· discover the precise trigonometric role values for angles that measure up 30°, 45°, and also 60° using the unit circle.

You are watching: A point on the terminal side of angle θ is given

· uncover the specific trigonometric role values of any type of angle whose referral angle steps 30°, 45°, or 60°.

· recognize the quadrants whereby sine, cosine, and also tangent are positive and also negative.


Mathematicians produce definitions because they have actually a usage in solving particular kinds of problems. Because that example, the 6 trigonometric features were originally identified in terms of ideal triangles since that was beneficial in addressing real-world troubles that affiliated right triangles, such as finding angles of elevation. The domain, or collection of entry values, that these features is the collection of angles between 0° and also 90°. You will now learn brand-new definitions for these functions in i beg your pardon the domain is the collection of every angles. The brand-new functions will have the exact same values together the original functions when the intake is an acute angle. In a right triangle you can only have acute angles, yet you will see the definition extended to incorporate other angles.

One usage for these brand-new functions is the they have the right to be offered to discover unknown next lengths and angle procedures in any kind that triangle. These brand-new functions deserve to be supplied in many cases that have actually nothing to perform with triangle at all.

Before looking in ~ the brand-new definitions, you require to come to be familiar v the standard way that mathematicians draw and also label angles.


General Angles


From geometry, you recognize that an angle is developed by two rays. The rays meet at a point called a vertex.

*

In trigonometry, angles are put on coordinate axes. The peak is constantly placed at the origin and one beam is always placed ~ above the hopeful x-axis. This ray is referred to as the The stationary beam that creates an angle in conventional position and also lies ~ above the optimistic x-axis.


")">initial side
that the angle. The various other ray is referred to as the terminal side the the angle. This placing of an angle is referred to as The placement of an edge upon a set of name: coordinates axes v its vertex in ~ the origin, its early side put along the optimistic x-axis, and also a directional arrowhead pointing come the angle’s terminal side.


")">standard position
. The Greek letter theta () is often used to stand for an edge measure. Two angles in standard position are displayed below.

*

When an angle is attracted in typical position, it has a direction. An alert that over there are small curved arrows in the above drawing. The one on the left walk counterclockwise and is defined to it is in a optimistic angle. The one top top the appropriate goes clockwise and also is defined to be a an unfavorable angle. If you supplied a protractor to measure the angles, you would acquire 50° in both cases. We refer to the first one as a 50° angle, and we describe the 2nd one as a  angle.

Why would certainly you also have an unfavorable angles? just like all definitions, that is a matter of convenience. A spaceship in a circular orbit approximately Earth’s equator might be travel in either of two directions. So you could say that it traveled v a  angle to indicate that that went in opposing direction of a spaceship the went v a 50° angle. Why is counterclockwise positive? This is just a convention—something the mathematicians have actually agreed on—because one way has to be positive and the other means negative.

To see just how positive angles result from counterclockwise rotation and negative angles result from clockwise rotation, shot the interactive practice below. Either go into an angle measure in the box labeled “Angle” and hit go into or use the slider to relocate the terminal side of edge θ with the quadrants.

This is a Java Applet produced using GeoGebra native www.geogebra.org - it looks like you don"t have actually Java installed, please go to www.java.com


Example

Problem

Draw a 160° edge in typical position.

The edge is positive, so you begin at the x-axis and go 160° counterclockwise.

Answer

*


Example

Problem

Draw a

*
angle in typical position.

The edge is negative, therefore you start at the x-axis and go 200° clockwise. Remember that 180° is a directly line. That will bring you come the negative x-axis, and also then you need to go 20° farther.

Answer

*


Notice the the terminal sides in the two examples above are the same, yet they represent different angles. Such pairs of angles are claimed to be coterminal angles.

For every angle drawn in typical position, over there is a associated angle well-known as a The angle developed by the terminal next of an angle in typical position and also the x-axis, whose measure is in between 0° and also 90°.


