Exercises. 880 Foundations of Trigonometry

Transcription

1 880 Foundations of Trigonometry 0.. Exercises For a link to all of the additional resources available for this section, click OSttS Chapter 0 materials. In Exercises - 0, find the exact value. For help with these exercises, click the resource below: The Inverse Trigonometric Circular Functions. arcsin. arcsin. arcsin 0. arcsin. arcsin 7. arcsin. arcsin 8. arcsin 9. arcsin 0. arccos. arccos. arccos. arccos. arccos 0. arccos. arccos 7. arccos 8. arccos 9. arctan 0. arctan. arctan. arctan 0. arctan. arctan. arctan. arccot 7. arccot 8. arccot 9. arccot 0 0. arccot. arccot. arccot. arcsec. arccsc. arcsec. arccsc 7. arcsec 8. arccsc 9. arcsec 0. arccsc

2 0. The Inverse Trigonometric Functions 88 In Exercises - 8, assume that the range of arcsecant is [ 0, π [ π, π and that the range of arccosecant is 0, π ] ] π, π when finding the exact value.. arcsec. arcsec. arcsec. arcsec. arccsc. arccsc 7. arccsc 8. arccsc In Exercises 9 -, assume that the range of arcsecant is [ 0, π π, π] and that the range of arccosecant is [ π, 0 0, π ] when finding the exact value. 9. arcsec 0. arcsec. arcsec. arcsec. arccsc. arccsc. arccsc. arccsc In Exercises 7-8, find the exact value or state that it is undefined. 7. sin arcsin 8. sin arcsin 0. sin arcsin 0.. sin arcsin. cos arccos. cos arccos 9. sin arcsin. cos arccos. cos arccos cos arccos π 7. tan arctan 8. tan arctan 9. tan arctan 70. tan arctan tan arctan π 7. cot arccot 7. cot arccot 7. cot arccot 7 7π 7. cot arccot cot arccot 77. sec arcsec 78. sec arcsec 79. sec arcsec 80. sec arcsec 0.7

3 88 Foundations of Trigonometry 8. sec arcsec 7π 8. csc arccsc 8. csc 8. csc arccsc arccsc π 8. csc arccsc csc arccsc In Exercises 87-0, find the exact value or state that it is undefined. π 87. arcsin sin π 90. arcsin sin π 9. arccos cos π 9. arccos cos 88. arcsin sin π π 9. arcsin sin π 9. arccos cos π 97. arctan tan π 99. arctan tan π 00. arctan tan π 0. arccot cot π 0. arccot cot 0. arccot cot π 0. arccot cot π π 89. arcsin sin π 9. arccos cos 9. arccos cos π 98. arctan tan π 0. arctan tan 0. arccot cot π π In Exercises 07-8, assume that the range of arcsecant is [ 0, π [ π, π and that the range of arccosecant is 0, π ] ] π, π when finding the exact value. π 07. arcsec sec 0. arcsec sec π π. arccsc csc. arccsc csc π π 08. arcsec sec π. arcsec sec π. arccsc csc π 7. arcsec sec 09. arcsec sec π π. arccsc csc. arccsc csc π 8. arccsc csc 9π 8

4 0. The Inverse Trigonometric Functions 88 In Exercises 9-0, assume that the range of arcsecant is [ 0, π π, π] and that the range of arccosecant is [ π, 0 0, π ] when finding the exact value. π 9. arcsec sec. arcsec sec π π. arccsc csc π 8. arccsc csc π 0. arcsec sec π. arcsec sec π. arccsc csc π 9. arcsec sec In Exercises -, find the exact value or state that it is undefined. For help with these exercises, click one or more of the resources below: The Inverse Trigonometric Circular Functions Using the Quotient, Reciprocal, and Pythagorean Identities. sin arccos. sin arccos. sin arccot. sin arccsc. cos π. arcsec sec π. arccsc csc 7. arccsc csc π 0. arccsc csc 9π 8. sin arctan arcsin 7. cos arctan 7 8. cos arccot 9. cos arcsec 0. tan arcsin. tan arccos. tan arcsec. tan arccot. cot arcsin. cot. cot arccsc 7. cot arctan sec 9. sec arcsin. csc arccot 9. csc arcsin arccos arccos 0 0. sec arctan 0. sec arccot 0. csc arctan

