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Wize High School Algebra II Textbook (Common Core) > Trigonometric Ratios (Radians)

Radians

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Radians vs. Degrees

Arc Length in Degrees

The length of the arc bounding the sector of a circle is proportional to the sector angle and is called the arc length.

To calculate arc length, in degrees, we use the following equation:

Arc Length = Sector Angle360∘⋅(Circumference)\boxed{\text{Arc Length = }\displaystyle\frac{\text{Sector Angle}}{\text{360}^\circ}\cdot\text{(Circumference)}}

Example 1

Calculate the arc length of a sector of a circle of radius20 cm20\text{ cm} if the sector angle is 120∘120^\circ. Round answer to the nearest tenth.

Arc Length=Sector Angle360∘⋅(Circumference)=120∘360∘⋅(2π(20))=41.9 cm\begin{array}{rcl}\text{Arc Length}&=&\displaystyle\frac{\text{Sector Angle}}{\text{360}^\circ}\cdot\text{(Circumference)}\\\\ &=&\displaystyle\frac{120^\circ}{360^\circ}\cdot(2\pi(20))\\\\ &=&41.9~\text{cm} \end{array}

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Up to this point, we’ve been measuring angles in degrees. However, just like how we can measure length in terms of meters or inches, there’s another way to measure angles that are more often used in physics and math applications: radians.

Arc Length in Radians

One radian is the measure of an angle that is subtended at the center of a circle by an arc equal in length to the radius of the circle.

Let's look at the following circle with radius 'RR'.
From the definition of arc length,

Arc Length=Sector Angle360∘⋅(Circumference)R=1 Radian360∘⋅(2π(R))360∘R2πR=1 Radian1 Radian=180∘π or approximately 57.29°\begin{array}{rcl}\text{Arc Length}&=&\displaystyle\frac{\text{Sector Angle}}{\text{360}^\circ}\cdot\text{(Circumference)}\\\\ R&=&\displaystyle\frac{\text{1 Radian}}{360^\circ}\cdot(2\pi(R))\\\\ \displaystyle\frac{360^\circ{}R}{2\pi{}R}&=&\text{1 Radian}\\\\ \text{1 Radian}&=&\displaystyle\frac{180^\circ}{\pi}\text{ or approximately }57.29\degree \end{array}

In conclusion, π radian=180∘.\pi~\text{radian}=180^\circ.
Therefore, arc length, in radians, is:
a=Rθ\boxed{a=R\theta}
where θ\theta is in radians.
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Example 2

A circle has a radius of 43 mm\displaystyle\frac{4}{3} ~\text{mm} and a arc length of 25681 mm\displaystyle\frac{256}{81}~\text{mm}. What is the sector angle, in radians?

a=Rθ25681=43θθ=25681⋅34θ=6427 radiansorθ≈2.37 radians\begin{array}{rcl} a&=&R\theta\\\\ \displaystyle\frac{256}{81}&=&\displaystyle\frac{4}{3}\theta\\\\ \theta&=&\displaystyle\frac{256}{81}\cdot \displaystyle\frac{3}{4}\\\\ \theta&=&\displaystyle\frac{64}{27}~\text{radians}&&&\text{or}&&&\theta&\approx&2.37~\text{radians} \end{array}

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Converting Between Degrees and Radians

Degrees to Radians

Let θ \theta~ be an angle in degrees. Then, the following equation
θ=π180∘\boxed{\theta=\displaystyle\frac{\pi}{180^\circ}}
converts degrees to radians.


Example 1

Let θ=95∘\theta=95^{\circ}. What is θ\theta, in radians? Leave answer in exact form.


95∘×π180∘=19π3695^\circ{}\times\displaystyle\frac{\pi}{180^\circ}=\displaystyle\frac{19\pi}{36}

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Radians to Degrees

Let θ \theta~ be an angle in radians. Then, the following equation
θ=180∘π\boxed{\theta=\displaystyle\frac{180^\circ}{\pi}}
converts radians to degrees.

Example 2

Let θ=5π3\theta=\displaystyle\frac{5\pi}{3}. What is θ\theta, in degrees?


5π3×180∘π=300∘\displaystyle\frac{5\pi}{3}\times\displaystyle\frac{180^\circ}{\pi}=300^\circ

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Example: Radians & Degrees

Calculate the arc length, in radians, of the sector of a circle with radius 10 cm10~\text{cm} and a sector angle of 30∘30^\circ. Round to the nearest hundredth of a centimeter.


First, find the angle in radians.