")">reference angle
. This is the angle created by the terminal side and the x-axis. The recommendation angle is always considered to be positive, and has a value all over from 0° come 90°. Two angles room shown below in standard position.

*
*

You can see the the terminal next of the 135° angle and also the x-axis kind a 45° angle (this is because the two angles must add up come 180°). This 45° angle, presented in red, is the recommendation angle for 135°. The terminal next of the 205° angle and also the x-axis form a 25° angle. It is 25° due to the fact that

*
. This 25° angle, shown in red, is the referral angle for 205°.

Here room two much more angles in typical position.

*
*

The terminal next of the 300° angle and also the x-axis form a 60° edge (this is since the two angles must add up come 360°). This 60° angle, presented in red, is the recommendation angle because that 300°. The terminal side of the 90° angle and the x-axis kind a 90° angle. The referral angle is the same as the original angle in this case. In fact, any angle indigenous 0° come 90° is the very same as its reference angle.


Example

Problem

What is the referral angle because that 100°?

*

The terminal next is in Quadrant II. The original angle and also the recommendation angle together kind a directly line follow me the x-axis, for this reason their amount is 180°.

Therefore, the reference angle is 80°.

Answer

The recommendation angle because that 100° is 80°.


Example

Problem

What is the recommendation angle because that

*
?

*

The terminal side and the x-axis type the “same” angle as the original. A referral angle is constantly a positive number, so the recommendation angle here is 70°, presented in red.

Answer

The recommendation angle because that

*
 is 70°.


What is the reference angle for 310°?

A) 40°

B) 50°

C)

D)


Show/Hide Answer

A) 40°

Incorrect. You may have correctly drawn the angle in typical position, turning counterclockwise and also landing in the fourth quadrant. However, girlfriend may have actually looked in ~ the angle developed by the terminal side and also the y-axis, instead of the x-axis. The exactly answer is 50°.

B) 50°

Correct. In standard position, the angle will revolve counterclockwise with the first, second, third, and into the fourth quadrant. The referral angle and also the original angle together make one complete circle, or 360°. Therefore, the recommendation angle is

*
.

C)

Incorrect. Girlfriend may have correctly attracted the edge in standard position, turning counterclockwise and also landing in the 4th quadrant. However, girlfriend may have actually looked in ~ the angle developed by the terminal side and the y-axis, rather of the x-axis. Also, the referral angle is constantly positive. The exactly answer is 50°.

D)

Incorrect. You may have correctly drawn the edge in conventional position, transforming counterclockwise and landing in the fourth quadrant. However, you may have perplexed the referral angle with a coterminal angle. Remember the the reference angle is constantly positive. The exactly answer is 50°.

A unit circle is a circle that is centered at the origin and has radius 1, as shown below.

*

If  are the works with of a allude on the circle, climate you can see from the appropriate triangle in the drawing and also the Pythagorean Theorem that

*
. This is the equation the the unit circle.


The General an interpretation of the Trigonometric Functions


The 30° - 60° - 90° triangle is seen listed below on the left. Next to that is a 30° angle attracted in conventional position together with a unit circle.

*

The 2 triangles have actually the same angles, so they room similar. Therefore, matching sides are proportional. The hypotenuse top top the right has actually length 1 (because that is a radius). Because this is fifty percent of the hypotenuse ~ above the left, every one of the sides on the ideal are fifty percent of the matching sides top top the left. For example, the side surrounding to the 30 degree angle on the left is ; thus the equivalent side on the triangle ~ above the right needs to be fifty percent that, or .

Look in ~ the appropriate triangle ~ above the left. Using the meanings of sine and also cosine:

*

Now look at the allude where the terminal next intersects the unit circle. The x-coordinate is same to

*
, and the y-coordinate is equal to
*
. This is not a coincidence. Stop look at a much more general case.

Suppose girlfriend draw any acute edge  in typical position together with a unit circle, as viewed below.

