Angles are defined as geometric objects - wherever two lines, segments or rays intersect. We can also define an angle as the action of rotating a ray about its endpoint.

A four quadrant coordinate plane. A ray with its vertex at the origin and points to the positive x axis is labeled initial side. A ray with its vertex at the origin and points in second quadrant is labeled terminal side. The angle formed by the initial and terminal sides is shown.

Using this rotational definition, we define the starting position of the ray as the initial side of the angle. The ray's position after the rotation forms the terminal side of the angle.

If we view the angle in the coordinate plane, we say that the angle shown is in standard position.

Standard position

An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis

Initial side

The ray on the x-axis

Terminal side

The other ray of an angle in standard position

A four quadrant coordinate plane. An counterclockwise angle labeled positive is drawn with its initial side on the positive x axis and the terminal side on the third quadrant.

By defining an angle as a rotation, we can also allow for the concept of positive and negative angles. A counterclockwise rotation creates a positive angle.

A four quadrant coordinate plane. Moving clockwise, an angle labeled negative is drawn with its initial side on the positive x axis and the terminal side on the second quadrant.

It is also true that a clockwise rotation creates a negative angle.

Examples

Example 1

Consider the angles shown.

Angle where the initial side is on the negative x-axis and terminal side is at positive x-axis.

Angle rotating clockwise where the initial side is on the positive x-axis and terminal side is on Quadrant 4.

Which angle is in standard position?

Worked Solution

Create a strategy

Use the rotation symbol to check if the angle's initial side is along the positive x-axis.

Apply the idea

At option A, the rotation starts from the negative x-axis not from positive x-axis.

At option B, the rotation starts from the positive x-axis.

The answer is option B.

Determine whether the angle in standard position has a positive or negative measure.

Worked Solution

Create a strategy

It is positive if the angle rotates counterclockwise, while negative if it rotates clockwise.

Apply the idea

The angle rotates in clockwise direction so the angle measure is negative.

Idea summary

An angle is in standard position when its vertex is at the origin and its initial side coincides with the positive xaxis.

The terminal side, which is free to rotate, determines the angle's measure in either positive (counterclockwise) or negative (clockwise) directions.

Radian angle measures

Exploration

Drag the sliders to explore the applet. The different colors show the different lengths of each radian measure around the arc of the circle.

Loading interactive...

Is the number of arc lengths around the circle consistent for different radius measures?
How many arcs fit on the circumference of the circle?
Express the circumference as a multiple of the radius.
Determine the circumference of the circle in terms of \pi and the radius r.
How many radians are there in the circumference of a circle with a radius of 1?

Recall that an angle in radians can be calculated as \theta = \dfrac{s}{r}, where s is the arc length and r is the radius of the circle.

If we now restrict our attention to circles of radius one unit, then 1 radian would be the angle subtended by an arc of length 1 unit.

The circumference of a circle of radius one unit is C= 2 \pi \left( 1 \right) = 2 \pi, so the angle represented by a full turn around a circle is 2 \pi radians =6.28 radians =360 \degree.

Radian

The measure of an angle \theta that, when drawn as a central angle, subtends an arc whose length equals the length of the radius of the circle

Two unit circles, each divided into 4 equal parts. In the first circle, an arc that has a length of 1 unit is highlighted. The central angle that subtends the arc measures 1 subscript c. In the second circle, the whole semi-circle is highlighted and is labeled pi units. The angle that subtends the semi-circle is labeled pi units.

Since a half-circle is equivalent to \pi radians, and \pi = 180 \degree, angles given in radian measure are commonly expressed as fractions of \pi.

Examples

Example 2

For each fraction of a complete circle shown in the table below, find the corresponding degree and radian measures of each angle. If possible, simplify the radian degree measures.

	1	\dfrac{3}{4}	\dfrac{2}{3}	\dfrac{1}{2}	\dfrac{1}{3}	\dfrac{1}{4}	\dfrac{1}{6}	\dfrac{1}{8}	\dfrac{1}{12}
Measure in degrees	360 \degree
Measure in radians

Worked Solution

Create a strategy

Since we know that one full revolution of a circle is 360 \degree and equivalently 2 \pi radians, we will multiply each fraction by 360 to calculate the measure in degrees, and multiply each fraction by 2 \pi to calculate the measure in radians and simplify where possible.

