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.