1. Unit 8: Modeling with Trigonometric Functions
Mathematics 3
Ms. C. Taylor

2. Warm-Up
Solve the following:
𝑥−1 2𝑥 + 𝑥−3 𝑥 = 3 2

4. The ratios have names & abbreviations…
sin
cos
tan
cot
sec
csc
Ratio Name
sine
cosine
tangent
cotangent
secant
cosecant

5. The Trigonometric (trig) ratios:
FUNCTION
INVERSE FUNCTION

6. Also true are…

7. In the ratios: x is an angle (not the 90 degree angle)
“adjacent”, “opposite” and “hypotenuse” are all side lengths, not angles
“Adjacent” is the side next to the known
angle
“Opposite” is the side across from the
known angle

8. !IMPORTANT! To solve trig functions in the calculator,
make sure to set your MODE to DEGREES
Directions:
Press MODE, arrow down to Radian
Arrow over to Degrees
Press ENTER

9. Warm-Up

10. Measuring Angles
The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.
There are two common ways to measure angles, in degrees and in radians.
We’ll start with degrees, denoted by the symbol º.
One degree (1º) is equivalent to a rotation of of one revolution.

11. Measuring Angles

12. Radian Measure Definition of Radian:
One radian is the measure of a central angle  that intercepts arc s equal in length to the radius r of the circle.
In general,

13. Conversions Between Degrees and Radians
To convert degrees to radians, multiply degrees by
To convert radians to degrees, multiply radians by

14. Ex 5. Convert the degrees to radian measure.
60
30
-54
-118
45

15. Ex 6. Convert the radians to degrees.
a) b) c) d)

16. Unit circle Radius of the circle is 1. x = cos(θ) y = sin(θ)
Pythagorean Theorem:
This gives the identity:
Zeros of sin(θ) are where n is an integer.
Zeros of cos(θ) are where n is an integer.

18. Warm-Up

19. Graphs of sine & cosine Fundamental period of sine and cosine is 2π.
Domain of sine and cosine is
Range of sine and cosine is [–|A|+D, |A|+D].
The amplitude of a sine and cosine graph is |A|.
The vertical shift or average value of sine and cosine graph is D.
The period of sine and cosine graph is
The phase shift or horizontal shift is

20. Sine graphs y = sin(x) y = sin(x) + 3 y = 3sin(x) y = sin(3x)

21. Graphs of cosine y = cos(x) y = cos(x) + 3 y = 3cos(x) y = cos(3x)

22. Tangent and cotangent graphs
Fundamental period of tangent and cotangent is π.
Domain of tangent is where n is an integer.
Domain of cotangent where n is an integer.
Range of tangent and cotangent is
The period of tangent or cotangent graph is

23. Graphs of tangent and cotangent
y = tan(x)
Vertical asymptotes at
y = cot(x)
Vertical asymptotes at

24. Graphs of secant and cosecant
y = csc(x)
Vertical asymptotes at
Range: (–∞, –1] U [1, ∞)
y = sin(x)
y = sec(x)
Vertical asymptotes at
Range: (–∞, –1] U [1, ∞)
y = cos(x)

25. Warm-Up

26. Do you remember the Unit Circle?
Where did our Pythagorean identities come from??
Do you remember the Unit Circle?
What is the equation for the unit circle?
x2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean Identity!

27. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ =
cos2θ cos2θ cos2θ
tan2θ = sec2θ
Quotient
Identity
Reciprocal
Identity
another Pythagorean Identity

28. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ =
sin2θ sin2θ sin2θ
cot2θ = csc2θ
Quotient
Identity
Reciprocal
Identity
a third Pythagorean Identity

29. Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
sin2q + cos2q = 1
tan2q + 1 = sec2q
cot2q + 1 = csc2q
sin2q = 1 - cos2q
tan2q = sec2q - 1
cot2q = csc2q - 1
cos2q = 1 - sin2q

30. Simplifying trig Identity
Example1: simplify tanxcosx
sin x
cos x
tanx cosx
tanxcosx = sin x

31. Simplifying trig Identity
sec x
csc x
Example2: simplify 1
cos x 1
cos x
sinx = x
sec x
csc x 1
sin x =
sin x
cos x
= tan x