Apply the idea

	1	\dfrac{3}{4}	\dfrac{2}{3}	\dfrac{1}{2}	\dfrac{1}{3}	\dfrac{1}{4}	\dfrac{1}{6}	\dfrac{1}{8}	\dfrac{1}{12}
Measure in degrees	360 \degree	270 \degree	240 \degree	180 \degree	120 \degree	90 \degree	60 \degree	45 \degree	30 \degree
Measure in radians	2 \pi	\dfrac{3\pi}{2}	\dfrac{4\pi}{3}	\pi	\dfrac{2\pi}{3}	\dfrac{\pi}{2}	\dfrac{\pi}{3}	\dfrac{\pi}{4}	\dfrac{\pi}{6}

Reflect and check

We can see the clear relationship between 180 \degree or \pi with 90 \degree, 60 \degree, 45 \degree, and 30 \degree since:

90 \degree is \frac{1}{2} of 180 \degree or \frac{1}{2} of \pi
60 \degree is \frac{1}{3} of 180 \degree or \frac{1}{3} of \pi
45 \degree is \frac{1}{4} of 180 \degree or \frac{1}{4} of \pi
30 \degree is \frac{1}{6} of 180 \degree or \frac{1}{6} of \pi

Example 3

For each of the following radian measures, sketch and label the corresponding angle on the unit circle in standard position. Explain how you chose to place the terminal side of the angle.

-1

\dfrac{\pi}{2}

Worked Solution

Create a strategy

On the coordinate plane, the unit circle is cut into four portions for each quadrant. We know that \dfrac{1}{4} of the entire unit circle is equivalent to 90 \degree or \dfrac{\pi}{2}.

Apply the idea

The axes on the coordinate plane form 90 \degree angles and we can see that the first quarter of the unit circle is equivalent to 90 \degree or \dfrac{\pi}{2}.

-1

Reflect and check

We can also describe the angle in terms of \pi since \pi is 180 \degree. If we need half of \pi or \dfrac{\pi}{2}, we know that this measure is half of a semicircle, which we have shown as the same as a quarter of a circle.

\dfrac{3 \pi}{2}

Worked Solution

Create a strategy

Place the initial side on the x-axis, then use the fact that each axis represents 90 \degree or \dfrac{\pi}{2} to choose what fraction of the entire circle \dfrac{3 \pi}{2} is:

Three circles, each plotted on a four quadrant coordinate plane and its center at the origin. On the left circle, the part of the circle on the first quadrant is shaded. The angle formed by the positive x axis and the positive y axis is labeled pi over 2. On the middle circle, the part of the circle on the first and second quadrants are shaded. The angle formed by the positive x axis and the negative x axis is labeled 2 times pi over 2. On the right circle, the part of the circle on the first, second, and third quadrants are shaded. The angle formed by the positive x axis and the negative y axis is labeled 3 times pi over 2.

Apply the idea

Since \dfrac{\pi}{2} represents 90 \degree or a quarter of one full circle, and we need to draw \dfrac{3 \pi}{2}, we would draw three portions of the unit circle we found in part (a).

-1

\dfrac{\pi}{3}

Worked Solution

Apply the idea

We know that \pi represents half of a revolution of the unit circle, so if we cut half of the circle into three, we know one of the three portions will be \dfrac{\pi}{3}.

-1

Reflect and check

Since \dfrac{\pi}{3} represents \dfrac{1}{3} of \pi or \dfrac{1}{3} of half of the unit circle, we know that there are a total of 6 of those portions throughout the unit circle. This makes sense since \dfrac{\pi}{3}=30 \degree and 360 \degree \div 6 = 30 \degree.

\dfrac{4 \pi}{3}

Worked Solution

Create a strategy

We can view this radian measure as four \dfrac{\pi}{3} of the unit circle. We know that \dfrac{\pi}{3} is a third of a semicircle, so in order to determine the location of \dfrac{4 \pi}{3}, we can imagine a total of four portions of the graph from part (c).

Apply the idea

A total of four \dfrac{\pi}{3} from part (c) would give us \dfrac{4 \pi}{3}.

-1

Example 4

Find the missing measure for each circle. Round your answers to two decimal places.

Given a circle with a 12 \text{ ft} arc subtended by the angle \dfrac{\pi}{4}, find the radius.

