1. Trigonometric Functions of Any Angle
Section 5.3
Trigonometric Functions of Any Angle

2. Trigonometric Functions of Any Angle
The point P = (x, y) is a point r units from the origin on the terminal side of θ.
A right triangle is formed by drawing a line segment from P = (x, y) perpendicular to the x-axis.
Note that y is the length of the side opposite θ and x is the length of the side adjacent to θ.

3. Definitions of Trigonometric Functions of Any Angle
Let θ be any angle in standard position and let P = (x, y) be a point on the terminal side of θ.
If 𝑟= 𝑥 2 + 𝑦 2 is the distance from (0, 0), the six trigonometric functions of θ are defined by the following ratios:

4. Because the point P(x, y) is any point on the terminal side of θ other than the origin, (0, 0), 𝑟= 𝑥 2 + 𝑦 2 cannot be zero.
Note that the denominator of the sine and cosine functions is r. Because r ≠ 0, the sine and cosine functions are defined for any angle θ.
This is not true for the other four trigonometric functions. Note that the denominator of the tangent and secant functions is x. These functions are not defined if x = 0.
Likewise, the denominator of the cotangent and cosecant functions is y. These functions are not defined for quadrantal angles with terminal sides on the positive or negative x-axis.

5. Example 1. Evaluating Trigonometric Functions
Let P = (-3, 5) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.

6. Page 525 #6
A point on the terminal side of angle θ is given. Find the exact value of each of the six trigonometric functions of
(5, -5)

7. Finding Values of Trigonometric Functions for Quadrantal Angles
First, draw the angle in standard position.
Second, choose a point P on the angle’s terminal side.
The trigonometric function values of depend only on the size of θ and not on the distance of point P from the origin. Thus, we will choose a point that is 1 unit from the origin.
Finally, apply the definitions of the appropriate trigonometric functions.

8. Example 2. Trigonometric Functions of Quadrantal Angles
Evaluate, if possible, the sine function and the tangent function at the following four quadrantal angles:
𝜃=0°=0
𝜃=90°= 𝜋 2
𝜃=180°=𝜋
𝜃=270°= 3𝜋 2

9. The Signs of the Trigonometric Functions
If θ is not a quadrantal angle, the sign of a trigonometric function depends on the quadrant in which θ lies.
In all four quadrants, r is positive. However, x and y can be positive or negative.
Example. If θ lies in quadrant II, x is negative and y is positive. Thus, the only positive ratios in this quadrant are 𝑦 𝑟 (sinθ) and its reciprocal 𝑟 𝑦 (cscθ). The other four trigonometric functions are negative.

11. Example 3. Finding the Quadrant in Which an Angle Lies
If 𝑡𝑎𝑛𝜃<0 and 𝑐𝑜𝑠𝜃>0, name the quadrant in which angle θ lies.

12. Example 4. Evaluating Trigonometric Functions
Given 𝑡𝑎𝑛𝜃=− 2 3 and 𝑐𝑜𝑠𝜃>0, find 𝑐𝑜𝑠𝜃 and 𝑐𝑠𝑐𝜃.

13. We use the quadrant in which θ lies to determine whether a negative sign should be associated with the numerator or the denominator.
Example. 𝑡𝑎𝑛𝜃= 3 5 and 𝑐𝑜𝑠𝜃<0 Because 𝑡𝑎𝑛𝜃 is positive and 𝑐𝑜𝑠𝜃 is negative, θ lies in quadrant III.
In quadrant III, x is negative and y is negative. Thus, 𝑡𝑎𝑛𝜃= 3 5 = 𝑦 𝑥 = −3 −5

14. Reference Angles
We will often evaluate trigonometric functions of positive angles greater than 90° and all negative angles by making use of a positive acute angle called a reference angle.
Let θ be a nonacute angle in standard position that lies in a quadrant. Its reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis.

15. Example 5. Finding Reference Angles
Find the reference angle, θ’, for each of the following angles:
𝜃=345°
𝜃= 5𝜋 6
𝜃=−135°
𝜃=2.5

16. Finding reference angles for angles that are greater than 360° (2π) or less than -360° (-2π) involves using coterminal angles.
Recall that coterminal angles can be obtained by increasing or decreasing an angle’s measure by an integer multiple of 360° or 2π.

18. Example 6. Finding Reference Angles
Find the reference angle for each of the following angles:
𝜃=580°
𝜃= 8𝜋 3
𝜃=− 13𝜋 6

19. Page 520 Find the reference angle for each of the following angles:
𝜃=665°
𝜃= 15𝜋 4
𝜃=− 11𝜋 3

20. Evaluating Trigonometric Functions Using Reference Angles
The values of the trigonometric functions of a given angle θ, are the same as the values of the trigonometric functions of the reference angle, θ’, except possibly for the sign.
A function value of the reference angle, θ’, is always positive, but the function value for θ may be positive or negative.
Example. We can use a reference angle to obtain an exact value for tan 120°.The reference angle for 𝜃=120° is 𝜃 ′ =180°−120°=60°. We know the exact value of 𝑡𝑎𝑛60°= 3 . Thus, we can conclude that tan 120° equals − 3 or 3 .

21. Using Reference Angles to Evaluate Trigonometric Functions
The value of a trigonometric function of any angle θ is found as follows: Find the associated reference angle, θ’, and the function value for θ’ Use the quadrant in which θ lies to prefix the appropriate sign to the function value in step 1.

22. Example 7. Using Reference Angles to Evaluate Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric functions:
𝑠𝑖𝑛135°
𝑐𝑜𝑠 4𝜋 3
cot⁡ − 𝜋 3

23. Example 8. Using Reference Angles to Evaluate Trigonometric Functions
Use reference angles to find the exact value of each of the following trigonometric functions:
𝑡𝑎𝑛 14𝜋 3
𝑠𝑒𝑐 − 17𝜋 4

24. Page 525 # 61
Use reference angles to find the exact value of each expression. Do not use a calculator.
𝑐𝑜𝑠225°

25. Page 523
Use reference angles to find the exact value of each of the following trigonometric functions:
𝑐𝑜𝑠 17𝜋 6
𝑠𝑖𝑛 − 22𝜋 3

26. Page 525 # 87
Find the exact value of the expression below. Write the answer as a single fraction. Do not use a calculator.
𝑠𝑖𝑛 𝜋 3 𝑐𝑜𝑠𝜋−𝑐𝑜𝑠 𝜋 3 𝑠𝑖𝑛 3𝜋 2

27. Homework
Blitzer Algebra & Trigonometry 4e textbook pages 525 – 526 # 2, 4, 8, 11, 13, 14, 18, 21, 22, 24, 28, 33, 47, 50, 64, 65, 88, 100

28. Explain why tan 90° is undefined.
Exit Slip
Explain why tan 90° is undefined.