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Inverse Trigonometric Functions

Inverse Trigonometric Functions. We will now formalize what we have been doing informally throughout this chapter. Review of Inverse Functions. They “undo” one another. Can you name some? What is the inverse of y = 2x + 3? x and y are interchanged They are symmetrical around the line y = x.

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Inverse Trigonometric Functions

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  1. Inverse Trigonometric Functions We will now formalize what we have been doing informally throughout this chapter.

  2. Review of Inverse Functions • They “undo” one another. Can you name some? • What is the inverse of y = 2x + 3? • x and y are interchanged • They are symmetrical around the line y = x. • To have an inverse a function must be “one-to-one”. How do we tell? • f(f-1(x)) = x

  3. Is the function y = sin x one-to one? So let’s restrict the function to –π/2 ≤ x ≤π/2. Why do you suppose we chose this interval?

  4. Now draw the inverse of this function. Call it y = sin-1 x. Properties of y = sin-1 x or y = arcsin x. • The domain is -1≤x≤ 1. Why? • The range is –π/2 ≤ y ≤π/2. • So this function gives the angle that has the given sine value. However, it will always give a value in which quadrants?

  5. Is the function y = cos x one-to one? So let’s restrict the function to 0 ≤ x ≤ π. Why do you suppose we chose this interval?

  6. Now draw the inverse of this function. Call it y = cos-1 x. Properties of y = cos-1 x or y = arccosx. • The domain is -1≤x≤ 1. Why? • The range is 0≤ y ≤ π. • So this function gives the angle that has the given cosine value. However, it will always give a value in which quadrants?

  7. Is the function y = tan x one-to one? So let’s restrict the function to –π/2 < x < π/2. Why do you suppose we chose this interval?

  8. Now draw the inverse of this function. Call it y = tan-1 x. Properties of y = tan-1 x or y = arctanx. • The domain is all reals. Why? • The range is -π/2< y < π/2. • So this function gives the angle that has the given tangent value. However, it will always give a value in which quadrants?

  9. Summary of Inverse Trig Functions domain range y = sin -1 x -1≤x≤1 –π/2 ≤ y ≤π/2 y = cos-1 x -1≤x≤1 0≤ y ≤ π y = tan-1 x all reals –π/2 < y <π/2

  10. Evaluate without a calculator • a) evaluate arccos (-√2/2) • b) evaluate arcsin (-1) • c) evaluate arctan (0) • d) evaluate arcsin (-1/2) • e) evaluate arctan (-1)

  11. Evaluate with a calculator. • A) arcsin (.2447) • B) arctan (-8.45) • C) arccos (2)

  12. Evaluate without a calculator. • A) sin (arctan 1) • B) csc ( arcsin ¾) • C) tan (arccos 8/17) • D) arcsin (sin π/6) • E) sin (arcsin √2/2) • F) sin (arcsin 2) • G) arcsin (sin π)

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