4.4 Trigonometric Functions of Any Angle
ObjectivesUse the definitions of trigonometric
functions of any angleUse the signs of the trigonometric
functionsFind reference anglesUse reference angles to evaluate
trigonometric functions
Applications
Cycles govern many aspects of life – heartbeats, sleep patterns, seasons, and tides all follow regular, predictable cycles.
Trigonometric functions are used to model phenomena that occur in cycles.
Example
Let P = (1, -3) be a point on the terminal side of θ. Find each of the six trigonometric functions of θ.
Examples
Evaluate, if possible, the six trigonometric functions of the four quadrantal angles.
Signs of Trigonometric Functions If θ is not a quadrantal angle, the sign of a
trigonometric function depends on the quadrant in which θ lies.
Quadrant IQuadrant
II
Quadrant IV
Quadrant IIIx
y
Signs of Trigonometric Functions If θ is not a quadrantal angle, the sign of a
trigonometric function depends on the quadrant in which θ lies.
Quadrant I
All positive
Quadrant II
Sin and cscpositive
Quadrant IV
Cos and sec positive
Quadrant III
Tan and cot positive
x
y
Example
If sin θ < 0 and cos θ < 0, name the quadrant in which angle θ lies.
Example
If tan θ < 0 and cos θ > 0, name the quadrant in which angle θ lies.
Example
Given that tan θ = and cos θ > 0, find cos θ and csc θ.
32
Example
Given that tan θ = and cos θ < 0, find sin θ and sec θ.
31
SAT PREP QUESTION OF THE DAYAn exhibitor is selling decorative wreaths at an arts and
crafts show. The net profit P, in dollars, from the sales of the wreaths is given by P(n) = 0.75n – 50, where n is the number of wreaths sold. How many wreaths must the exhibitor sell in order to earn a net profit of $100?
(A)25
(B)75
(C)150
(D)175
(E) 200
SAT PREP QUESTION OF THE DAYAn exhibitor is selling decorative wreaths at an arts and
crafts show. The net profit P, in dollars, from the sales of the wreaths is given by P(n) = 0.75n – 50, where n is the number of wreaths sold. How many wreaths must the exhibitor sell in order to earn a net profit of $100?
(A)25
(B)75
(C)150
(D)175
(E) 200
Assignment
Page 499, #s 1-33 odd
Reference Angles
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.
Reference Angles
When considering a reference angle, the angle MUST be:
• Acute• Positive• With respect to the x-axis !!
Examples
Find the reference angle for each of the following angles:
a) 210o b) c) -240o d) 3.6
47
Examples
Find the reference angle for each of the following angles:
a) 665o b) c) 4
154
11
Using Reference Angles
Use the following procedure:1. Sketch the angle2. Find the associated reference angle
(remember it is with respect to the x-axis)
3. Find the trig function of the reference angle
4. Use the quadrant in which the original angle lies to determine the appropriate sign.
Examples
Use reference angles to find the exact value of the following trigonometric functions:
a) sin 300o b) c) 4
5tan
6sec
Examples
Use reference angles to find the exact value of the following trigonometric functions:
a) b) 3
14tan
417
sec
Examples
Use reference angles to find the exact value of the following trigonometric functions:
a) b) 6
17cos
322
sin
SAT PREP QUESTION OF THE DAYIf x2 + y2 = 73 and xy = 24, what is the value of (x + y)2 ?
(A)73(B)97(C)100(D)121(E) 144
SAT PREP QUESTION OF THE DAYIf x2 + y2 = 73 and xy = 24, what is the value of (x + y)2 ?
(A)73(B)97(C)100(D)121(E) 144
Assignment
Page 499, #s 35-85 odd
Insert video clip “Hand Trick”http://www.youtube.com/watch?v=OBmhgiLi77k&feature=related
Review for Test 4A (4.1-4.4)
Page 498; Page 500, #s 1-30; Pages 560-562Page 563, #s 1-67; Page 567, #s 1-13, 20-
22,CD; “Writing in Math” at the end of each
section
If 30 percent of 40 percent of a positive number is equal to 20 percent of w percent of the same number, what is the value of w?
(A)80(B)60(C)50(D)15(E)10
SAT PREP QUESTION OF THE DAY
If 30 percent of 40 percent of a positive number is equal to 20 percent of w percent of the same number, what is the value of w?
(A)80(B)60(C)50(D)15(E)10
SAT PREP QUESTION OF THE DAY
Insert video clip “Trig Modeling”• http://player.discoveryeducation.com/inde
x.cfm?guidAssetId=DEE474F0-A257-4276-89D3-8DF3A73C8C01&blnFromSearch=1&productcode=US
Introduction – not necessary to view.