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The relation between angles and their associated trigonometric ratios can be shown using circles of any size. For simplicity, circles with a radius equal to one unit are used when showing those relations. This lesson will introduce the concept of unit circle and how it can be used to represent angles.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Trigonometric Ratios of Non-Acute Angles

Trigonometric ratios are defined for acute angles, however, sometimes it is necessary to find the trigonometric ratio of a non-acute angle
Sine and cosine of any angle applet
Is it possible to define the sine and cosine of any angle? Consider the other trigonometric ratios. Can those be defined for a non-acute angle
Explore

Angles With Vertex at the Origin

Many angles can be drawn on a coordinate plane by placing the vertex in the origin and choosing two rays. By this method, it is possible to have different angles with the same measure. In the applet, move the rays around to form an angle with a measure of Then, try getting the same measure, but with the rays in entirely different positions than the first time.

Angles with vertex at the origin in a coordinate plane applet
Discussion

Angles Centered at the Origin

Different angles with a shared vertex at the origin can be formed such that they all have the same measure. This can be somewhat vague, however. To remove this ambiguity, one of the sides of the angle is fixed and an orientation is chosen.

Concept

Standard Position of an Angle

An angle in the coordinate plane is in standard position if its vertex is at the origin and one ray is on the positive axis. The ray on the axis is called the initial side. The other ray is the terminal side of the angle.
Angle in Standard Position
The measure of an angle can be positive or negative.
  • If the terminal side is rotated counterclockwise, then the measure of the angle is positive.
  • If the terminal side is rotated clockwise, then the measure of the angle is negative.

Move the slider to see how the angle measure changes.

Example

Placing the Numerals of a Clock

Dylan always wakes up late for school and misses the bus. He wants to design a clock with an alarm on it. Dylan brings a rough sketch of the design to his Auntie Wilma, an engineer. Seeing the numbers are not accurately placed, she teaches Dylan how to properly distribute the numbers of the clock.

Uncle Will telling Dylan that the twelve numerals are 30 degrees apart

Knowing this, Dylan erases all the numbers on his design and starts over. Auntie Wilma recommends that he draw a coordinate plane on top of the design so that the origin is at the center of the clock.

Coordinate plane on a clock without hands and numbers
Which numeral should Dylan draw at in standard position?
Which numeral should he draw at in standard position?

Hint

An angle of in standard position is on the axis. Use the fact that the numerals are apart from each other.

Solution

Dylan starts by remembering that the initial side of an angle in standard position is on the axis. Since the measure of the angle is the terminal side of the angle lies on top the initial side.

Coordinate plane on a clock showing an angle of 0 degrees in standard position

The numeral at is the one on the right hand side of a clock. By looking at one of Auntie's clocks, Dylan figures out that the numeral is He goes ahead and writes it down on his clock design.

Dylan's new design with only numeral 1
According to Auntie Wilma, numerals are apart from each other. Thus, to find the numeral at in standard position, Dylan counts how many intervals are in
Auntie Wilma reminds Dylan here that positive angles are measured counterclockwise.
Coordinate plane on a clock without hands showing angles of 30 and 60 degrees in standard position

Once more, Dylan checks one of Auntie's clocks and finds that is the numeral at So, he writes it down. So far, his design looks as follows.

Dylan's new design with only numerals 1 and 3
Dylan is off to a great start in his process of making a clock. He is optimistic that should he make an accurate clock, he will finally be able to wake up on time for school!
Example

A Long Painting Session

Dylan sets out some art supplies to begin painting the clock's housing. He checks another one of Auntie Wilma's clocks as he begins painting. It is

Clock showing the time at 3:00 PM

Finished and exhausted, Dylan checks the clock again. He is surprised to see that it is now What a long painting session!

Clock showing the time at 8:00 PM
Auntie Wilma sees this as an opportunity to teach Dylan about measurements and asks him to find the measure of the angle, in degrees and radians, made by the displacement of the hour hand.

Hint

Clockwise rotations are defined to be negative. To find the measure in radians, use the fact that equals radians.

Solution

The painting session lasted from to This means that it was hours long.
Since the numerals in the clock are apart from each other, it is possible to find the displacement of the hour hand of the clock.
Because the hands of the clock rotate clockwise, the angle measure is negative. This means that the hour hand was displaced by
Clock showing the time at 8 PM
To find the measure in radians, the relation between radians and degrees will be used.

Relation between Radians and Degrees

This relation can be used to find the displacement of the hour hand in radians.
Consequently, the hour hand rotated radians.
Earlier, Dylan could not correctly place the numerals on his clock design. Now he can find the measure of the angles made by the displacements of the hands in both degrees and radians on a functioning clock. Talk about progress!
Discussion

Angles With the Same Terminal Side

Depending on how the rotation of its terminal side was performed, it is possible to assign different measures to an angle in standard position. These angles, which have the same terminal side, have a special name.