*

The terminal side of the edge intersects the unit circle at the suggest . Let’s write the interpretations of the six trigonometric functions and then rewrite castle by referring to the triangle over and making use of the variables x and y.

*
*

The first equation and also the one listed below it, through the center steps cut out, phone call you:

*

Now you have the right to see that the y-coordinate that this suggest is constantly equal to the sine that the angle, and also the x-coordinate of this suggest is constantly equal come the cosine of the angle.

Now consider any angle  in traditional position in addition to a unit circle. The terminal side will intersect the circle at some suggest . Depending upon the angle, that suggest could it is in in the first, second, third, or fourth quadrant.

The sine that the edge  is same to the y-coordinate the this allude and the cosine the the edge  is same to the x-coordinate the this point. In fact, the 6 trigonometric functions for any kind of angle  are now defined by the 6 equations provided above.

The next couple of examples will help you check that when  is an acute angle, these brand-new definitions offer you the same outcomes as the initial definitions.


Example

Problem

Draw a 45° edge in conventional position in addition to a unit circle. Confirm that the x- and y-coordinates of the allude of intersection that the terminal side and the circle are equal to

*
 and
*
.

Start with the 45° - 45° - 90° triangle. Then draw the 45° edge in conventional position.

*

This unit one triangle is similar to the 45° - 45° - 90° triangle. Every side length have the right to be acquired by splitting the lengths of the 45° - 45° - 90° triangle by

*
. So each leg top top the unit one triangle is:
*

From the works with on the unit circle:

*

From the triangle:

*

From the works with on the unit circle:

*

From the triangle:

*

Look at the x- and also y-coordinates of the allude on the unit circle, then usage the triangle to discover  and .

The x-coordinate is same to , and also the y-coordinate is same to .

Answer

Yes,

*
 and
*
.


In the following two examples, the edge labels the 37° and also 53° are actually really close approximations. The main idea that the examples (that those fractions entailing x and y room equal come the assorted trigonometric functions) still holds true.


Example

Problem

Draw a 37° angle in typical position together with a unit circle. Usage the triangle listed below to discover the x- any y-coordinates of the suggest of intersection that the terminal side and the circle. Compute  and

*
. Check that they space equal to
*
 and
*
.

*

*

Draw the 37° angle in conventional position. The unit one triangle is similar to the 3-4-5 best triangle. Because this hypotenuse equals the initial hypotenuse split by 5, you can uncover the foot lengths by splitting the initial leg lengths by 5.

*

Find the x- and also y-coordinates.

*

*

*

*

Compute the ratios. Compare the results to what friend would obtain for

*
 and
*
 using the initial triangle. They room the same!

Answer

So yes,

*
 and
*
.


Example

Problem

Draw a 53° angle in traditional position together with a unit circle. Use the triangle below to uncover the x- any kind of y-coordinates of the suggest of intersection that the terminal side and also the circle. Compute

*
 and
*
. Confirm that they are equal come
*
 and
*
.

*

*

Draw the 53° edge in conventional position. The unit one triangle is similar to the 3-4-5 best triangle. Because this hypotenuse amounts to the original hypotenuse split by 5, friend can discover the foot lengths by splitting the original leg lengths through 5.

*

Find the x- and y-coordinates.

*

*

*

*

Compute the ratios. Compare the results to what girlfriend would acquire for

*
 and
*
 using the initial triangle. They are the same!

Answer

Yes,

*
 and
*
.


The very first three of our new definitions lead us to one much more important identity:

*

We deserve to replace y by

*
and x through
*
in to acquire the trigonometry identification .

*

Since cotangent is the mutual of tangent, this provides you one more trigonometric identity.

*

Remember, an identity is true for every possible value of the variable. Therefore no issue what angle you are using, the worths of tangent and cotangent are given by this quotients.

Although some textbooks offer slightly different general meanings of the trigonometric functions, the important thing to understand is the they finish up offering you the exact same values as the definitions currently given you.

