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## Trigonometric Functions on Any Angle

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### 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 Angle

A 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.

continued on next slide

Reference Angle

There 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.

continued on next slide

Reference Angle

There is a straight-forward process for finding reference angles.

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

continued on next slide

Reference Angle

There is a straight-forward process for finding reference angles.

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

continued on next slide

Reference Angle

Directions for quadrant I

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

continued on next slide

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

This distance is the reference angle.

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.

continued on next slide

Reference Angle

Directions for quadrant III

This distance is the reference angle.

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.

Thus

continued on next slide

Reference Angle

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.

continued on next slide

Reference Angle

Directions for quadrant IV

This distance is the reference angle.

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.

Thus

continued on next slide

Reference Angle

Directions for quadrant IV

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.

continued on next slide

Quadrant I

Quadrant IV

Quadrant III

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

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

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

In which quadrant is the angle?

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

This blue part is one whole π in the clockwise direction

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

continued on next slide

What is the reference angle, , for the angle

?

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.

continued on next slide

What is the reference angle, , for the angle

?

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.

continued on next slide

What is the reference angle, , for the angle ?

Quadrant II

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

Thus the reference angle is

Evaluate each of the following for .

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.

continued on next slide

Our next step is to figure out

what quadrant is in. You can

see from the picture that we are in quadrant II.

Evaluate each of the following for .

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

This will give us a reference angle of

continued on next slide

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.

Evaluate each of the following for .

continued on next slide

Evaluate each of the following for .

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.

continued on next slide

Evaluate each of the following for .

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.

continued on next slide

Evaluate each of the following for .

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.

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

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

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