Lesson 7.1 : Angles

New Concept

This lesson frames an angle as a rotation from an initial side to a terminal side. You'll learn to draw angles in standard position, convert between degrees and radians, and apply these skills to measure circular arcs and speed.

What’s next

Next, you'll master these concepts through a series of practice cards on drawing angles, converting units, and solving for arc length.

Property

An angle is in standard position if its vertex is located at the origin, and its initial side extends along the positive x-axis. If the angle is measured in a counterclockwise direction from the initial side to the terminal side, the angle is said to be a positive angle.

If the angle is measured in a clockwise direction, the angle is said to be a negative angle. An angle is a quadrantal angle if its terminal side lies on an axis, including $0^\circ$, $90^\circ$, $180^\circ$, $270^\circ$, or $360^\circ$.

To draw an angle in standard position, express the angle measure as a fraction of $360^\circ$ and reduce it. This fraction represents the portion of the circle to rotate from the positive x-axis.

Property

One radian is the measure of the central angle of a circle such that the length of the arc between the initial side and the terminal side is equal to the radius of the circle. A full revolution ($360^\circ$) equals $2\pi$ radians. To convert between degrees and radians, use the proportion where $\theta$ is the measure in degrees and $\theta_R$ is the measure in radians:

Examples

• To convert $\frac{\pi}{3}$ radians to degrees, we use the proportion: $\frac{\theta}{180} = \frac{\pi/3}{\pi}$, which simplifies to $\theta = \frac{180}{3} = 60^\circ$.
• To convert $270^\circ$ to radians, we set up the proportion: $\frac{270}{180} = \frac{\theta_R}{\pi}$. This gives $\theta_R = \frac{270\pi}{180} = \frac{3\pi}{2}$ radians.
• To convert $2$ radians to degrees, we solve $\frac{\theta}{180} = \frac{2}{\pi}$. This results in $\theta = \frac{2(180)}{\pi} = \frac{360}{\pi} \approx 114.6^\circ$.

Explanation

Degrees and radians are just different units for measuring angles, like feet and meters for distance. Radians are based on the radius of a circle, which makes them incredibly useful in higher-level math formulas involving circles and waves.

Property

Coterminal angles are two angles in standard position that have the same terminal side. To find a coterminal angle, you can add or subtract full rotations.
For an angle in degrees, add or subtract multiples of $360^\circ$. For an angle in radians, add or subtract multiples of $2\pi$.
An angle’s reference angle is the size of the smallest acute angle, $t'$, formed by the terminal side of the angle $t$ and the horizontal axis.

Examples

• To find a positive angle coterminal with $820^\circ$, we subtract $360^\circ$ repeatedly. $820^\circ - 360^\circ = 460^\circ$. Since $460^\circ$ is still too large, we subtract again: $460^\circ - 360^\circ = 100^\circ$. So, $100^\circ$ is coterminal with $820^\circ$.
• To find a positive angle coterminal with $-210^\circ$, we add $360^\circ$: $-210^\circ + 360^\circ = 150^\circ$.
• To find an angle coterminal with $\frac{17\pi}{4}$ that is between $0$ and $2\pi$, we subtract $2\pi$. $\frac{17\pi}{4} - 2\pi = \frac{17\pi}{4} - \frac{8\pi}{4} = \frac{9\pi}{4}$. Since this is still greater than $2\pi$, we subtract again: $\frac{9\pi}{4} - 2\pi = \frac{\pi}{4}$.

Explanation

Coterminal angles are angles that 'land' in the same spot after spinning around a circle. Think of 3 o'clock and 15 o'clock on a watch—they point the same way. Finding a coterminal angle simplifies work by using an equivalent angle within one revolution.

Property

In a circle of radius $r$, the length of an arc $s$ subtended by an angle with measure $\theta$ in radians is given by the formula:

Examples

• In a circle with a radius of $8$ cm, the arc length subtended by a central angle of $\frac{3\pi}{4}$ radians is $s = 8 \times \frac{3\pi}{4} = 6\pi$ cm.
• To find the arc length in a circle with a radius of $10$ meters and a central angle of $90^\circ$, first convert the angle to radians: $90^\circ = \frac{\pi}{2}$ radians. Then, $s = 10 \times \frac{\pi}{2} = 5\pi$ meters.
• A circle has a radius of $4$ inches. If the arc length is $6\pi$ inches, the central angle is $\theta = \frac{s}{r} = \frac{6\pi}{4} = \frac{3\pi}{2}$ radians.

Explanation

This formula finds the length of a piece of a circle's edge, like the crust on a slice of pizza. The length of the arc depends on how big the circle is (radius) and how wide the angle of the slice is (theta).

Property

The area of a sector of a circle with radius $r$ subtended by an angle $\theta$, measured in radians, is:

Examples

• The area of a sector with a radius of $10$ feet and an angle of $\frac{\pi}{5}$ radians is $A = \frac{1}{2} (\frac{\pi}{5}) (10^2) = \frac{1}{10} \pi (100) = 10\pi$ square feet.
• A sprinkler waters a sector with a radius of $30$ feet and rotates $60^\circ$. First, convert to radians: $60^\circ = \frac{\pi}{3}$ rad. The area is $A = \frac{1}{2} (\frac{\pi}{3}) (30^2) = \frac{1}{6} \pi (900) = 150\pi$ square feet.
• If a sector has an area of $8\pi$ square meters and a radius of $4$ meters, the angle is found by solving $8\pi = \frac{1}{2} \theta (4^2)$, so $8\pi = 8\theta$, which means $\theta = \pi$ radians.

Explanation

A sector is a 'slice' of a circle, like a piece of pie. This formula calculates the area of that slice. A bigger slice (larger angle) or a pie from a bigger pan (larger radius) will have more area.

Property

As a point moves along a circle of radius $r$, its angular speed, $\omega$, is the angular rotation $\theta$ per unit time, $t$:

Examples

• A wheel completes 4 rotations in 2 seconds. Since one rotation is $2\pi$ radians, the total angle is $4 \times 2\pi = 8\pi$ radians. The angular speed is $\omega = \frac{8\pi}{2} = 4\pi$ rad/s.
• A point on the edge of a spinning disk with radius $0.5$ m has an angular speed of $10$ rad/s. Its linear speed is $v = r\omega = (0.5)(10) = 5$ m/s.
• A car with wheels 30 inches in radius travels at a speed of 40 mi/h. Its linear speed is $v$. The angular speed of its wheels is $\omega = \frac{v}{r}$. We must use consistent units to calculate this.

Explanation

Think of a spinning carousel. Everyone has the same angular speed (one rotation takes the same time). But a person on the outer edge travels a bigger circle, so their linear speed (how fast they are actually moving) is greater.