For example, start with a circle of radius r (in ar of radius 1) and also an angle  in traditional position. The terminal side will certainly intersect the circle at some point . That suggest could it is in in any kind of quadrant, but we display one in the an initial quadrant.

*

Now compose down the initial definitions and then rewrite them making use of the variables x, y, and r.

*
*

These six fractions are supplied as the general meanings of the trigonometric functions for any type of angle , in any type of quadrant.


Example

Problem

Compute  using the diagram below. Check that this is the same as the worth of

*
.

*

*

The over diagram has a 30° - 60° - 90° triangle. The hypotenuse amounts to the radius, so the is 10. The side opposite 30° is half of 10, or 5. The adjacent side is  times the contrary side, or

*
.

*

*

The political parties of the triangle give you the values of x and y in the first diagram.

*

Substitute these into the definition.

*

Here is our traditional 30° - 60° - 90° triangle.

*

Compute

*
 using the ideal triangle definition.

Answer

So yes,

*
.


You have actually been given brand-new or “general” definitions of the 6 trigonometric functions and have seen that, when you compute these features using acute angles, the an outcome is the exact same as the result you would obtain from using the initial definitions. Currently you will learn how to use these interpretations to angle that are not acute and to negative angles.

Given any angle , draw it in conventional position together with a unit circle. The terminal side will intersect the circle in ~ some allude , as presented below.

Here again room the general definitions of the six trigonometric attributes using a unit circle.

*

Now let’s use these meanings with the angles 30°, 150°, 210°, and 330°. You have currently done this because that 30°. Here is that drawing:

*

The angle 150°, 210°, and 330° have something in common. Every one of them has actually a recommendation angle the 30°, together you deserve to see from the drawings below. Because of this, that is simple to discover the coordinates of the points wherein the terminal sides intersect the unit circle using the drawing above.

*

For the angle 150°, this intersection suggest is the mirror picture of  over the y-axis, for this reason the collaborates for 150° are.

For the edge 210°, this allude is the mirror photo of  over the x-axis, therefore the works with for 210° are

*
.

For the angle 330°, this allude is the mirror picture of  over the x-axis, therefore the works with for 330° room

*
.

You deserve to now find the values of all 6 trigonometric attributes for 150°, 210°, and 330°. Let’s pick a few trigonometric functions and also evaluate them using these angles. For example, utilizing the leftmost diagram over and the an interpretation of cosine:

*

Using the center diagram and the definition of cotangent:

*

Using the rightmost diagram and the an interpretation of cosecant:

*

If you take the drawing over with the 30° edge in traditional position, and also turn the triangle so the the much shorter leg is on the x-axis, you acquire a illustration of a 60° angle in typical position, as seen below.

*

You deserve to use the details in this diagram to uncover the worths of the 6 trigonometric attributes for any type of angle that has a reference angle of 60°.


Example

Problem

Determine

*
 and
*
.

*

Draw 300° in traditional position and find the reference angle. Find the x-coordinate of the point  where the terminal side intersects the unit circle. Due to the fact that cos 60° = ½, we recognize x = ½.

*

Use the definition of cosine. Instead of the worth of the x-coordinate that you uncovered above.

*

Use the definition of secant. Instead of the value of the x-coordinate the you uncovered above. Keep in mind that, just just like acute angles, secant and also cosine are reciprocals.

Answer

*
,
*


The procedure is the same even if the angle is negative. Remember that a negative angle is simply one who direction is clockwise.


Example

Problem

Find the worths of

*
 and
*
.

*

Draw  in standard position and also find the recommendation angle. Uncover the y-coordinate that the suggest  where the terminal next intersects the unit circle.

Because

*
 and we are in the third quadrant, we recognize
*
.

*

Use the definition of sine. Instead of the worth of the y-coordinate the you discovered above.

*

Use the definition of cosecant. Substitute the value of the y-coordinate the you found above. Rationalize the denominator. Note that, just as with acute angles, cosecant and sine room reciprocals.

