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Trig Inverses: Functions, Graphs & Review

Learn how to find inverses of functions, explore inverse trig functions, and understand domain restrictions for invertibility. Complete tables, graph functions, and evaluate expressions.

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Trig Inverses: Functions, Graphs & Review

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  1. 6.8 – Trig Inverses and their graphs

  2. Quick Review • How do you find inverses of functions? • Are inverses of functions always functions? • How did we test for this?

  3. Inverse Trig Functions

  4. Consider the graph of y = sin x • What is the domain and range of sin x? • What would the graph of y = arcsin x look like? • What is the domain and range of arcsin x? Domain: all real numbers Range: [-1, 1] Domain: [-1, 1] Range: all real numbers

  5. Is the inverse of sin x a function? • This will also be true for cosine and tangent. • Therefore all of the domains are restricted in order for the inverses to be functions.

  6. How do you know if the domain is restricted for the original functions? • Capital letters are used to distinguish when the function’s domain is restricted.

  7. Original Domains  Restricted Domains

  8. Complete the following table on your own

  9. Table of Values of Sin x and Arcsin x Why are we using these values?

  10. Graphs of Sin x and Arcsin x

  11. Table of Values of Cos x and Arccos x Why are we using these values?

  12. Graphs of Cos x and Arccos x

  13. Table of Values of Tan x and Arctan x Why are we using these values?

  14. Graphs of Tan x and Arctan x

  15. Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse. • To write the equation: • Exchange x and y • Solve for y Let’s graph 2Tan x = y first. Complete the table: Then graph! • x = Arctan ½y • Tan x = ½y • 2Tan x = y Now graph the original function, y = Arctan ½x by switching the table you just completed!

  16. Write an equation for the inverse of y = Sin(2x). Then graph the function and its inverse. • To write the equation: • Exchange x and y • Solve for y Let’s graph y = Sin(2x)first. Why are these x-values used? • x = Sin(2y) • Arcsin(x) = 2y • Arcsin(x)/2 = y Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!

  17. Evaluate each expression

  18. Evaluate each expression

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