4-3: Trigonometric Functions of Any Angle What you’ll learn about ■ Trigonometric Functions of Any Angle ■ Trigonometric Functions of Real Numbers ■ Periodic Functions ■ The 16-point unit circle... and why Extending trigonometric functions beyond triangle ratios.

Definitions of Trigonometric Functions of Any Angle

(x, y) y x r opposite adjacent hypotenuse

Definitions of Trigonometric Functions of Any Angle (x, y) y x r adjacent opposite hypotenuse

Let (-3, 4) be a point on the terminal side of Ѳ. Find the sin, cos and tan. (-3, 4) y x r adjacent opposite hypotenuse x = -3 y = 4 r = 5

Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x - x + y + y - sin Ѳ + sin Ѳ - cos Ѳ - cos Ѳ + tan Ѳ - tan Ѳ +tan Ѳ - csc Ѳ + csc Ѳ - sec Ѳ - sec Ѳ + cot Ѳ - cot Ѳ + cot Ѳ -

How do you get a negative? One is positive and one is negative

How do you get a negative? One is positive and one is negative

How do you get a negative? One is positive and one is negative

How do you get a negative? One is positive and one is negative cos Ѳ + cos Ѳ - cos Ѳ +

How do you get a negative? One is positive and one is negative cos Ѳ +

How do you get a negative? One is positive and one is negative cos Ѳ +

0 (1, 0)(-1, 0) (0, 1) (0, -1)

Ranges of Trigonometric Functions We know that If the measure of increases toward 90 o, then y increases The value of y approaches r, and they are equal when So, y cannot be greater than r. Using the convenient point (0,1) y can never be greater than 1. x y r

Ranges Continued Using a similar approach, we get:

Determining if a Value is Within the Range Evaluate (calculator) (not possible) (not possible)

Reference Angles Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ ref angle Think of the reference angle as a “distance”—how close you are to the closest x-axis.

Definition of a Reference Angle Let Ѳ be an angle in standard position. Its reference angle is the acute angle α formed by the terminal side of Ѳ and the horizontal axis. Ѳ α α=180⁰ - Ѳ α=π - Ѳ Ѳ α α=Ѳ - 180⁰ α=Ѳ - π Ѳ α α=360⁰ - Ѳ α=2π - Ѳ

Find the reference angle for Ѳ=300⁰ Ѳ What quadrant is the terminal side in? α α=360⁰ - 300⁰ α=60⁰ α=360⁰ - Ѳ

Find the reference angle for Ѳ=2.3 Ѳ What quadrant is the terminal side in? α α=3.14 – 2.3 α≈ α= π - Ѳ

Find the reference angle for Ѳ=-135⁰ Ѳ What quadrant is the terminal side in? α Reference Angle : α= Ѳ - 180⁰ α=45⁰ α=225⁰ - 180⁰ Find the positive coterminal angle to -135⁰ Coterminal angle =-135⁰ + 360⁰ Coterminal angle = 225⁰

Common Trigonometric Functions Ѳ(degrees) 0⁰30⁰45⁰60⁰90⁰180⁰270⁰ Ѳ(radians) sin Ѳ cos Ѳ tan Ѳ 0 π und und

Positive Trig Function Values r r r r x-x y y -y ALL STUDENTS TAKE CALCULUS All functions are positive Sine and its reciprocal are positive Tangent and its reciprocal are positive Cosine and its reciprocal are positive

Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x - x + y + y - sin Ѳ + sin Ѳ - cos Ѳ - cos Ѳ + tan Ѳ - tan Ѳ +tan Ѳ - csc Ѳ + csc Ѳ - sec Ѳ - sec Ѳ + cot Ѳ - cot Ѳ + cot Ѳ -

Ѳ What quadrant is the terminal side in? α α= Ѳ - π Is cos positive or negative in quadrant III?

Positive Trig Function Values r r r r x-x y y -y ALL STUDENTS TAKE CALCULUS All functions are positive Sine and its reciprocal are positive Tangent and its reciprocal are positive Cosine and its reciprocal are positive

Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x - x + y + y - sin Ѳ + sin Ѳ - cos Ѳ - cos Ѳ + tan Ѳ - tan Ѳ +tan Ѳ - csc Ѳ + csc Ѳ - sec Ѳ - sec Ѳ + cot Ѳ - cot Ѳ + cot Ѳ -

Ѳ What quadrant is the terminal side in? α Is cos positive or negative in quadrant III?

Ѳ What quadrant is the terminal side in? α α= 180⁰ - 150⁰ α=30⁰ Is tan positive or negative in quadrant II? Find the coterminal angle for -210⁰ coterminal= -210⁰ + 360⁰ coterminal= 150⁰

Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x - x + y + y - sin Ѳ + sin Ѳ - cos Ѳ - cos Ѳ + tan Ѳ - tan Ѳ +tan Ѳ - csc Ѳ + csc Ѳ - sec Ѳ - sec Ѳ + cot Ѳ - cot Ѳ + cot Ѳ -

What quadrant is the terminal side in? Is tan positive or negative in quadrant II? Ѳ α

Ѳ What quadrant is the terminal side in? α Is csc positive or negative in quadrant II?

Positive and Negative Quadrants Quadrant I Quadrant III Quadrant IV Quadrant II x + y + sin Ѳ + cos Ѳ + tan Ѳ + sec Ѳ + csc Ѳ + cot Ѳ + x - x + y + y - sin Ѳ + sin Ѳ - cos Ѳ - cos Ѳ + tan Ѳ - tan Ѳ +tan Ѳ - csc Ѳ + csc Ѳ - sec Ѳ - sec Ѳ + cot Ѳ - cot Ѳ + cot Ѳ -

What quadrant is the terminal side in? Is csc positive or negative in quadrant II? Ѳ α

Finding Exact Measures of Angles Find all values of Sine is negative in QIII and QIV Using the values we found earlier, we know

Finding Exact Measures of Angles – Cont. Our reference angle is 60 o. We must be 60 o off of the closest x-axis in QIII and QIV.

Note: there is other way to remember special angle, radian and point of unit circle