Trigonometric Function Precalculus PREC111 PDF

MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY TRIGONOMETRIC FUNCTION Precalculus PREC111 MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY Recall: Standard equation of a circle with radius and 2 2 2 center (h,k): (𝑥...

MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY TRIGONOMETRIC FUNCTION Precalculus PREC111 MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY Recall: Standard equation of a circle with radius and 2 2 2 center (h,k): (𝑥 − ℎ) +(𝑦 − 𝑘) = 𝑟. Unit Circle: A circle with (h,k) = (0,0) and r = 1 2 2 𝑥 +𝑦 =1 Circumference: 2𝜋 MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY THE POINT FUNCTION The point function P is a function that sends a real number 𝜃 to a point P(𝜃) = (xy) on the unit circle. P: ℝ ℝxℝ 𝜃 P(𝜃) = (x,y) How does P send 𝜃 to P(𝜃) MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY P(𝜃)is the point on the unit circle making an arclength of 𝜃 from the point (1,0), measured counterclockwise if 𝜃 > 0 , and clockwise if 𝜃 < 0. MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY Examples: 𝜋 1. P(0) = (1, 0) 6.P(- ) = 𝜋 2 2. P( )= 7. P(-π)= 2 3. P(𝜋) = 3𝜋 8. P(- ) = 3𝜋 2 4. P( ) = 9. P(-2𝜋) = 2 5. P(2𝜋) = MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY THE POINT FUNCTION AND CIRCULAR FUNCTIONS Recall: The point function P: ℝ ℝxℝ is a function that sends a 𝜃𝜖ℝ to a point P(𝜃) = (x,y) on the unit 2 2 circle 𝑥 + 𝑦 = 1. MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY THE POINT FUNCTION AND CIRCULAR FUNCTIONS Let 𝜃𝜖ℝ and P(𝜃) = (x , y) we define the ff: x:=cos𝜃 y: = sin𝜃 We also write P(𝜃) = (cos 𝜃, sin𝜃) Example: P(0) = (1,0) cos 0 = 1 sin 0 = 0 MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY THE POINT FUNCTION AND CIRCULAR FUNCTIONS 𝜋 Example: P( ) = (0,1) 2 𝜋 cos 2 = 0 𝜋 sin 2 = 1 Domain P = all real numbers Range P = {(x,y)/𝑥 2 + 𝑦 2 =1} MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY THE POINT FUNCTION AND CIRCULAR FUNCTIONS Some properties of P(𝜃). Let 𝜃𝜖ℝ. P(𝜃) = (cos 𝜃, 𝑠𝑖𝑛𝜃), k𝜖Z 1. P(𝜃 ± 2𝜋)= P(𝜃) P(𝜃 ± 2𝜋k)= P(𝜃) cos(𝜃 ± 2𝜋)= cos𝜃 sin(𝜃 ± 2𝜋)= sin𝜃 cos(𝜃 ± 2𝜋k)= cos𝜃 sin(𝜃 ± 2𝜋𝑘)= sin𝜃 MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY THE POINT FUNCTION AND CIRCULAR FUNCTIONS Some properties of P(𝜃). Let 𝜃𝜖ℝ. P(𝜃) = (cos 𝜃, 𝑠𝑖𝑛𝜃), k𝜖Z 2. 𝑐𝑜𝑠 2 𝜃 + 𝑠𝑖𝑛2 𝜃= 1 3. P(𝜃) = (cos 𝜃, sin 𝜃) Q1 (+,+) Q2 ( -, +) Q3 (-, -) Q4 (+, -) MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY Example: Which quadrant does P(𝜃 ) lie? 3𝜋 1. P( ) 4 7𝜋 2. P( ) 6 3. P(-4) MANDALUYONG C OLLEGE OF S CIENCE &T ECHNOLOGY Example: 3 Given sin𝜃 = and P(𝜃) is in quadrant 2. what 4 is cos 𝜃?