Worked Solution

Create a strategy

Use the formula for calculating the radian measure of an arc in a circle: \theta = \dfrac{s}{r}

Apply the idea

\displaystyle \theta	\displaystyle =	\displaystyle \dfrac{s}{r}
\displaystyle \dfrac{\pi}{4}	\displaystyle =	\displaystyle \dfrac{12}{r}	Substitution
\displaystyle \dfrac{\pi}{4} \cdot r	\displaystyle =	\displaystyle 12	Multiply both sides of the equation by r
\displaystyle r	\displaystyle =	\displaystyle 12 \cdot \dfrac{4}{\pi}	Multiply both sides of the equation by \dfrac{4}{\pi}
\displaystyle 15.28 \text{ ft}	\displaystyle =	\displaystyle r	Evaluate the multiplication

Reflect and check

Since the radius is unknown, we could start by rearranging the formula in terms of r and then substitute the values into the equation to solve for the radius:

\displaystyle \theta	\displaystyle =	\displaystyle \dfrac{s}{r}
\displaystyle \theta \cdot r	\displaystyle =	\displaystyle s	Multiply both sides of the equation by r
\displaystyle r	\displaystyle =	\displaystyle \dfrac{s}{\theta}	Divide both sides of the equation by \theta
\displaystyle r	\displaystyle =	\displaystyle \dfrac{12}{\frac{\pi}{4}}	Substitution
\displaystyle r	\displaystyle =	\displaystyle 15.28 \text{ ft}	Evaluate the division

Given a circle with a radius of 9 \text{ m}, find the measure of the arc subtended by a 2.09 \text{ radian} angle.

Worked Solution

Apply the idea

\displaystyle \theta	\displaystyle =	\displaystyle \dfrac{s}{r}
\displaystyle 2.09	\displaystyle =	\displaystyle \dfrac{s}{9}	Substitution
\displaystyle 2.09 \cdot 9	\displaystyle =	\displaystyle s	Multiply both sides of the equation by 9
\displaystyle 18.81 \text{ m}	\displaystyle =	\displaystyle s	Evaluate the multiplication

Reflect and check

Note that radians are not always written in terms of \pi.

Given a circle with a radius of 6 \text{ cm}, find the radian measure of the angle subtended by a 33 \text{ cm} arc.

Worked Solution

Create a strategy

By using the formula for calculating the radian measure of an arc, we know that the solution will be in radians, not degrees.

Apply the idea

\displaystyle \theta	\displaystyle =	\displaystyle \dfrac{s}{r}
\displaystyle \theta	\displaystyle =	\displaystyle \dfrac{33}{6}	Substitution
\displaystyle \theta	\displaystyle =	\displaystyle 5.5	Evaluate the division

Reflect and check

If we wanted to determine how many degrees 5.5 radians is, we could first determine the fraction of a circle that 5.5 radians makes up. This is \dfrac{5.5}{6.28} \approx 0.876. Then, we can multiply this by 360 \degree and see that the angle measure in degrees is about 315 \degree.

As we become more fluent with understanding radian measures in terms of \pi, we may notice that 315 \degree is a multiple of 45 \degree, which is equivalent to \dfrac{\pi}{4}. Since \dfrac{315 \degree}{45 \degree}= 7, the radian measure of this angle can also be written as 7 \cdot \dfrac{\pi}{4} = \dfrac{7 \pi}{4}.

Idea summary

A radian is the measure of an angle \theta that, when drawn as a central angle, subtends an arc whose length equals the length of the radius of the circle.

We use \pi = 180 \degree and equivalent fractions of a unit circle to convert between degrees and radians.

Coterminal angles

It is also possible to have an angle that rotates more than once around the circle. Rotations of this type will have measures with a magnitude greater than 2 \pi.

Because of the rotation definition of an angle, it's possible to have two angles with the same initial and terminal sides but different measures. Angles that are related in this way are called coterminal angles.

Two circles, each plotted on a four quadrant coordinate plane and its center at the origin. On the left circle, an angle is drawn with its initial side on the positive x axis, and the terminal side in the first quadrant. Moving counterclockwise, the angle is labeled theta with a spiral that makes one complete revolution around the circle before landing on the terminal side. This angle is labeled theta. Another counterclockwise angle with the same initial and terminal side but does not make a full revolution is shown and is labeled theta minus 2 pi. On the right circle, an angle is drawn with its initial side on the positive x axis and a terminal side on the fourth quadrant. Another angle with the same terminal and initial sides but moving clockwise is labeled theta plus 2 pi.

Coterminal angles

Angles in standard position that have a common terminal side

In general, an angle coterminal with another angle differs from it by an integer multiple of 2 \pi.

For the unit circle, we measure angles of any magnitude between the positive x-axis and the radius to a point that moves on the circle. The trigonometric functions of those angles are defined in a manner that guarantees that a function of any angle will be related to the same function of an angle in the first quadrant.

A unit circle with the four quadrants labeled. An angle is drawn with its initial side on the positive x axis, and the terminal side in the third quadrant. Counterclockwise the angle is labeled 7 pi over 6. Another angle labeled reference angle or relative acute angle is drawn with its initial side on the negative x axis and the same terminal side as the 7 pi over 6 angle.