Answer

*
,
*


Look at the outcomes from the last 2 examples and also observe the following:

In every case, the value of the trigonometric function was either the exact same as the value of that function for the recommendation angle (60°), or the negative of the value of that function for the reference angle. Why did this happen? The computations because that 60° were done using the allude

*
. The computations because that 300° and also  were done using the points
*
 and
*
. The x-coordinates have the same absolute value. The y-coordinates additionally have the same absolute value. When you substitute right into the expressions x,
*
, y, and

*
, the an outcome will it is in the same, or have actually a negative sign.

You will obtain a similar result with various other angles. So the procedure because that finding the value of a trigonometric role simplifies to the following:

recognize the referral angle. Calculation the trigonometric duty value that the recommendation angle. Determine if the value of the duty is positive or negative.

Let’s shot this procedure in the adhering to example.


Example

Problem

Compute

*
 and
*
.

*

Draw the edge in conventional position. The recommendation angle is 45°.

Use the 45° - 45° - 90° triangle. The worth of

*
 is 1.

*

In the very first diagram, we placed a

*
 sign to indicate that x is positive, and also a
*
 sign to show that y is negative. Usage this to identify the authorize of .

*

Since the result was negative, the worth of  is negative.

*

You have the right to go v a similar procedure v cotangent or usage the truth that it is the reciprocal of tangent.

Answer

*
,
*


The angle whose measures are a lot of of 90° have terminal political parties on the axes. This deserve to be confusing, due to the fact that the terminal next is not in one quadrant, however rather top top a border between quadrants. So let’s look in ~ these angles separately. The drawing listed below shows the point out of intersection the the terminal political parties of 0°, 90°, 180°, and 270° with the unit circle.

*

You deserve to use this drawing and also the interpretations to find the trigonometric features for 0°, 90°, 180°, and also 270°. For example:

*

For all 6 functions, you substitute the values of x and y together you walk earlier. However, what happens if you try to compute utilizing the meaning ?

*

You cannot divide by 0, therefore  is simply undefined. Similarly

*
 is undefined, because if you try to use the definition, friend will end up splitting by 0. The same is true any type of time among the interpretations leads to department by 0: the trigonometric function is unknown for the angle.

What room the values of

*
 and ?

A)

B)

C)

D)


A)

Incorrect. The value for cosine is correct. However, when assessing cotangent, girlfriend may have switched the x- and y-coordinates. The exactly answer is B.

B)

Correct. Due to the fact that  and 90° coincides to the allude

*
,
*
. Currently
*
 and 180° corresponds to the suggest
*
. If friend tried to apply the an interpretation you would be splitting by 0, due to the fact that
*
. This way that  is undefined.

C)

Incorrect. When assessing both of this functions, friend may have actually switched the x- and y-coordinates. The exactly answer is B.

D)

Incorrect. The worth for cotangent is correct. However, when examining cosine, friend may have actually switched the x- and y-coordinates. The exactly answer is B.

You have the right to use the adhering to charts to aid you mental the worths of the trigonometric functions for the reference angles 0°, 30°, 45°, 60°, and 90° because that sine and cosine. As soon as you have these, girlfriend can obtain the value of tangent indigenous the identity , and also the values of the other three trigonometric functions using reciprocals.

Make a table together follows:


As an initial step, put the numbers 0, 1, 2, 3, and 4 in the “sine” row and also 4, 3, 2, 1, and 0 in the “cosine” row. Do this in pencil. You space going to replace these numbers!


Now change the numbers 0 with 4 by taking their square roots and also dividing by 2. The rows now contain the correct, yet unsimplified, worths for sine and cosine.


30°

45°

60°

90°

sine

cosine


You deserve to simplify

*
 to 0,
*
 to 1, and also
*
 to 2, and then divide by 2. This will provide you the final table through the correct worths of sine and cosine at these angles.


