Trigonometric Functions on Any Angle

Section 4.4

Objectives• Determine the quadrant in which the

terminal side of an angle occurs. • Find the reference angle of a given angle. • Determine the sine, cosine, tangent,

cotangent, secant, and cosecant values of an angle given one of the sine, cosine, tangent, cotangent, secant, or cosecant value of the angle.

Vocabulary

• quadrant • reference angle • sine of an angle • cosine of an angle • terminal side of an angle • initial side of an angle • tangent of an angle • cotangent of an angle • secant of an angle • cosecant of an angle

Reference AngleA reference angle is the smallest distance between the terminal side of an angle and the x-axis.

All reference angles will be between 0 and π/2.

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Reference AngleThere is a straight-forward process for finding reference angles.

Step 1 – Find the angle coterminal to the given angle that is between 0 and 2π.

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Reference AngleThere is a straight-forward process for finding reference angles.

Step 2 – Determine the quadrant in which the terminal side of the angle falls.

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Reference AngleThere is a straight-forward process for finding reference angles.

Step 3 – Calculate the reference angle using the quadrant-specific directions.

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θ

Reference Angle

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For quadrant I, the shortest distance from the terminal side of the angle to the x-axis is the same as the angle θ.

Thus

where the reference angle is

Directions for quadrant I

Reference Angle

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θ

This distance is the reference angle.

Directions for quadrant II

For quadrant II, the shortest distance from the terminal side of the angle to the x-axis is shown in blue. This is the rest of the distance from the terminal side of the angle to π.

Thus

Note: Here put subtracted the angle from π since the angle was smaller than π. This gave us the positive reference angle. If we had subtracted π from the angle, we would have needed to take the absolute value of the answer.

Reference Angle

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θ

For quadrant III, the shortest distance from the terminal side of the angle to the x-axis is shown in blue. This is the distance from the π to the terminal side of the angle.

Directions for quadrant III

This distance is the reference angle.

Thus

Reference Angle

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θ

Directions for quadrant III

Note: Here put subtracted π from the angle since the angle was larger than π. This gave us the positive reference angle. If we had subtracted the angle from π, we would have needed to take the absolute value of the answer.

θ

Reference AngleDirections for quadrant IV

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For quadrant IV, the shortest distance from the terminal side of the angle to the x-axis is shown in blue. This is the rest of the distance from the terminal side of the angle to 2π.

This distance is the reference angle.

Thus

2

θ

Reference AngleDirections for quadrant IV

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2

Note: Here put subtracted the angle from 2π since the angle was smaller than 2π. This gave us the positive reference angle. If we had subtracted 2π from the angle, we would have needed to take the absolute value of the answer.

Reference Angle Summary

2Quadrant IV

Quadrant II

Quadrant III

Quadrant I

Step 3 – Calculate the reference angle using the quadrant-specific directions indicated to the right.

Step 2 – Determine the quadrant in which the terminal side of the angle falls.

Step 1 – Find the angle coterminal to the given angle that is between 0 and 2π.

In which quadrant is the angle ? 6

7

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To find out what quadrant θ is in, we need to determine which direction to go and how far. Since the angle is negative, we need to go in the clockwise direction. The distance we need to go is one whole π and 1/6 of a π further.

This blue part is one whole π in the clockwise direction

This red part is approximately 1/6 of a π further.

Now that we have drawn the angle, we can see that the angle θ is in quadrant II.

What is the reference angle, , for the angle

?

67

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Using our summary for finding a reference angle, we start by finding an angle coterminal to θ that is between 0 and 2π. Thus we need to start by adding 2π to our angle.

65

anglecoterminala

612

67

anglecoterminala

267

anglecoterminala

What is the reference angle, , for the angle

?

67

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65

anglecoterminala

The next step is to determine what quadrant our coterminal angle is in. We really already did this in the first question of the problem. Coterminal angles always terminate in the same quadrant. Thus our coterminal angle is in quadrant II.

65

What is the reference angle, , for the angle ?

67

65

anglecoterminala

Quadrant II

Finally we need to use the quadrant II directions for finding the reference angle.

6

65

66

65

Thus the reference angle is 6

Evaluate each of the following for .

sin.1

411

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To solve a problem like this, we want to start by finding the reference angle for θ.

Since our angle is bigger than 2π, we need to subtract 2π to find the coterminal angle that is between 0 and 2π.

43

48

411

24

11

Evaluate each of the following for .

Our next step is to figure out

what quadrant is in. You can

see from the picture that we are in quadrant II.

43

sin.1

411

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To find the reference angle for an angle in quadrant II, we subtract the coterminal angle from π.

43

This will give us a reference angle of

443

44

43

Evaluate each of the following for .

We will now use the basic trigonometric function values for

The only thing that we will need to change might be the signs of the basic values. Remember that the sign of the cosine and tangent functions will be negative in quadrant II. The sign of the sine will still be positive in quadrant II.

4

sin.1

411

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43

22

411

sin

22

411

cos

Evaluate each of the following for .

cos.24

11

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Once again, we will use our reference angle to determine the basic trigonometric function value. The only difference between the basic value and the value for our angle may be the sign.

tan.3

Evaluate each of the following for .

1

22

22

411

cos

411

sin

411

tan

411

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Once again, we will use our reference angle to determine the basic trigonometric function value. The only difference between the basic value and the value for our angle may be the sign.

Evaluate each of the following for .

sec.4 411

22

2

22

1

411

cos

14

11sec

Once again, we will use our reference angle to determine the basic trigonometric function value. The only difference between the basic value and the value for our angle may be the sign.

tan.3

For , find the values of the trigonometric functions based on .

sec.4

sin.1

cos.2

910

csc

20

cot.5

tan.3

Evaluate the following

expressions if and

sec.4

sin.1

csc.2

72

cos 0tan

cot.5

cot.3

Evaluate the following

expressions if and

sec.4

sin.1

cos.2

34

tan 0sin

csc.5

22 cossin.3

If and θ is in quadrant IV, then find the following.

cottan.1

tancsc.2

86

cos