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1 lgebra Chapter 8: nalytical Trigonometry 8- Inverse Trigonometric Functions Chapter 8: nalytical Trigonometry Inverse Trigonometric Function: - use when we are given a particular trigonometric ratio and we are asked to solve for the original angle measure. - it is sometimes referred to as arc-sine sin, arc-cosine cos, or arc-tangent tan. Note: sin x sin x sin x sin x sin x sin x csc x Examples: For angles between 0 and, sin sin 6 6 cos cos 6 tan tan 6 To access Inverse Trigonometric Function on most calculators: nd IN IN or nd CO CO Graph of Inverse ine Function and its Domain and Range: or Recall that a function, fx, has to pass the vertical line test. Hence, for an inverse function, f x has to pass the horizontal line test one to one. y sin x nd TN TN y sin x Calculator is in Radian Mode y sin x revised Domain: x Range: y 6 Note that y sin x does NOT pass the horizontal line test between [, ]. However, if we take the interval at [ ], it will pass the horizontal line test. Hence, we can graph y sin x for y:[ ].,, y sin x Therefore, the output from a calculator of a sin x input, where x, is always between[ ], y sin x Domain: x Range: y sinsin x x for x sin sin x x for x Copyrighted by Gabriel Tang B.Ed., B.c. Page 67.

2 Chapter 8: nalytical Trigonometry lgebra Graph of Inverse Cosine Function and its Domain and Range: Recall that a function, fx, has to pass the vertical line test. Hence, for an inverse function, f x has to pass the horizontal line test one to one. y cos x y cos x y cos x revised Domain: 0 x Range: y Note that y cos x does NOT pass the horizontal line test between [, ]. However, if we take the interval at [0, ], it will pass the horizontal line test. Hence, we can graph y cos x for y: [0, ]. Therefore, the output from a calculator of a cos x input, where x, is always between [0, ] y cos x y cos x Domain: x Range: 0 x coscos x x for x cos cos x x for 0 x Graph of Inverse Tangent Function and its Domain and Range: Recall that a function, fx, has to pass the vertical line test. Hence, for an inverse function, f x has to pass the horizontal line test one to one. y tan x y tan x y tan x revised Domain: x Range: y R Note that y tan x does NOT pass the horizontal line test between [, ]. However, if we take the. interval at [ ], it will pass the horizontal line test. Hence, we can graph y tan x for y:[ ],, y tan x Therefore, the output from a calculator of a sin x input, where x, is always between[ ] y tan x Domain: x R Range: y, tantan x x tan tan x x for x R for x Page 68. Copyrighted by Gabriel Tang B.Ed., B.c.

3 lgebra Chapter 8: nalytical Trigonometry Example : Find the exact value of each expression, if it is defined. a. sin sin b. tan tan c. cos Example : Find the exact value of each expression, if it is defined. Calculator gives the a. sin sin b. sin same decimal value. sin Hence, sin interval allowed. sin e. tan From the unit circle, at, the y- value is. Verify: sin and is within the range of y sin x, which is y. sinsin x x for x and is within [, ] coscos x x for x and is within [, ] Verify: tan and range of y tan x, which is y. Copyrighted by Gabriel Tang B.Ed., B.c. Page 69. No olution tan sinsin x x for x and is within [, ] Calculator gives the c. cos cos d. cos 7 cos same decimal value. 6 Hence, cos interval allowed. cos 6 tantan x x for x R. cos cos x x for 0 x 7 and 6 is not within [0, ] Calculator gives the tan f. tan same decimal value. tan Hence, interval allowed. tan 6 tan tan tan x x for x and is within [, ] g. sin cos ½ h. tan sin 6 sin tan ½ 0.66 rad The range of y tan x is The domain of y cos x [, ], and 0.66 rad is [, ], and ½ is within is within this interval. this interval. The domain of y cos x is [, ]. is outside this interval. Calculator gives the same decimal value. Hence, interval allowed. Calculator does not give the same decimal value. 7 Hence, 6 is not within the interval allowed. Calculator gives the same decimal value. Hence, interval allowed.