We often use \theta for the angle and \alpha for the acute reference angle between 0 and \dfrac{\pi}{2} in the first quadrant.

Reference angle

An acute angle formed by the terminal side of an angle and the x-axis

To find a reference angle, first, if necessary, add or subtract multiples of 2 \pi to obtain an angle between 0 and 2 \pi. Then, use the symmetry of the unit circle to find the size of the acute angle in relation to the x-axis.

Examples

Example 5

Consider the angle - \dfrac{5 \pi}{4}.

Graph the angle in standard form.

Worked Solution

Create a strategy

We know that since the angle is negative, the terminal side of the angle will move in the clockwise direction.

First, draw a circle and since each semicircle is equivalent to \pi, cut the semicircles into four so that the circle has a total of eight equal portions.

A circle with a center at the origin plotted on a four quadrant coordinate plane. The circle is divided into 8 equal parts.

Use five measures of \dfrac{\pi}{4} in the clockwise direction to graph the angle in standard form.

A circle with a center at the origin plotted on a four quadrant coordinate plane. The circle is divided into 8 equal parts. An angle is drawn with its initial side on the positive x axis, and the terminal side in the second quadrant, just between the third and fourth part starting from the positive x axis. Clockwise an angle is drawn.

Apply the idea

-1

Find the measure of the coterminal angle to - \dfrac{5 \pi}{4} that is between 0 and 2 \pi.

Worked Solution

Create a strategy

Add 2 \pi to -\dfrac{5\pi}{4} until the coterminal angle is between 0 and 2 \pi.

Apply the idea

-\dfrac{5 \pi}{4} + 2 \pi = - \dfrac{ 5 \pi}{4} + \dfrac{8 \pi}{4} = \dfrac{3 \pi}{4}.

The measure of the coterminal angle to -\dfrac{5 \pi}{4} that is between 0 and 2 \pi is \dfrac{3 \pi}{4}.

Reflect and check

We could also have used the angle graphed in standard form on the unit circle to see that \dfrac{3 \pi}{4} is equivalent to -\dfrac{5 \pi}{4}.

Find the measure of the reference angle for - \dfrac{5 \pi}{4}.

Worked Solution

Create a strategy

Since the coterminal angle to -\dfrac{5 \pi}{4} is \dfrac{3 \pi}{4} which is in the second quadrant, we can use the symmetry of the unit circle to find the reference angle to \dfrac{3 \pi}{4} in the first quadrant.

Apply the idea

A circle with a center at the origin plotted on a four quadrant coordinate plane. The circle is divided into 8 equal parts. Starting from the positive x axis, the first and fourth parts are shaded. The first part is labeled reference angle for 3 pi over 4 in quadrant 1. The label negative 5 pi over 4 equals 3 pi over 4 is shown at the second quadrant.

The reference angle for -\dfrac{5 \pi}{4} is \dfrac{\pi}{4}.

Example 6

For each of the following angle measures, determine the coterminal angle between 0 and 2\pi and graph the angle in standard form on a coordinate plane.

\dfrac{35 \pi}{6} radians

Worked Solution

Create a strategy

The given angle is more than one full revolution, 360 \degree or 2 \pi, so subtract 2 \pi from \dfrac{35 \pi}{6} until we get to an angle coterminal to it between 0 and 2 \pi.

Apply the idea

\dfrac{35 \pi}{6} - \dfrac{12 \pi}{6} = \dfrac{23 \pi}{6} - \dfrac{12 \pi}{6} = \dfrac{11 \pi}{6}

-1

Reflect and check

Since \dfrac{35 \pi}{6} was a counterclockwise or positive rotation with its initial side along the positive x-axis, we know to find its coterminal side between 0 and 2 \pi by moving clockwise or in the negative direction, leading to subtracting two full revolutions.

-\dfrac{8 \pi}{3}

Worked Solution

Create a strategy

The given angle is negative, so to find the angle coterminal to it between 0 and 2 \pi, add 2 \pi to -\dfrac{8 \pi}{3} until we get to an angle coterminal to it between 0 and 2 \pi.

Apply the idea

-\dfrac{8 \pi}{3} + \dfrac{6 \pi}{3} = -\dfrac{2 \pi}{3} + \dfrac{6 \pi}{3} = \dfrac{4 \pi}{3}

-1

Reflect and check

Since -\dfrac{8 \pi}{3} was a clockwise or negative rotation with its initial side along the positive x-axis, we know to find its coterminal side between 0 and 2 \pi by moving counterclockwise or in the positive direction, leading to adding two full revolutions.

