How To Find The Degree Of An Angle Using Trigonometry

Right Triangles and the Pythagorean Theorem

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

Learning Objectives

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

Key Takeaways

Central Points

• The Pythagorean Theorem, [latex]{\displaystyle a^{two}+b^{2}=c^{2},}[/latex] is used to notice 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 [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^{two}+b^{2}=c^{2}}[/latex].

Key Terms

• legs: The sides adjacent to the right angle in a right triangle.
• right triangle: A [latex]3[/latex]-sided shape where one angle has a value of [latex]90[/latex] degrees
• hypotenuse: The side contrary 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 ([latex]a[/latex] and [latex]b[/latex]) is equal to the expanse of the foursquare on the hypotenuse ([latex]c[/latex]). The formula is [latex]a^2+b^2=c^2[/latex].

Right Triangle

A right angle has a value of 90 degrees ([latex]90^\circ[/latex]). A correct triangle is a triangle in which one angle is a right bending. The relation between the sides and angles of a right triangle is the basis for trigonometry.

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

Right triangle: The Pythagorean Theorem tin be used to discover the value of a missing side length in a right triangle.

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

The Pythagorean Theorem

The Pythagorean Theorem, too known equally Pythagoras' Theorem, is a fundamental relation in Euclidean geometry. It defines the relationship amid the iii sides of a right triangle. It states that the foursquare 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 every bit an equation relating the lengths of the sides [latex]a[/latex], [latex]b[/latex] and [latex]c[/latex], ofttimes called the "Pythagorean equation":^[1]

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

Although it is frequently said that the cognition of the theorem predates him,^[two] 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 ([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^2+b^2=c^two[/latex].

Finding a Missing Side Length

Example 1:  A right triangle has a side length of [latex]ten[/latex] feet, and a hypotenuse length of [latex]20[/latex] feet.  Detect the other side length.  (round to the nearest 10th 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{ \brainstorm{align} a^{two}+b^{two} &=c^{2} \\ (10)^2+b^2 &=(20)^2 \\ 100+b^ii &=400 \\ b^2 &=300 \\ \sqrt{b^two} &=\sqrt{300} \\ b &=17.three ~\mathrm{feet} \end{align} }[/latex]

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

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

[latex]\displaystyle{ \begin{align} a^{2}+b^{2} &=c^{2} \\ three^two+4^2 &=c^2 \\ 9+sixteen &=c^2 \\ 25 &=c^two\\ c^2 &=25 \\ \sqrt{c^two} &=\sqrt{25} \\ c &=five~\mathrm{cm} \stop{align} }[/latex]

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 issues well-nigh right triangles, and place their inputs and outputs

Key Takeaways

Central Points

• A right triangle has one bending with a value of ninety degrees ([latex]90^{\circ}[/latex])The 3 trigonometric functions most often used to solve for a missing side of a correct triangle are: [latex]\displaystyle{\sin{t}=\frac {reverse}{hypotenuse}}[/latex], [latex]\displaystyle{\cos{t} = \frac {adjacent}{hypotenuse}}[/latex], and [latex]\displaystyle{\tan{t} = \frac {opposite}{next}}[/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 "adjacent to.") The contrary side is the side beyond from the bending. The hypotenuse  is the side of the triangle opposite the right angle, and information technology is the longest.

Correct triangle: The sides of a correct triangle in relation to angle [latex]t[/latex].

When solving for a missing side of a right triangle, simply the only given information is an acute 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 {next}{hypotenuse}}[/latex]
• Tangent    [latex]\displaystyle{\tan{t} = \frac {opposite}{next}}[/latex]

The trigonometric functions are equal to ratios that chronicle certain side lengths of a  right triangle.  When solving for a missing side, the first footstep is to identify what sides and what angle are given, and then select the appropriate function to apply 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 detect other measurements.  Use one of the trigonometric functions ([latex]\sin{}[/latex], [latex]\cos{}[/latex], [latex]\tan{}[/latex]), place the sides and angle given, prepare the equation and use the calculator and algebra to find the missing side length.

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

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

Looking at the effigy, solve for the side opposite the acute bending of [latex]34[/latex] degrees.  The ratio of the sides would be the reverse side and the hypotenuse.  The ratio that relates those two sides is the sine function.

