Angles Represented In Degrees And Radians

For the purpose of future trigonometric problems and applications, it may sometimes be easier to represent angles in terms of degrees or radians depending on the problem. We will learn how to convert a value in degrees to a value in radians and vice versa.

Given some angle Θ in degrees, we can multiply this angle in degrees by a specific factor in order to obtain an equivalent angle in radians. Let's first begin by recognizing the following equivalency:

(1)
\begin{align} 180^\circ = \pi^r \end{align}

Thus we can obtain the following as an equivalency:

(2)
\begin{align} 1 = \frac{\pi^r}{180^\circ} \end{align}

Hence, if Θ is represented in degrees, we can multiply Θ by the equivalency relationship above in order to convert the value of Θ from degrees to radians such that:

(3)
\begin{align} \theta_{rad.} = \theta_{deg.} * \frac{\pi^r}{180^\circ} \end{align}

## Example 1

To solve this problem, we will simply apply our formula to obtain the solution:

(4)
\begin{align} \theta_{rad.} = 160^\circ * \frac{\pi^r}{180^\circ} \end{align}

Which when simplified, we obtain:

(5)

We will use the same equivalency from earlier:

(6)
\begin{align} 180^\circ = \pi^r \end{align}

We can thus rearrange this equivalency to get a new multiplication factor equivalency such that:

(7)
\begin{align} 1 = \frac{180^\circ}{\pi^r} \end{align}

Thus, converting from radians to degrees follows a similar equation:

(8)
\begin{align} \theta_{deg.} = \theta_{rad.} * \frac{180^\circ}{\pi^r} \end{align}

In conclusion, we can acknowledge a simply rule for determining which formula to utilize. If we want to convert from one unit to the other, then we will choose the conversion formula where the units we want to convert to is on the numerator of the conversion factor, while at the same time, the denominator will be the same units that we are converting out of.

## Example 2

Once again, we will apply the appropriate formula to obtain:

(9)
\begin{align} \theta_{deg.} = \pi^r * \frac{180^\circ}{\pi^r} \end{align}

Which when simplified we obtain:

(10)
\begin{align} \theta_{deg.} = 180^\circ \end{align}

# Summary of Important Degree and Radian Conversions

Below is a table that expresses the major angle conversions necessary to know by heart.

Angle Θ in Degrees Angle Θ in Radians
Θ = 0 deg. Θ = 0 rad.
Θ = 30 deg. Θ = π/6 rad.
Θ = 45 deg. Θ = π/4 rad.
Θ = 60 deg. Θ = π/3 rad.
Θ = 90 deg. Θ = π/2 rad.
Θ = 120 deg. Θ = 2π/3 rad.
Θ = 135 deg. Θ = 3π/4 rad.
Θ = 150 deg. Θ = 5π/6 rad.
Θ = 180 deg. Θ = π rad.
Θ = 210 deg. Θ = 7π/6 rad.
Θ = 225 deg. Θ = 5π/4 rad.
Θ = 240 deg. Θ = 4π/3 rad.
Θ = 270 deg. Θ = 3π/2 rad.
Θ = 300 deg. Θ = 5π/3 rad.
Θ = 315 deg. Θ = 7π/4 rad.
Θ = 330 deg. Θ = 11π/6 rad.
Θ = 360 deg. Θ = 2π rad.

It is important to note that a full revolution is equivalent to equal to 360 degrees or 2π radians. Hence, we can thus state that a rotation of some ray S from the origin by 360 degrees or 2π radians will have the ray be positioned back at its original starting position. Therefore we can state that when evaluating cosΘ and sinΘ, we know that:

(11)
\begin{align} cos(\theta) = cos(\theta + 2\pi^r) = cos(\theta + 4\pi^r) = ... = cos(\theta + 2n\pi^r) , \quad n \in \mathbb{I} \end{align}

And for sine:

(12)
\begin{align} sin(\theta) = sin(\theta + 2\pi^r) = sin(\theta + 4\pi^r) = ... = sin(\theta + 2n\pi^r) , \quad n \in \mathbb{I} \end{align}

This equivalencies above may seem complicated, however, it is basically saying that the cosine and sine evaluated at theta is equivalent to some integer value {…, -2, -1, 0, 1, 2 …} multipled by 2 and then multiplied by Θ.

## Example 3

Given that Θ = 20 degrees, determine three other values of Θ where sinΘ is equivalent.

The answer is rather simple. If Θ = 20, then Θ = 20 + 360, Θ = 20 + 2(360), Θ = 20 + 3(360) will result in angles that when evaluated by sinΘ will be equivalent. Thus, three values of Θ for this solution can be:

Θ = 380 deg.
Θ = 740 deg.
Θ = 1100 deg.