4.5 (Day 1) Inverse Sine & Cosine

Remember: the inverse of a function is found by switching the x & y values (reflect over line y = x) Domains become ranges…. ranges become domains We want the inverse sine & inverse cosine to be functions (pass vertical line test) so we need to restrict their domains – can’t be all real numbers To denote we want the inverses to be functions, we use capital letters Sin –1 x and Cos –1 x OR Arcsin x and Arccos x

y = Sin –1 x D = [–1, 1] Want range to include (–) & (+) values Choose QI for (+) values Which quad is closest to QI that contains (–) values? y = Cos –1 x D = [–1, 1] (the range for sin θ) (the range for cos θ) I II III IV (+) (–) close I II III IV (–) (+) close (+) These restrictions tell us where we draw the reference θ & △

*What type of answer is required (1) sin x cos x (2) Sin –1 x Cos –1 x trig function of an angle is a ratio trig (angle) = ratio a)Sin – b)Arcsin Ex 1) Evaluate to 4 decimal places (radian mode) inverse function of a ratio is an angle trig –1 (ratio) = angle  Find the θ and draw it’s picture in the correct quadrant! c)Cos –1 (–0.56)

short △  QIII Ex 2) Evaluate. Find exact value if possible. a) b) ratio  θ 0 c) ratio  θ tall △  QIV c) ratio  θ

picture Ex 3) Determine the exact value. a) θ θ  θ picture b) θ  θ 9 5 (a ratio!) –3 5 4 (a ratio!)

Optics: Light is refracted when it travels from air to water. i is the angle of incidence (in air) and r is the angle of refraction (in water). Equation is: Ex 4) If a light ray makes a 30° angle with the vertical in air, determine the angle with the vertical in water. *Degree mode*

Homework #405 Pg 220 #1–13 all, 15–18, 22, 25, 28–31