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1. 4.6 Other Inverse Trig Functions

2. To get the graphs of the other inverse trig functions we make similar efforts we did to get inverse sine & cosine. We will also do the same type of computational problems! y = Tan–1x and y = Cot –1x Restricting domains: want (+) and (–) values AND no asymptotes in between y = tan x y = cot x close und und und I I II II close (+) (+) (–) (–) So y = Tan–1x Domain = R Range = So y = Cot–1x Domain = R Range = (0, π) IV IV III III (–) (–) (+) (+) und

3. Let’s trace with Sharpie on our WS of graphs to discover what the inverse trig functions look like • Trace the axes & tick marks • Write a (+) where x & y are positive • Trace asymptotes that are “pinning” in our values • Trace the graph between the asymptotes • Flip paper “over line y = x” • Label on new graph y = Cot–1 x y = Tan–1 x + + y = π + + y = 0

4. y = Csc–1x and y = Sec –1x Same domain as reciprocal function y = csc x y = sec x close und und I II (+) close (+) I II (+) (–) IV III (–) (–) IV III (+) (–) So y = Csc–1x Domain = Range = So y = Sec–1x Domain = Range = [0, π] & & y≠ 0

5. Time for more tracing & flipping • Trace the axes & tick marks • Write a (+) where x & y are positive • Now think about domains to make it a function • y = sec x asymptote at • y = cscx asymptote at x = 0 • - Reflect over y = x • Label on new graph trace values between [0, π] trace values between y = Sec–1 x y = Csc–1 x (–1, π) y = 0 (1, 0)

6. Calculator & Reference Triangle work * Remember what type of answer we are going for! Ex 1) Evaluate to 4 decimal places. Tan–1 1.54 0.9949 in range ? Yes! c) y = Arccsc (–3.86) –0.2621 in range ? Yes! b) y = Arccot (–5.1) –0.1936 in range [0, π]? No So, –0.1936 + π = 2.9480

7. Ex 2) Evaluate to nearest tenth of a degree. Arcsec (–1.433) 134.3° in range [0, 180°]? Yes! b) y = Cot–1 4.317 13.0° in range (0, 180°)? Yes!

8. Ex 3) Determine the exact value. a) (Draw those pictures!!) 12 b) θ θ 5 ratio  angle = θ c) 2 θ angle = θ

9. Ex 4) Rewrite y = sin (Cos–1t) as an algebraic expression. angle = θ 1 θ t

10. Homework #407 Pg 226 #1–15 odd, 16–24 all, 32, 35–39