5 88 Foundations of Trigonometry In Exercises -, find the exact value or state that it is undefined. For help with these exercises, click one or more of the resources below: The Inverse Trigonometric Circular Functions Using the Quotient, Reciprocal, and Pythagorean Identities Using the Double Angle Identities Using the Half Angle Identities. sin arcsin + π 7. tan arctan + arccos. cos arcsec + arctan 8. sin arcsin 9. sin arccsc 0. sin arctan. cos arcsin. cos arcsec 7. cos arccot arctan. sin In Exercises - 8, rewrite the quantity as algebraic expressions of x and state the domain on which the equivalence is valid.. sin arccos x. cos arctan x 7. tan arcsin x 8. sec arctan x 9. csc arccos x 70. sin arctan x 7. sin arccos x 7. cos arctan x 7. sinarccosx x x 7. sin arccos 7. cos arcsin 7. cos arctan x 77. sin arcsin7x 78. sin arcsin x 79. cos arcsinx 80. secarctanx tanarctanx 8. sin arcsinx + arccosx 8. cos arcsinx + arctanx

6 0. The Inverse Trigonometric Functions tan arcsinx 8. sin arctanx 8. If sinθ = x for π < θ < π, find an expression for θ + sinθ in terms of x. 8. If tanθ = x 7 for π < θ < π, find an expression for θ sinθ in terms of x. 87. If secθ = x for 0 < θ < π, find an expression for tanθ θ in terms of x. In Exercises 88-07, solve the equation using the techniques discussed in Example 0..7 then approximate the solutions which lie in the interval [0, π to four decimal places. 88. sinx = cosx = sinx = cosx = sinx = cosx = tanx = 7 9. cotx = 9. secx = 97. cscx = cosx = tanx = sinx = 8 0. tanx = sinx = sinx = cosx = cosx = cotx = tanx = 0.09 In Exercises 08-0, find the two acute angles in the right triangle whose sides have the given lengths. Express your answers using degree measure rounded to two decimal places. 08., and 09., and 0., 7 and For help with Exercises -, click one or more of the resources below: Solving right triangles Solving application problems with right triangle trigonometry. A guy wire 000 feet long is attached to the top of a tower. When pulled taut it touches level ground 0 feet from the base of the tower. What angle does the wire make with the ground? Express your answer using degree measure rounded to one decimal place.

7 88 Foundations of Trigonometry. At Cliffs of Insanity Point, The Great Sasquatch Canyon is 77 feet deep. From that point, a fire is seen at a location known to be 0 miles away from the base of the sheer canyon wall. What angle of depression is made by the line of sight from the canyon edge to the fire? Express your answer using degree measure rounded to one decimal place.. Shelving is being built at the Utility Muffin Research Library which is to be inches deep. An 8-inch rod will be attached to the wall and the underside of the shelf at its edge away from the wall, forming a right triangle under the shelf to support it. What angle, to the nearest degree, will the rod make with the wall?. A parasailor is being pulled by a boat on Lake Ippizuti. The cable is 00 feet long and the parasailor is 00 feet above the surface of the water. What is the angle of elevation from the boat to the parasailor? Express your answer using degree measure rounded to one decimal place.. A tag-and-release program to study the Sasquatch population of the eponymous Sasquatch National Park is begun. From a 00 foot tall tower, a ranger spots a Sasquatch lumbering through the wilderness directly towards the tower. Let θ denote the angle of depression from the top of the tower to a point on the ground. If the range of the rifle with a tranquilizer dart is 00 feet, find the smallest value of θ for which the corresponding point on the ground is in range of the rifle. Round your answer to the nearest hundreth of a degree. In Exercises -, rewrite the given function as a sinusoid of the form Sx = A sinωx + φ using Exercises and in Section 0. for reference. Approximate the value of φ which is in radians, of course to four decimal places.. fx = sinx + cosx 7. fx = cosx + sinx 8. fx = cosx sinx 9. fx = 7 sin0x cos0x 0. fx = cosx sinx. fx = sinx cosx In Exercises -, find the domain of the given function. Write your answers in interval notation.. fx = arcsinx x. fx = arccos. fx = arcsin x x. fx = arccos x. fx = arctanx 7. fx = arccot x 9 8. fx = arctanlnx 9. fx = arccot x 0. fx = arcsecx