4 Chapter 8: nalytical Trigonometry lgebra Example : Evaluate cos hyp θ adj x sin opp by sketching a triangle. Recall OH CH TO opp sin θ hyp Example : Rewrite sintan x as an algebraic expression in x adj cos θ hyp opp tan θ adj When we simplify sin, we are solving for the angle θ. Therefore, we can let θ sin cos sin cos θ adj x cos θ hyp Using Pythagorean Theorem: x + x 69 x x cos sin We can let θ tan opp x x, which can be rewrite as tan θ x. adj. Now drawing and labelling the triangle, we can find an expression for the hypotenuse. hyp x + hyp x + Going back to the original expression, sintan opp x x sin θ hyp x + hyp? θ adj sintan x x x + opp x 8- ssignment: pg #,,,, 7, 9,,,, 7 Honours #9,,,, 8 Page 70. Copyrighted by Gabriel Tang B.Ed., B.c.

5 lgebra Chapter 8: nalytical Trigonometry 8- Trigonometric Equations ome Basic Trigonometric Definitions and Identities proven equations tan θ cot θ sinθ cosθ tanθ cosθ sinθ Using regular factoring technique common factor and factoring trinomials, simplifying rational expressions, substituting with basic trigonometric definitions listed above, and referring to the unit circle, we can solve for the solution of various type of trigonometric equations. Example : Find the solutions for 0 x <. csc θ sinθ c. cos x + cos x + 0 d. cscx sec θ cos θ + sin θ Copyrighted by Gabriel Tang B.Ed., B.c. Page 7. cosθ a. tan x 0 b. sin x cos x sin x, tan x sin x cos x sin x 0 sin x cos x 0 x or sin x 0 cos x 0 cos x Let a cos x a + a + 0 a + a + 0 a + 0 a + 0 a ½ a cos x ½ cos x,,, 0, x 0 or Let a x a x csc a sin a, 0 x No solutions sin a ½ 7 9 a 6, 6,,,, x 6, 6,,,, , 0 0 x < solve x in rev 0 a < 0 a < 6 solve a in rev 6, x or x 7 9 x,,,,, e. sin x f. sin x 9 cos x round to the th decimal place 0 sin x 9cos x [ubstitute identity: x Let a 0 x < solve x in rev 0 cos x 9cos x sin x cos x] a x 0 a < 0 a < solve a in ½ rev, y sin a sin a ½ sin a ± sin a ± ± a, x 6, x, x,,, 0 cos x 9cos x 0 cos x 9cos x + cos x + 9cos x 0 cos x cos x + 0, cos x 0 cos x + 0 cos x cos x cos x x No olution x.69 rad,.97 rad, y

6 Chapter 8: nalytical Trigonometry lgebra Example : Using a graphing calculator, determine the solutions of sin x x. In Radian Mode, enter each side of the equation as Y and Y in Y screen ZOOM elect ZTrig Run Intersect twice from nd TRCE x ±.0 Example : The height of a tidal wave above the average sea level is related to time by the function, dt. sin 0.6t. +., where d represents depth in metres, above sea level and t is the time in hours t 0 means midnight. When within one whole day is the tidal wave at a depth of at least m for a ship to dock safely? mplitude a. m Vert. Disp. c. Maximum. m +. m.9 m Minimum. m. m 0.9 m ince we are given the output m to find the input time, we need to enter two equations and find the intersecting points. In Radian Mode, enter the equations as Y and Y in Y screen Water depth will be above m at 0.6 hrs < t < 8.7 hrs and. hrs < t < 0.96 hrs : M < t < 8: M and : PM < t < 8:6 PM 8- ssignment: pg #,, 7, 9,,,,, b, b, 7, 79, 8; Honour # Page 7. Copyrighted by Gabriel Tang B.Ed., B.c.

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