Concept

Coterminal Angles

Two or more angles in standard position with the same terminal side are called coterminal angles. For example, angles measuring and are coterminal angles.
Angles of 50, 410, and -310 degrees in standard position
If two angles are coterminal, the measure of one of them is equal to the measure of the other angle plus a multiple of or in the case of radians, plus a multiple of Therefore, for any angle and any integer coterminal angles can be found by the following formula.
Example

Finding Coterminal Angles

Although Dylan has just measured angles counterclockwise, Auntie Wilma suggests that he should be able to measure angles clockwise. Dylan knows that the angle between the and numeral measures his goal is to express that angle in the same direction as the movement of the clock's hands.

Clock showing the angle between the 1 and 3 numerals
Find the negative coterminal angle, in degrees, that is closest to
Now, in radians, find the negative coterminal angle that is closest to

Hint

Coterminal angles can be found by adding or subtracting multiples of or rad, from the given angle.

Solution

Dylan wants to find an angle that is coterminal to a angle that is measured in the same direction as the movement of the clock's hands. This means that he is looking to measure it clockwise. That means the measure will be negative in value.

Clock showing the angle between the 1 and 3 numerals
Coterminal angles can be found by adding or subtracting multiples of to a given angle. Since Dylan is looking for the negative coterminal angle closest to the coterminal angle can be found by subtracting from
To find the measure in radians, the relation between radians and degrees will be used.

Relation between Radians and Degrees

This relation can be used to convert into radians.
Therefore, the coterminal angle measures radians.
Dylan has added another skill in his understanding of clocks. Previously he could find the angle of displacement of hands of the clock counting counterclockwise, and now he can do it clockwise. He is one step closer to building an awesome alarm clock.
Discussion

Quadrantal and Reference Angles

One of the goals of this lesson is to associate trigonometric ratios to any angle. However, these are defined for acute angles, so a way around must be chosen. Quadrantal and reference angles are used for this purpose.

Concept

Quadrantal Angle

An angle in standard position is called a quadrantal angle if its terminal side lies on the or axis.
Quadrantal angles
A quadrantal angle has a measure that is a multiple of or radians. Therefore, the measure of a quadrantal angle can be written in the following form, where is a certain integer.
Concept

Reference Angle

For an angle that is not quadrantal, the acute angle formed by the terminal side of and the axis is called a reference angle. Examine the diagram to see the relationship between and for
Calculating the corresponding reference angle of an angle in any quadrant
Since angles can also be measured in radians, the above relationships can also be written in radians.
Degrees Radians
Quadrant I
Quadrant II
Quadrant III
Quadrant IV

If is greater than or less than then finding its coterminal angle with a positive measure between and will be helpful to find the reference angle.

Angle Coterminal Angle Reference Angle
Reference angles are useful when evaluating a trigonometric function for any angle
Example

Finding Quadrantal and Reference Angles

Dylan is still working on pinpointing the clock numerals. To help himself, he draws a coordinate plane on top of his design, just as Auntie Wilma suggested.

A clock without hands, but with the x and y-axis.
Auntie Wilma says to Dylan that now he should draw the numerals corresponding to the quadrantal angles. What numerals should Dylan draw?
Once done, Auntie Wilma asks Dylan to draw the numerals that form a reference angle. What numerals should Dylan draw?

Hint

Identify which numerals are on the axes of the coordinate plane. Find the numerals that are next to the ones on the axis.

Solution

The terminal side of a quadrantal angle lies on the or axis. Therefore, the numerals that Dylan should draw are those that lie on top of the coordinate axes, except for the numeral which he drew earlier.

A handless clock with numerals 1 and 3.

Once more, Dylan looks at one of Auntie's clock and sees that the numerals corresponding to quadrantal angles are and However, he already drew Hence, he proceeds to draw these three new numerals in his design.

A handless clock with the numerals 1, 3, 6, 9, and 12

To find the numerals that form reference angles of Dylan first reviews the definition of reference angle.

Reference Angle

For an angle that is not quadrantal, the acute angle formed by the terminal side of and the axis is called a reference angle.

Note that this is the angle between the axis and the corresponding numeral. This means that it does not have to be an angle in standard position; it can also be either clockwise or counterclockwise.

Reference angles of 30 degrees marked in a handless clock

By looking at the clock on the living room wall of Auntie's house, Dylan finds that the numerals which form a reference angle are and He goes ahead and adds those numerals to the design.

A handless clock with numerals 1,2,3,4,6,8,9,10, and 12
Dylan continues to make progress in his design. Thanks to quadrantal angles and reference angles he added seven more numerals to the clock. He just has three more to go.
Pop Quiz

Finding Reference Angles

An angle in standard position is given in the following applet.

reference angle random applet
Discussion

Relating Coordinates to Right Triangles

By placing a right triangle with one vertex at the origin and one of their legs along the axis, the coordinates of can be found using the legs of the triangle.

