2. Chapter 8: The Unit Circle and the Functions of Trigonometry
8.1 Angles, Arcs, and Their Measures
8.2 The Unit Circle and Its Functions
8.3 Graphs of the Sine and Cosine Functions
8.4 Graphs of the Other Circular Functions
8.5 Functions of Angles and Fundamental Identities
8.6 Evaluating Trigonometric Functions
8.7 Applications of Right Triangles
8.8 Harmonic Motion

3. 8.5 Functions of Angles and Fundamental Identities
To define the six trigonometric functions, start with an angle  in standard position. Choose any point P having coordinates (x,y) on the terminal side as seen in the figure below.
Notice that r > 0 since distance is never negative.

4. 8.5 Trigonometric Functions
The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant.
Trigonometric Functions
Let (x,y) be a point other than the origin on the terminal side of an angle  in standard position. The distance from the
point to the origin is The six trigonometric functions of angle  are as follows.

5. 8.5 Finding Function Values of an Angle
Example The terminal side of angle  (beta) in
standard position goes through (–3,–4). Find the
values of the six trigonometric functions of .
Solution

6. 8.5 Finding Function Values of an Angle
The six trigonometric functions
can be found from any point on
the line. Due to similar triangles,
so sin = y/r is the same no matter which point is used to find it.

7. 8.5 Function Values of Quadrantal Angles
Undefined Function Values
If the terminal side of a quadrantal angle lies along the
y-axis, the tangent and secant functions are undefined.
x-axis, the cotangent and cosecant functions are undefined.

8. 8.5 Function Values of Quadrantal Angles

9. 8.5 Reciprocal Identities
Since sin = y/r and csc = r/y,
Similarly, we have the following reciprocal identities for any angle  that does not lead to a
0 denominator.

10. 8.5 Using the Reciprocal Identities
Example Find sin if csc =
Solution

11. 8.5 Signs and Ranges of Function Values
In the definitions of the trigonometric functions, the distance r is never negative, so r > 0.
Choose a point (x,y) in quadrant I, then both x and y will be positive, so the values of the six trigonometric functions will be positive in quadrant I.
A point (x,y) in quadrant II has x < 0 and y > 0. This makes sine and cosecant positive for quadrant II angles, while the other four functions take on negative values.
Similar results can be obtained for the other quadrants.

12. 8.5 Signs and Ranges of Function Values
Example Identify the quadrant (or quadrants) of any angle 
that satisfies sin > 0, tan < 0.
Solution Since sin > 0 in quadrants I and II, while tan < 0
in quadrants II and IV, both conditions are met only in quadrant II.

13. 8.5 Signs and Ranges of Function Values
The figure shows angle  as it increases from 0º to 90º.
The value y increases as  increases, but never exceeds r, so y  r. Dividing both sides by r gives
In a similar way, angles in quadrant IV suggests

14. 8.5 Signs and Ranges of Function Values
Since
for any angle .
In a similar way,
sec and csc are reciprocals of sin and cos, respectively, making

15. 8.5 Ranges of Trigonometric Functions
Example Decide whether each statement is possible or
impossible.
(b) tan = (c) sec = 0.6
Solution
Not possible since
Possible since tangent can take on any value.
Not possible since sec  –1 or sec  1.
For any angle  for which for which the indicated function
exists:
–1  sin  and –1  cos  1;
tan and cot may be equal to any real number;
sec  –1 or sec  and csc  –1 or csc  1.

16. 8.5 Pythagorean Identities
Three new identities from x2 + y2 = r2
Divide by r2
Since cos = x/r and sin = y/r, this result becomes
Divide by x2
Dividing by y2 leads to 1 + cot2 = csc2.

17. 8.5 Pythagorean Identities

18. 8.5 Quotient Identities
Recall that Consider the quotient of sin and cos where cos  0.
Similarly
Quotient Identities

19. 8.5 Quotient Identities
Example Find sin and cos, if tan = 4/3 and  is in
quadrant III.
Solution Since  is in quadrant III, sin and cos will
both be negative.