From the time of the ancient Babylonians, it has been the practice to divide circles into $360$360 small arcs. The angle subtended at the centre by any one of those arcs, is called one degree. In effect, an arc of the circle is used as a measure of the angle it subtends.
In a similar way, we could define the angle subtended by an arc at the centre of a circle, by the ratio of the arc length divided by the radius. Angles defined this way are called radians and an angle in radians can be calculated as $\theta=\frac{s}{r}$θ=sr, where $s$s is the arc length and $r$r is the radius of the circle. Hence, the angle subtended by an arc whose length is equal to the radius is $1$1 radian. Radians are an alternate way to describe angles and are the international standard unit for measuring angles. Because angles in radian measure are in essence just fractions of the circle, they do not require a unit, although some writers indicate that radian measure is being used by adding a superscript c after a number or the abbreviation rad, like this: $2^c$2c or $2$2$rad$rad. (The c is short for circular-measure.)
If we now restrict our attention to circles of radius one unit, then $1$1 radian would be the angle subtended by an arc of length $1$1 unit.
Do you remember how to find the circumference of a circle? We use the formula $C=2\pi r$C=2πr. So if the radius ($r$r) is $1$1, then the circumference is $2\pi$2π.
The angle represented by a full turn around the circle is $2\pi$2π radians. This is equivalent to $360^\circ$360°.
A half-circle makes an angle of $\pi$π radians or $180^\circ$180° and a right-angle is $\frac{\pi}{2}$π2 radians.
Since:
| $\pi^c$πc | $=$= | $180^\circ$180° |
| $1^c$1c | $=$= | $\frac{180}{\pi}^\circ$180π° |
| $\approx$≈ | $57.3^\circ$57.3° |
In practice, angles given in radian measure are commonly expressed as fractions of $\pi$π.
Converting degrees to radians
Since $\pi^c=180^\circ$πc=180°, to change from degrees to radians we must divide by $180^\circ$180° and multiply by $\pi$π .
Converting radians to degrees
Method 1: If the angle in radians is in exact form (that means it contains a $\pi$π), substitute $180^\circ$180° for $\pi$π.
Method 2: If the angle in radians does not contain a $\pi$π, multiply by $180^\circ$180° and divide by $\pi$π, as shown in the conversion diagram above.
$\pi=180^\circ$π=180°
$\frac{\pi}{2}=90^\circ$π2=90°
$\frac{\pi}{3}=60^\circ$π3=60°
$\frac{\pi}{4}=45^\circ$π4=45°
$\frac{\pi}{6}=30^\circ$π6=30°
Worked example
Example 1
Convert $5.25^c$5.25c to degrees.
Think: Multiply by $\frac{180^\circ}{\pi}$180°π.
Do:
| $5.25^c$5.25c | $=$= | $\frac{5.25\times180^\circ}{\pi}$5.25×180°π |
| $\approx$≈ | $300.80^\circ$300.80° |
Reflect: Always check that your answer makes sense. We know that $2\pi$2π is approximately $6.28$6.28 radians which is $360^\circ$360°.
$5.25$5.25 is a little less than $6.28$6.28 so we would expect the angle to be a little less than $360^\circ$360°. Which it is!
Example 2
Convert $\frac{8\pi}{6}$8π6 radians to degrees.
Think: Substitute $180^\circ$180° for $\pi$π.
Do:
| $\frac{8\pi}{6}^{^c}$8π6c | $=$= | $\frac{8\times180^\circ}{6}$8×180°6 |
| $=$= | $\frac{1440^\circ}{6}$1440°6 | |
| $=$= | $240^\circ$240° |
Reflect: Alternatively, it's handy to know that $\frac{\pi}{6}$π6 is $30^\circ$30°. Therefore we have $8$8 lots of $30^\circ$30° which is $240^\circ$240°.
Trigonometric functions with radians
Just as we looked at the trigonometric functions on angles using degrees in our last lesson, we can apply these to angles using radians.
Let's look at the sign and symmetry of the trigonometric functions in terms of radians.
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Worked example
Example 3
Express $\cos\frac{13\pi}{20}$cos13π20 in terms of a first quadrant angle.
Think: The fraction $\frac{13}{20}$1320 is a little more than $\frac{1}{2}$12, therefore the angle is between $\frac{\pi}{2}$π2 and $\pi$π, so it is in the second quadrant. In the second quadrant cosine is negative.
Do: Therefore, $\cos\frac{13\pi}{20}$cos13π20 must be the same as $-\cos\left(\pi-\frac{13\pi}{20}\right)=-\cos\frac{7\pi}{20}$−cos(π−13π20)=−cos7π20.
Example 4
A point $P$P on the unit circle is at at angle of $\frac{4\pi}{3}$4π3 from the positive $x$x-axis, find the coordinates of point $P$P.
Think: The coordinates will be in the form $\left(\cos\theta,\sin\theta\right)$(cosθ,sinθ)
Do: With calculator (in radians):
| $\cos(\theta)$cos(θ) | $=$= | $\cos\left(\frac{4\pi}{3}\right)$cos(4π3) |
| $=$= | $-\frac{1}{2}$−12 |
| $\sin(\theta)$sin(θ) | $=$= | $\sin\left(\frac{4\pi}{3}\right)$sin(4π3) |
| $=$= | $-\frac{\sqrt{3}}{2}$−√32 |
Do: Without calculator:
The angle $\frac{4\pi}{3}$4π3 is $4$4 lots of $60^\circ$60° which places point $P$P in the third quadrant, hence both coordinates will be negative.
Our related acute angle is $\frac{\pi}{3}$π3. Use our exact value triangle for $60^\circ$60° to find $\cos(\frac{\pi}{3})=\frac{1}{2}$cos(π3)=12 and $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$sin(π3)=√32
Hence, $P$P is located at $\left(-\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$(−12,−√32).
Practice questions
Question 1
Given that $x=\pi$x=πc represents half a circle, what fraction of the circumference of the unit circle does $x=\frac{\pi}{4}$x=π4c represent?
Question 2
Convert $-300^\circ$−300° to radians.
Give your answer in exact form.
Question 3
Convert $\frac{2\pi}{3}$2π3 radians to degrees.
Question 4
We want to evaluate $\sin\frac{7\pi}{6}$sin7π6 by first rewriting it in terms of the related acute angle. What is the related acute angle of $\frac{7\pi}{6}$7π6?
Question 5
Using the approximations $\cos\frac{\pi}{5}=0.81$cosπ5=0.81 and $\sin\frac{\pi}{5}=0.59$sinπ5=0.59, write down the approximate value of each of the following correct to two decimal places.
- $\cos\frac{4\pi}{5}$cos4π5
- $\sin\left(-\frac{4\pi}{5}\right)$sin(−4π5)