 2 side lengths.
• To find a missing angle value, use the trigonometric functions sine, cosine, or tangent, and the changed central on a calculator to apply the inverse 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 angle is as simple as identifying the sides in relation to the acute angle, choosing the correct function, setting up the equation and solving.  Finding the missing astute angle when given two sides of a correct triangle is just as uncomplicated.

Changed Trigonometric Functions

In order to solve for the missing acute bending, use the same three trigonometric functions, but, use the inverse primal ([latex]^{-i}[/latex]on the reckoner) to solve for the angle ([latex]A[/latex]) when given ii sides.

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

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

[latex]\displaystyle{ A^{\circ} = \tan^{-1}{\left(\frac {\text{reverse}}{\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 contrary side length [latex]12~\mathrm{feet}[/latex], detect the acute angle to the nearest degree:

Right triangle: Observe the mensurate of bending [latex]A[/latex], when given the opposite side and hypotenuse.

From bending [latex]A[/latex], the sides contrary 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{ \begin{align} \sin{A^{\circ}} &= \frac {\text{reverse}}{\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} }[/latex]

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

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