1. Chapter 6: Trigonometry
Section 6.4: Trigonometric Functions

2. Day 1

3. Trigonometric Ratios
P (x.y)
Let θ be an angle in standard position and let P(x,y) be any point on the terminal side of θ. Let r be the distance from (x,y) to the origin:
Then the trigonometric ratios of θ are defined as follows: r y θ x

4. Trigonometric ratios
Find Sin, Cos, and Tan of the angle θ, whose terminal side passes through the point (-3,-2).
Find Sin, Cos, and Tan of the angle θ, whose terminal side passes through the point (-2,3).

5. Trigonometric Functions
Trigonometric ratios have been defined for all angles. But modern applications of trigonometry deal with functions whose domains consist of real numbers. The basic idea is quite simple: If t is a real number then:
sin t is defined to be the sine of an angle of t radians;
cos t is defined to be the cosine of an angle of t radians;
and so on. Instead of starting with angles, as was done up until now, this new approach starts with a number and only then moves to angles
Form an angle
of t radians
Begin with a
Number t
Determine sint, cost, tant

6. Trigonometric Ratios
(x.y)
Let t be a real number. Choose any point (x,y) on the terminal side of an angle t radians in standard position.
Then the trigonometric ratios of t radians are defined as follows: r y t x

7. Trigonometric ratios
Find Sin t, Cos t, and Tan t when the terminal side of t radians passes through the point (5,-1).
Find Sin t, Cos t, and Tan t when the terminal side of t radians passes through the point (-4,4).

8. Trigonometric ratios
The terminal side of an angle of t radians lies in quadrant 1 on the line through the origin parallel to -2y+5x=12. Find Sin t, Cos t, and Tan t.
YOU TRY!! The terminal side of an angle of t radians lies in quadrant 1 on the line through the origin parallel to 3y-4x=12. Find Sin t, Cos t, and Tan t.

9. Homework!!!
Page 452: 1-6
6.4 Worksheet #1

10. Day 2

11. Trigonometry and The Unit Circle
In the unit circle, the radius is always 1. So if r = 1, then: 1 1 -1 1 -1

12. Domain and Range sin and cos: Domain is the set of all real numbers!
Range is the set of all real numbers between -1 and 1.
tan:
Domain is the set of all real numbers except ±π/2 + kπ, where k = 0.±1,±2,…
Range is the set of all real numbers!

13. Exact Values of Our Special Angles
Square root of finger over palm! t
30o
45o
60o
Sin t
1/2
√2/2
√3/2
Cos t
Tan t
√3/3 1 √3
Csc t 2 √2
2√3/3
Sec t
Cot t
Cos
90o
60o 2
45o
30o 0o
Sin
Flip hand over for Tangent!!

14. Exact Values of Our Special Angles
Square root of finger over palm!
Without using a calculator, Find the sin, cos, and tan of 30o.
Cos
90o
60o 2
45o
30o 0o
Sin
Flip hand over for Tangent!!

15. Exact Values of Our Special Angles
Square root of finger over palm!
Without using a calculator, Find the sin, cos, and tan of 45o.
Cos
90o
60o 2
45o
30o 0o
Sin
Flip hand over for Tangent!!

16. Trigonometry and The Unit Circle
Find sint, cost, and tant when the terminal side of an angle of t radians passes through the given point on the unit circle.
Find sint, cost, and tant when the terminal side of an angle of t radians passes through the given point on the unit circle.

17. Trig Function Signs

18. Homework!!!
Page 452:
7-10
Create a poster of Trig Signs in different quadrants. 50 pts
1) Colorful 10 pts
2) All correct signs 30 pts
3) Neatness 10 pts

19. Day 3

20. Reference Angles
Reference Angle is the positive acute angle formed by the terminal side of θ and the x- axis. t t
t’=t
t’=π-t t t
t’=t-π
t’=2π-t

21. Reference Angles Find the reference angle to the given angle:
Now find sin, cos, and tan for each problem and append the appropriate sign.

22. Practice
Find the exact value of the sin, cos, and tan of the number without using a calculator.

23. Complete the Chart T Sin Cos Tan Csc Sec Cot 30o. π/6 1/2 √3/2 √3/3 2
2√3/3 √3
45o, π/4
√2/2 1 √2
60o, π/3
90o, π/2
Undef

24. Homework!!! Pg.452 11-14, 15-23 part a only, 24-35.
Complete the chart!

25. Day 4

26. Evaluating Expressions
Write the expression as a single real number.

27. Evaluating Expressions
YOU TRY!!! Write the expression as a single real number.

28. Homework!!
Pg.452:
6.4 Worksheet #2