32. Example Simplify: = cot x (csc2 x - 1) Factor out cot x
= cot x (cot2 x)
Use pythagorean identity
= cot3 x
Simplify

33. Warm-Up

34. Types of Angles
There are four different types of angles in any given circle. The type of angle is determined by the location of the angles vertex.
1. In the Center of the Circle: Central Angle
2. On the Circle: Inscribed Angle
3. In the Circle: Interior Angle
4. Outside the Circle: Exterior Angle
* The measure of each angle is determined by the Intercepted Arc

35. Intercepted Arc
Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds:
1. The endpoints of the arc lie on the angle.
2. All points of the arc, except the endpoints, are in the interior of the angle.
3. Each side of the angle contains an endpoint of the arc.

36. Central Angle
Definition: An angle whose vertex lies on the center of the circle.
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)
* The measure of a central angle is equal to the measure of the intercepted arc.

37. Measuring a Central Angle
The measure of a central angle is equal to the measure of its intercepted arc.

38. Inscribed Angle
Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).
Examples: 3 1 2 4
Yes!
No!
Yes!
No!

39. Measuring an Inscribed Angle
The measure of an inscribed angle is equal to half the measure of its intercepted arc.

40. Corollaries
If two inscribed angles intercept the same arc, then the angles are congruent.

41. An angle inscribed in a semicircle is a right angle.
Corollary #2
An angle inscribed in a semicircle is a right angle.

42. Corollary #3
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
** Note: All of the Inscribed Arcs
will add up to 360

43. Warm-Up

44. Area of a Sector Formula
measure of the central angle or arc m
πr2
Area of a sector =
360
The area of the entire circle!
The fraction of the circle! .

45. 6. The area of sector AOB is 48π and
. Find the radius of ○O. m
πr2
Area of a sector =
360
270
48π =
πr2
360 4 16 3 4 r2
48 = 3 4 3 r2
64 = r
= 8

46. 7. The area of sector AOB is
and
. Find the radius of ○O. m
πr2
Area of a sector =
360 9 40
π =
πr2 4
360 9 9 1 9 = r2 1 4 9 1 81 = r2 4 9
r = 2

47. The standard form of the equation of a circle with its center at the origin is
r is the radius of the circle so if we take the square root of the right hand side, we'll know how big the radius is.
Notice that both the x and y terms are squared. Linear equations don’t have either the x or y terms squared. Parabolas have only the x term was squared (or only the y term, but NOT both).

48. Let's look at the equation
This is r2 so r = 3
The center of the circle is at the origin and the radius is 3. Let's graph this circle. 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
Count out 3 in all directions since that is the radius
Center at (0, 0)

49. The center of the circle is at (h, k). This is r2 so r = 4
If the center of the circle is NOT at the origin then the equation for the standard form of a circle looks like this:
The center of the circle is at (h, k).
This is r2 so r = 4
Find the center and radius and graph this circle.
The center of the circle is at (h, k) which is (3,1). 2 -7 -6 -5 -4 -3 -2 -1 1 5 7 3 4 6 8
The radius is 4

50. If you take the equation of a circle in standard form for example:
This is r2 so r = 2
(x - (-2))
Remember center is at (h, k) with (x - h) and (y - k) since the x is plus something and not minus, (x + 2) can be written as (x - (-2))
You can find the center and radius easily. The center is at (-2, 4) and the radius is 2.
But what if it was not in standard form but multiplied out (FOILED)
Moving everything to one side in descending order and combining like terms we'd have:

51. 4 16 4 16 Move constant to the other side
If we'd have started with it like this, we'd have to complete the square on both the x's and y's to get in standard form.
Move constant to the other side
Group x terms and a place to complete the square
Group y terms and a place to complete the square 4 16 4 16
Complete the square
Write factored and wahlah! back in standard form.

52. Warm-Up

53. Warm-Up

54. Law of Sines & Law of Cosines
Use when you have a
complete ratio: SSA.
Use when you have SAS, SSS.