30°

45°

60°

90°

sine

0

1

cosine

1

0


First you learned the definitions for the trigonometric attributes of one acute angle. Then you learned the general interpretations of this functions, which have the right to be supplied for any angle, and also the technique for applying them. Finally, friend learned a less complicated procedure because that finding the worths of trigonometric functions:

recognize the recommendation angle. Calculation the trigonometric duty value that the recommendation angle. Recognize if the worth of the role is hopeful or negative.

Now you’ll find out an easy way to remember where the trigonometric features are positive and where they are negative.

The sine function: because

*
, sine is positive once
*
. This wake up in Quadrants I and also II.

The cosine function: because , cosine is positive when

*
. This wake up in Quadrants I and also IV.

The tangent function: because , tangent is positive when x and y space both confident or both negative. This occurs in Quadrants I and also III.

We can summarize this info by quadrant:

Quadrant I: sine, cosine, and tangent space positive.

Quadrant II: sine is positive (cosine and tangent room negative).

Quadrant III: tangent is hopeful (sine and also cosine space negative).

Quadrant IV: cosine is hopeful (sine and tangent are negative).

Let A stand for all (three functions, sine, cosine, and also tangent), S was standing for sine, T was standing for tangent, and also C was standing for cosine. Currently you can use these single letters to remember in which quadrant sine, cosine, and also tangent room positive. The last step is to replace each letter by a indigenous to offer you a expression that’s basic to remember: “All Students take Calculus.” going counterclockwise, place these words in the 4 quadrants.

*

Now if you look in Quadrant II, for example, you see words Students. The S tells you that sine is optimistic (while cosine and also tangent are negative).


Example

Problem

What indicators are

*
 and
*
?

Since

*
, 200° is in Quadrant III.

The indigenous “Take” represents the truth that tangent is positive, for this reason

*
.

Sine and cosine are an adverse in Quadrant III, for this reason

*
.

Answer

*


Example

Problem

In i beg your pardon quadrant have to an angle lie if the sine is positive and its tangent is negative?

The native “All” and “Students” tell united state that sine is confident in Quadrants I and II.

Tangent is optimistic in Quadrant I, but an unfavorable in Quadrant II.

Answer

Quadrant II


This an equipment applies come the functions sine, cosine, and also tangent. The various other three trigonometric features are reciprocals of these three. Remind the simple fact that the mutual of a positive number is positive, and the reciprocal of a an unfavorable number is negative. This indicates that sine and also cosecant have actually the exact same sign, cosine and secant have actually the same sign, and also tangent and cotangent have the very same sign. Therefore if you want to recognize the sign of cosecant, secant, or cotangent, discover the authorize of sine, cosine, or tangent, respectively.

What indicators are

*
 and
*
?

A) They space both positive.

B) They room both negative.

C)

D)


A) They room both positive.

Correct. Edge  is in Quadrant IV. Words “Calculus” speak you that cosine is positive. Due to the fact that secant is the mutual of cosine, that is additionally positive.

B) They are both negative.

Incorrect. Possibly you believed that due to the fact that the edge is negative, the values of these features would it is in negative. The values rely on the quadrant in i beg your pardon the edge lies. Edge  lies in Quadrant IV. The correct answer is A.

C)

Incorrect. The authorize of cosine is correct. Maybe you assumed that once you take it a reciprocal, the sign changes, but the authorize doesn’t change. Therefore, secant and cosine have the same sign. The exactly answer is A.

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D)

Incorrect. The authorize of secant is correct. Maybe you thought that because the angle is an adverse the worth of cosine would certainly be negative. The values count on the quadrant in which the edge lies. Angle  lies in Quadrant IV. The exactly answer is A.

The trigonometric functions were originally defined for acute angles. There space general definitions of these functions, which apply to angles of any kind of size, including an unfavorable angles. Worths of trigonometric functions are computed by finding the referral angle, determining the value of the trigonometric function of the reference angle, and also then determining if the value of the function is hopeful or negative. A useful means to remember this last action is “All Students Take Calculus.”