Right triangle with two vertices on the x-axis and the other on the first quadrant
If said vertex is in the first quadrant, then its coordinates are the lengths of the triangle's legs. Considering that lengths cannot be negative, the sign of the corresponding coordinate changes accordingly if the vertex is not on the first quadrant.
Applet with right triangle with two vertices on the x-axis and movable vertex
If the length of the hypotenuse is then the length of the legs for special right triangles can be found.
Three special right triangles with hypotenuse equal to 1

Since the hypotenuse of these triangles is the marked points all lie on a circle with radius whose center is at the origin.

Three special right triangles and a circle with radius 1

This circle receives a special name and is related to trigonometric ratios, particularly to sine and cosine.

Concept

Unit Circle

The unit circle is a circle with radius and whose center lies on the origin of a coordinate plane.

A right triangle can be associated to a point on the circle. This can be done by choosing its hypotenuse equal to the radius of the circle. The legs of the triangle are parallel to the axes.

The coordinates of a point on the unit circle can be related to trigonometric ratios by selecting as the angle that is at the origin.

The length of the hypotenuse is equal to one because it is the radius of the unit circle, so the point on the unit circle can be used to generalize sine and cosine functions.
Because of this, sine and cosine are also called circular functions.
Discussion

Sine and Cosine of Any Angle

The values of sine and cosine are defined for acute angles, however, for angles that are not acute, an appropriate reference angle can be used. The sign is chosen depending on which quadrant lies the terminal side.
If the angle is a quadrantal angle, the following values are assigned to each function.
Quadrantal Angle Sine Cosine
Example

Placing Non-Quadrantal Numerals of a Clock

Dylan is ready to finish the clock. He places it, without the housing, in his works place, then sets out a huge sheet of pink graph paper over the already placed numerals. He placed each numeral foot away from the clock's center.

Handless clock with only the 3, 6, 9, and 12 numerals

Dylan wants to place the rest of the numerals, but he has forgotten a protractor to measure the angles! Auntie Wilma points out that his clock can be seen as a unit circle, therefore, the rest of the numerals can be placed by finding their coordinates.

Handless clock with only the 3, 6, 9, and 12 numerals showing its radius of 1 foot
Find the coordinates where Dylan should place the numeral Round each coordinate to two decimal places.
Find the coordinates where Dylan should place the numeral Round each coordinate to two decimal places.

Hint

Find the sine and cosine of the angle corresponding to each numeral.

Solution

Dylan begins by finding the angle corresponding to each of the numerals. Going clockwise, the numeral is units away from the reference numeral. Dylan notices that since each numeral is apart from each other, the angle corresponding to the numeral can be found.
Dylan thinks that the angle corresponding to the numeral can be found in a similar way. It is units away clockwise from the numeral, which is at
Auntie Wilma recommends to summarize the information in a table, along with the corresponding reference angle.
Numeral Angle Reference Angle
Knowing that the numerals are placed in a unit circle, Dylan will find their coordinates using circular functions.
Dylan knows that Auntie Wilma would suggest that this information be added into the table.
Numeral Angle Reference Angle coordinate coordinate
Next, using a calculator, Dylans finds the sine and cosine of He then rounds each result to two decimal places.
Dylan then writes the coordinates of the and numerals into the table.
Numeral coordinate coordinate

Now that Dylan knows the coordinates of the numerals, he proceeds and draws them. Right before drawing the last numeral, Auntie Wilma asks him to let her do it, and Dylan happily agrees.

Handless clock with all the numerals

Dylan feels so proud to have made it this far in the clock design. Thanks to viewing his clock as a unit circle, he could draw the last three numerals. Dylan is supremely confident that they are correctly placed. He puts on the finishing touches, removes some of the tics for a more modern look, and sets up this huge and funny looking alarm clock by his bed.

A very cool alarm clock
Closure

Trigonometric Ratios of Any Angle

This lesson discussed a way of measuring angles without ambiguity as well as introducing the concepts of unit circle and circular functions. These concepts can be used to solve the challenge presented at the beginning.
Sine and cosine of any angle applet
It was found that, by using the unit circle, it is possible to define the sine and cosine of any angle, also known as circular functions. The rest of the trigonometric ratios will be explored using the same idea.
Unit Circle trigonometric ratios
Note how the rest of the trigonometric ratios have either or in their denominator. Since division by is not allowed, this implies that the tangent and the secant are undefined when that is, for the quadrantal angles on the axis, as well as for any of their coterminal angles.

The tangent and the secant are undefined for or any of their coterminal angles.

Likewise, the cotangent and cosecant are undefined when that is, for the quadrantal angles on the axis, as well as for any of their coterminal angles.

The cotangent and the cosecant are undefined for or any of their coterminal angles.