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Inverse Trigonometric Functions 4.7

Inverse Trigonometric Functions 4.7. How do you determine if a function has an inverse?. It must be one to one … pass the horizontal line test Will a sine, cosine, or tangent function have an inverse? Their inverses are defined over the following intervals: Sine: [ - π /2, π /2 ]

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Inverse Trigonometric Functions 4.7

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  1. Inverse Trigonometric Functions4.7

  2. How do you determine if a function has an inverse? • It must be one to one … pass the horizontal line test • Will a sine, cosine, or tangent function have an inverse? • Their inverses are defined over the following intervals: • Sine: [ -π/2, π/2 ] • Cosine: [ 0, π ] • Tangent: [ -π/2, π/2 ]

  3. Notation for inverse trig functions: • y = sin-1x or y = arcsin x • i.o.i sin y = x • y = cos-1x or y = arccos x • y = tan-1x or y = arctan x • Their graphs are on pg 324 if you would like to reference them

  4. Chart from page 324

  5. Let’s evaluate inverse trig functions: • 1.) 2.)

  6. Try some on your own… • 1.) • 2.) • 3.) It will be helpful to remember: sinθ = y then arcsin y = θ cosθ = x then arccos x = θ

  7. Calculator…. • To do this on your calculator… • 2nd shift then trig function • The steps are on page 325 if you need a refresher  • Let’s practice…. • Pg. 328 #’s 2 – 26 even

  8. Day 1 HW…. • Pg. 328 #’s 1 – 41 odd

  9. Inverse Trig Day 2!! • Compositions of Functions • f (f-1 (x) ) = ? • f-1 ( f(x) ) = ? • Therefore: • sin(arcsin x) = x arcsin(sin y) = y • cos(arccos x) = x arccos(cos y) = y • tan(arctan x) = x arctan(tan y) = y • *remember - only works over certain intervals… • Refer to page 326

  10. What should you do if it lies outside the range? • Use the co-terminal angles that are in the range! • Let’s practice: • 1) arcsin[sin (π/2) ] = ? • 2) arccos[cos (π/6) ] = ? • 3) tan[arctan (-5)] = ? • 4) arcsin[sin (-π/4)]= ?

  11. A few more…. • 5) arcsin(sin 5π/3 ) = ? 6) sin(arcsinπ ) = ? • 7) arctan (tan π/6) = ? 8) tan(arcsin √2/2)=? • Pg. 328 #’s 44, 46, 48

  12. Using right triangles to find exact values of compositions of inverse functions. • Ex. 1) Find the exact value of tan(arccos 2/3) • Use a right triangle

  13. Some more practice…. • 2.) sin(arccos √5/ 5) 3.) csc[arctan(-5/12)]

  14. Writing algebraic expressions… • Ex.1) sin(arccos 3x) ; 0 ≤ x ≤ 1/3 • Ex 2) cot(arcsin 2x) ; 0 ≤ x ≤ 1/3 • Practice pg. 328 #’s 60, 64, 66, 68

  15. HW DAY 2….. • Pg. 328 #’s 47-73 odd, 91, 95

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