[latex]\displaystyle{ \begin{align} \sin{t} &=\frac {opposite}{hypotenuse} \\ \sin{\left(34^{\circ}\correct)} &=\frac{10}{25} \\ 25\cdot \sin{ \left(34^{\circ}\right)} &=x\\ ten &= 25\cdot \sin{ \left(34^{\circ}\right)}\\ 10 &= 25 \cdot \left(0.559\dots\right)\\ x &=xiv.0 \end{align} }[/latex]

The side opposite the acute angle is [latex]14.0[/latex] feet.

Example two:
Given a right triangle with an acute angle of [latex]83^{\circ}[/latex] and a hypotenuse length of [latex]300[/latex] feet, detect the hypotenuse length (round to the nearest tenth):

Correct 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 acute angle of [latex]83[/latex] degrees. The ratio of the sides would be the adjacent side and the hypotenuse.  The ratio that relates these ii sides is the cosine part.

[latex]\displaystyle{ \begin{marshal} \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} \stop{marshal} }[/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

Utilise the acronym SohCahToa to define Sine, Cosine, and Tangent in terms of correct 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 "Southine is opposite over hypotenuse (Soh), Cosine is adjacent over hypotenuse (Cah), Tangent is opposite over next (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 {opposite}{hypotenuse} }[/latex]
• Cosine        [latex]\displaystyle{ \cos{t} = \frac {adjacent}{hypotenuse} }[/latex]
• Tangent      [latex]\displaystyle{ \tan{t} = \frac {opposite}{adjacent} }[/latex]

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

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

Evaluating a Trigonometric Function of a Right Triangle

Example 1:
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 opposite side length. (circular to the nearest tenth)

Right triangle: Given a right triangle with an acute angle of [latex]62[/latex] degrees and an side by side side of [latex]45[/latex] anxiety, solve for the opposite side length.

First, determine which trigonometric function to apply when given an side by side side, and you lot need to solve for the opposite side.  Ever determine which side is given and which side is unknown from the acute bending ([latex]62[/latex] 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.

[latex]\displaystyle{ \begin{marshal} \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( i.8807\dots \right) \\ x &=84.6 \end{marshal} }[/latex]

Example 2:  A ladder with a length of [latex]30~\mathrm{feet}[/latex] is leaning confronting a edifice.  The bending the ladder makes with the basis is [latex]32^{\circ}[/latex].  How high upwards the building does the ladder reach? (round to the nearest tenth of a foot)

Correct triangle: Later on sketching a picture of the trouble, nosotros have the triangle shown. The bending given is [latex]32^\circ[/latex], the hypotenuse is xxx feet, and the missing side length is the contrary leg, [latex]10[/latex] feet.

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

[latex]\displaystyle{ \begin{align} \sin{t} &= \frac {reverse}{hypotenuse} \\ \sin{ \left( 32^{\circ} \right) } & =\frac{ten}{thirty} \\ 30\cdot \sin{ \left(32^{\circ}\right)} &=x \\ ten &= 30\cdot \sin{ \left(32^{\circ}\right)}\\ 10 &= 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 astute bending measurement of a right triangle.

Learning Objectives

Use changed trigonometric functions in solving problems involving correct triangles

Fundamental Takeaways

Key Points

• A missing acute angle value of a right triangle tin can be establish when given two side lengths.
• To observe a missing bending value, apply the trigonometric functions sine, cosine, or tangent, and the inverse key on a figurer to apply the inverse function ([latex]\arcsin{}[/latex], [latex]\arccos{}[/latex], [latex]\arctan{}[/latex]), [latex]\sin^{-1}[/latex], [latex]\cos^{-ane}[/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 part, setting upwards the equation and solving.  Finding the missing astute angle when given two sides of a right triangle is just as elementary.

Changed Trigonometric Functions

In order to solve for the missing astute angle, employ the same 3 trigonometric functions, merely, use the changed cardinal ([latex]^{-ane}[/latex]on the calculator) to solve for the angle ([latex]A[/latex]) when given two sides.

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

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

[latex]\displaystyle{ A^{\circ} = \tan^{-1}{\left(\frac {\text{opposite}}{\text{adjacent}} \correct) }}[/latex]

Example

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

Correct triangle: Discover the measure of bending [latex]A[/latex], when given the opposite side and hypotenuse.

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

[latex]\displaystyle{ \begin{marshal} \sin{A^{\circ}} &= \frac {\text{contrary}}{\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} \cease{marshal} }[/latex]

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

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