4.4 Trigonometric Functions of Any Angle Objectives Use the definitions of trigonometric functions of any angle Use the signs of the trigonometric functions Find reference angles Use 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. y Quadrant II Quadrant I x Quadrant III Quadrant IV

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. y Quadrant II Sin and csc positive Quadrant I All positive x Quadrant III Tan and cot positive Quadrant IV Cos and sec positive

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.

Example Given that tan θ = and cos θ < 0, find sin θ and sec θ.

SAT PREP QUESTION OF THE DAY An 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? 25 75 150 175 200

SAT PREP QUESTION OF THE DAY An 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? 25 75 150 175 200

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

Examples Find the reference angle for each of the following angles: a) 665o b) c)

Using Reference Angles Use the following procedure: Sketch the angle Find the associated reference angle (remember it is with respect to the x-axis) Find the trig function of the reference angle 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)

Examples Use reference angles to find the exact value of the following trigonometric functions: a) b)

Examples Use reference angles to find the exact value of the following trigonometric functions: a) b)

SAT PREP QUESTION OF THE DAY If x2 + y2 = 73 and xy = 24, what is the value of (x + y)2 ? 73 97 100 121 144

SAT PREP QUESTION OF THE DAY If x2 + y2 = 73 and xy = 24, what is the value of (x + y)2 ? 73 97 100 121 144

Assignment Page 499, #s 35-85 odd

Insert video clip “Hand Trick” http://www. youtube. com/watch

Review for Test 4A (4.1-4.4) Page 498; Page 500, #s 1-30; Pages 560-562 Page 563, #s 1-67; Page 567, #s 1-13, 20-22, CD; “Writing in Math” at the end of each section

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? 80 60 50 15 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? 80 60 50 15 10

Insert video clip “Trig Modeling” http://player.discoveryeducation.com/index.cfm?guidAssetId=DEE474F0-A257-4276-89D3-8DF3A73C8C01&blnFromSearch=1&productcode=US Introduction – not necessary to view.