Degrees→Radians30∘⋅(π180∘)→π6 Radians\begin{array}{rcl} \text{Degrees}&{\color{red}\rightarrow}&\text{Radians}\\\\ 30^\circ\cdot\Bigg(\displaystyle\frac{\pi}{180^\circ}\Bigg)&{\color{red}\rightarrow}&\displaystyle\frac{\pi}{6}~\text{Radians} \end{array}

Then,

a=Rθa=10(π6)a=5π3≈ 5.24 cm\begin{array}{rcl} a&=&R\theta\\\\ a&=&10\Bigg(\displaystyle\frac{\pi}{6}\Bigg)\\\\ a&=&\displaystyle\frac{5\pi}{3}\approx{}~5.24~\text{cm} \end{array}

Practice: Radians & Degrees

Convert:
  1. 330∘ 330^\circ~ to radians.
  2. 7π4\displaystyle\frac{7\pi}{4} to degrees.

Practice: Radians & Degrees

Match the correct angle in degrees to its equivalent angle in radians.
A.
θ=78∘\theta=78^\circ
B.
θ=328∘\theta=328^\circ
C.
θ=145∘\theta=145^\circ
D.
θ=265∘\theta=265^\circ
13π30\displaystyle\frac{13\pi}{30}
82π45\displaystyle\frac{82\pi}{45}
29π36\displaystyle\frac{29\pi}{36}
53π36\displaystyle\frac{53\pi}{36}

Practice: Radians & Degrees

Match the correct angle in radians to its equivalent angle in degrees.
A.
θ=10π11\theta=\displaystyle\frac{10\pi}{11}
B.
θ=7π5\theta=\displaystyle\frac{7\pi}{5}
C.
θ=5π14\theta=\displaystyle\frac{5\pi}{14}
D.
θ=π12\theta=\displaystyle\frac{\pi}{12}
15∘15^\circ
64.29∘64.29^\circ
252∘252^\circ
163.64∘163.64^\circ

Practice: Radians & Degrees

The arc length of a circle is 354 mm3\sqrt{54}~\text{mm} and the radius is 6 mm\sqrt{6}~\text{mm}. What is the sector angle, in degrees? Round the answer to the nearest hundredth of a degree.

Practice: Radians & Degrees

A circle with center OO and arc length XYXY measuring 4.5 μm4.5~\mu{m} long have angles ∡ OXY = ∡ OYX = π6 radians\measuredangle{}~OXY~=~\measuredangle{}~OYX~=~\displaystyle\frac{\pi}{6}~\text{radians}, shown below.
What is the radius of the circle?
Extra Practice
Trigonometric Ratios
Determine whether the trigonometric expression is positive or negative.
Arc Length, Radians & Degrees
Practice: Radians & Degrees
A circle with center C\text{C} and arc length AB\text{AB} measuring 24πm24\pi{}\text{m} long have angles ∡ CAB = ∡ CBA = π8 radians\measuredangle{}~\text{CAB}~=~\measuredangle{}~\text{CBA}~=~\displaystyle\frac{\pi}{8}~\text{radians}, shown below.
What is the radius of the circle?
Arc Length, Radians & Degrees
Practice: Radians & Degrees
A circle with center C\text{C} and arc length AB\text{AB} measuring 15m15\text{m} long have angles ∡ CAB = ∡ CBA = π3 radians\measuredangle{}~\text{CAB}~=~\measuredangle{}~\text{CBA}~=~\displaystyle\frac{\pi}{3}~\text{radians}, shown below.
What is the radius of the circle?
Arc Length, Radians & Degrees
Practice: Radians & Degrees
The arc length of a circle is 418 mm4\sqrt{18}~\text{mm} and the radius is 2 mm\sqrt{2}~\text{mm}. What is the sector angle, in degrees? Round the answer to the nearest hundredth of a degree.
Arc Length, Radians & Degrees
Practice: Radians & Degrees
The arc length of a circle is 247 mm2\sqrt{47}~\text{mm} and the radius is 17 mm\sqrt{17}~\text{mm}. What is the sector angle, in degrees? Round the answer to the nearest hundredth of a degree.
Radians & Degrees
Practice: Radians & Degrees
Match the correct angle in degrees to its equivalent angle in radians.
Radians & Degrees
Practice: Radians & Degrees
Match the correct angle in degrees to its equivalent angle in radians.
Arc Length, Radians & Degrees
Practice: Radians & Degrees
Convert:
450∘ 450^\circ~ to radians.
Arc Length, Radians & Degrees
Practice: Radians & Degrees
Convert:
200∘ 200^\circ~ to radians.