4.6 Other Inverse Trig Functions

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 –1 x and y = Cot –1 x Restricting domains: want (+) and (–) values AND no asymptotes in between y = tan x y = cot x So y = Tan –1 x Domain = R Range = I II III IV (+) (–) (+) (–) close und I II III IV (+) (–) (+) (–) und So y = Cot –1 x Domain = R Range = (0, π) close

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 = Tan –1 x y = Cot –1 x y = π y =

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

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 = csc x asymptote at x = 0 - Reflect over y = x - Label on new graph y = Sec –1 x y = Csc –1 x (–1, π) y = 0 trace values between [0, π] trace values between (1, 0)

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

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

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

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

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