Idea summary

Coterminal angles have two angles with the same initial and terminal sides but different measures. To find coterminal angles, add or subtract an integer multiple of 2 \pi.

A reference angle is an acute angle formed by the terminal side of an angle and the x-axis.

Sine, cosine, and tangent on the unit circle

The image below shows a point P \left(x, y \right) with a terminal side length r.

A point P at (x, y) plotted on a first quadrant coordinate plane without numbers. A segment with length r is drawn from the origin to P. A vertical segment with length y is drawn from P to a point on the positive x axis. A horizontal segment with length x is drawn from the origin to a point along the positive x axis, and directly below point P. The segment with length r makes an angle labeled theta with respect to the positive x axis.

Once we place trigonometric ratios on the coordinate plane, we can now define trigonometric functions with respect to the acute reference angle \theta as:\begin{aligned} \text{The sine ratio: } \sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}} & = \dfrac{y}{r} \\\\ \text{The cosine ratio: } \cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} & = \dfrac{x}{r} \\\\ \text{The tangent ratio: } \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}} & = \dfrac{y}{x} \end{aligned}

Counterclockwise rotations of the terminal side of the angle represent positive angle measures, while clockwise rotations represent negative angle measures.

Angles may be measured in degrees or radians, and may be rational or irrational numbers. Angles larger than 360 \degree or 2 \pi result from continuing to rotate around the circle in multiple rotations. Thus, we can extend the definition of trigonometric functions to include values for all angles \theta in the set of real numbers.

Exploration

Explore the applet by dragging the triangle and checking the box.

Loading interactive...

What do you notice about the trigonometric ratio of an angle and its reference angle?
What do you notice about the signs of the ratios in different quadrants if you were to simplify them?

Examples

Example 7

The point on the following graph has coordinates \left( -7, -24 \right).

-20

-10

-20

-10

Find r, the distance from the point to the origin.

Worked Solution

Create a strategy

Use the distance formula to calculate the length of r.

Apply the idea

\displaystyle d	\displaystyle =	\displaystyle \sqrt{ \left ( x_2 - x_1 \right)^2 + \left( y_2 - y_1 \right)^2}	Distance formula
\displaystyle r	\displaystyle =	\displaystyle \sqrt{ \left( -7 - 0 \right)^2 + \left( -24 - 0 \right)^2}	Substitution
\displaystyle r	\displaystyle =	\displaystyle \sqrt{49 + 576}	Evaluate the subtraction and exponents
\displaystyle r	\displaystyle =	\displaystyle 25	Evaluate the addition and square root

Reflect and check

We could draw a right triangle from the point to the x-axis. The length of r could be calculated with the Pythagorean theorem, which is equivalent to the distance formula.

-20

-10

-20

-10

Evaluate the sine, cosine, and tagent functions for \theta. Leave the values of the functions as ratios.

Worked Solution

Create a strategy

Use \left(-7, -24 \right) and r=25 to write trigonometric ratios.

Apply the idea

\sin \theta = \dfrac{-24}{25}

\cos \theta = \dfrac{-7}{25}

\tan \theta = \dfrac{-24}{-7} = \dfrac{24}{7}

Idea summary

The sine ratio is represented by the following.

\displaystyle \sin \theta=\dfrac{\text{opposite}}{\text{hypotenuse}}=\dfrac yr

\bm{y}

is the y-coordinate of a point or the vertical distance from the origin

\bm{r}

is the terminal side length

The cosine ratio is represented by the following.

\displaystyle \cos \theta=\dfrac{\text{adjacent}}{\text{hypotenuse}}=\dfrac xr

\bm{y}

is the x-coordinate of a point or the horizontal distance from the origin

\bm{r}

is the terminal side length

The tangent ratio is represented by the following.

\displaystyle \tan \theta=\dfrac{\text{opposite}}{\text{adjacent}}=\dfrac yx

\bm{y}

is the y-coordinate of a point or the vertical distance from the origin

\bm{y}

is the x-coordinate of a point or the horizontal distance from the origin

Outcomes

3.2.A

Determine the sine, cosine, and tangent of an angle using the unit circle.

3.2 Sine, cosine, and tangent

Introduction

Ideas

Angles in standard position

Examples

Example 1

Radian angle measures

Exploration

Examples

Example 2

Example 3

Example 4

Coterminal angles

Examples

Example 5

Example 6

Sine, cosine, and tangent on the unit circle

Exploration

Examples

Example 7

Outcomes

3.2.A

What is Mathspace

About Mathspace