4.5 (Day 1) Inverse Sine & Cosine

1. 4.5 (Day 1) Inverse Sine & Cosine

2. 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–1x and Cos–1x OR Arcsinx and Arccosx

3. 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 cosθ) close I I II II (+) (+) (+) (–) These restrictions tell us where we draw the reference θ & △ close IV IV III III (–) (+) (–) (–)

4. *What type of answer is required • sin x • cosx • Sin–1x • Cos–1x trig function of an angle is a ratio trig (angle) = ratio inverse function of a ratio is an angle trig–1 (ratio) = angle  Find the θ and draw it’s picture in the correct quadrant! Ex 1) Evaluate to 4 decimal places (radian mode) Sin–1 0.3240 0.3300 Arcsin 0.5681 0.6042 Cos–1 (–0.56) 2.1652

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

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

7. 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*

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