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Ch 6.3: Trigonometric functions of any angle In this section, we will 1. extend the domains of the trig functions (as well as the need for it) using the unit circle 2. look at some properties of the unit circle 3. evaluate trig. values using reference angles 4. investigate the relationships between trig. functions Why extend the right triangle definition? 1) 0 < θ since it’s an interior angle of a right triangle. I No sin(0◦ ) or cos(−50◦ ) can be defined through the right triangle definition. 2) θ < 90 since we have a right triangle and the sum of the interior angles in a triangle is 180. I I In fact, if θ = 90, then we have two right angles. ⇒ No triangle at all!! If θ > 90, Same problem!! Conclusion: With right triangle definition, we can define sine, cosine, and tangent, but ONLY for θ ∈ (0, 90). ⇒ We need to extend our definition (and will use the unit circle to do so..) The Unit Circle definition - for sine, cosine, and tangent Definition Let (x, y ) be a point on the unit circle and θ is the angle between the positive x-axis (initial side) and the ray from 0 which goes through (x, y ) (terminal side). Then, x = cos θ , y = sin θ , y = tan θ. x Examples Use the unit circle definition to evaluate sin θ and cos θ for the following angles: θ = 0◦ , 90◦ , 180◦ , 270◦ , 360◦ Let’s check! Let θ ∈ (0, 90) and (x, y ) be corresponding point on the unit circle. New definition says cos θ = x and sin θ = y . Q: If θ ∈ (0, 90), can we say the same using the old definition? (Do we even have a right triangle?) Properties of the unit circle (UC) Since any point P(x, y ) on the unit circle (UC) has the following relation: x2 + y2 = 1 if we’re given one of x, y in (x, y ) on the UC, then we know the other coordinate. Example) If y = 12 , and (x, y ) in 2nd Q, what is x? Properties of the unit circle (UC) - Continued If point P(x, y ) on UC, then due to UC being symmetric to both x and y axis (and also to the origin): (x, −y ), (−x, y ), and (−x, −y ) are all on the UC as well. −4 Example) If x = −3 5 and y = 5 , a) verify that (x, y ) is on the UC, and b) find three other points on the UC? Reference Angle Definition For a non-quadrantal angle θ in standard position, the acute angle θr formed by the terminal side and the nearest x-axis is called the reference angle. Examples - Reference angles Find the reference angle for a) θ = 315◦ , b) θ = 1900◦ , c) θ = −120◦ . More Examples Find the reference angle for each rotation given. a. θ = 5π b. θ = 4π c. θ = − 7π 6 3 4 . Signs of the Trig functions Knowing that cos θ = x and sin θ = y , (and tan θ = yx ) Ex1) Let sin α > 0 and tan α < 0. Which quadrant is α in? Ex2) cos β < 0 and cot β > 0. Examples - Evaluating trig. values using reference angles Evaluate the following: 1. cos 330◦ 2. sin( 7π 6 ) What if (x, y ) is NOT on the unit circle? I sin θ I cos θ I tan θ I csc θ I sec θ I cot θ Definition Definition Given (x, y ) with r = p x 2 + y 2 and the corresponding θ, sin θ = csc θ = y , r cos θ = x , r tan θ = y x 1 r 1 r 1 x = , sec θ = = , cot θ = = sin θ y cos θ x tan θ y (Note: denominators must not be 0.) Q: For which (quadrantal) angles are these functions not defined? Examples Find the values of the 6 trig functions, given the following. 1. (7, 24) 2. (−3, −1) More example Given sin θ = 5/13 and cos θ < 0, find the values of other ratios. Fundamental Trigonometric Identities Reciprocal Identities: csc θ = 1 , sin θ sec θ = 1 , cos θ cot θ = 1 tan θ Ratio Identities: tan θ = sin θ sec θ = , cos θ csc θ cot θ = cos θ sin θ Cofunction Identities : From the unit circle definition Theorem cos θ = sin( π2 − θ), csc θ = sec( π2 − θ), cot θ = tan( π2 − θ) Examples Write the following in terms of its cofunction. 1. sin −60◦ 2. cos π3 3. cot 120◦ 4. sec π2 Homework for Ch 6.3: pg. 497 4, 18, 19, 44, 51, 65, 70, 71, 91, 92, 94