8 0. The Inverse Trigonometric Functions 887 x. fx = arccscx +. fx = arcsec 8. Show that arcsecx = arccos of fx = arcsecx.. Show that arccscx = arcsin of fx = arccscx.. Show that arcsinx + arccosx = π 7. Discuss with your classmates why arcsin for x as long as we use x. fx = arccsc e x [ 0, π π ], π as the range for x as long as we use [ π x, 0 0, π ] as the range for x Use the following picture and the series of exercises on the next page to show that arctan + arctan + arctan = π y D, A0, α β γ x O0, 0 B, 0 C, 0 a Clearly AOB and BCD are right triangles because the line through O and A and the line through C and D are perpendicular to the x-axis. Use the distance formula to show that BAD is also a right triangle with BAD being the right angle by showing that the sides of the triangle satisfy the Pythagorean Theorem. b Use AOB to show that α = arctan c Use BAD to show that β = arctan d Use BCD to show that γ = arctan e Use the fact that O, B and C all lie on the x-axis to conclude that α + β + γ = π. Thus arctan + arctan + arctan = π.

9 888 Foundations of Trigonometry. Simplify: a arcsin sin π b sin arcsin0. c cos arcsin0.. Simplify: sin arcsec + arctan. Simplify: sin arcsinx. Solve for the following equations: Checkpoint Quiz 0. a sinθ = b cost = 0. c tanx =. Write fx = cos7x sin7x in the form Sx = A sinωx + φ. For worked out solutions to this quiz, click the links below: Quiz Solution Part Quiz Solution Part Quiz Solution Part Quiz Solution Part

10 0. The Inverse Trigonometric Functions Answers. arcsin = π. arcsin = π. arcsin = π. arcsin = π. arcsin 0 = 0. arcsin = π 7. arcsin = π 8. arcsin = π 9. arcsin = π 0. arccos = π. arccos = π. arccos = π. arccos = π. arccos = π. arccos 0 = π 7. arccos = π 9. arctan = π 0. arctan = π. arctan 0 = 0. arctan = π. arccos = π 8. arccos = 0. arctan = π. arctan = π. arctan = π 8. arccot = π. arccot = π 9. arccot 0 = π 7. arccot = π 0. arccot = π. arccot = π. arccot = π. arcsec = π. arccsc = π 7. arcsec = π. arcsec = π 8. arccsc = π. arccsc = π 9. arcsec = 0 0. arccsc = π. arcsec = π. arcsec = π

11 890 Foundations of Trigonometry. arcsec. arccsc = π 9. arcsec = π = 7π. arcsec = π. arccsc = 7π 7. arccsc 0. arcsec = π = π 8. arccsc = π. arcsec = π. arcsec = π. arccsc = π. arccsc = π. arccsc = π. arccsc = π 7. sin arcsin = 8. sin arcsin = 9. sin arcsin = 0. sin arcsin 0. = 0.. sin arcsin is undefined.. cos arccos =. cos arccos =. cos arccos =. cos arccos = cos arccos π is undefined. 7. tan arctan = 8. tan arctan = 9. tan arctan = 70. tan arctan 0.9 = tan arctan π = π 7. cot arccot = 7. cot arccot = 7. cot arccot 7 = 7 7. cot arccot 0.00 = π 7. cot arccot = 7π 77. sec arcsec = 78. sec arcsec =

12 0. The Inverse Trigonometric Functions sec arcsec 8. sec arcsec 8. csc arccsc is undefined. π = π = 80. sec arcsec 0.7 is undefined. 8. csc arccsc = 8. csc arccsc is undefined. π 8. csc arccsc.000 = csc arccsc is undefined. π 87. arcsin sin = π π 89. arcsin sin = π π 9. arcsin sin = π 9. arccos cos π = π 9. arccos cos π = π π 97. arctan tan = π 88. arcsin sin π = π π 90. arcsin sin = π π 9. arccos cos = π π 9. arccos cos = π π 9. arccos cos = π 98. arctan tan π = π π 99. arctan tan π = arctan tan is undefined π 0. arctan tan = π 0. arccot cot π = π π 0. arccot cot = π π 07. arcsec sec = π π 09. arcsec sec = 7π. arcsec sec π = π π 0. arccot cot = π 0. arccot cot π is undefined π 0. arccot cot 08. arcsec sec π = π = π 0. arcsec sec π is undefined. π. arccsc csc = π

13 89 Foundations of Trigonometry π. arccsc csc = π. arccsc csc π = π 7. arcsec sec π = π π 9. arcsec sec = π π. arcsec sec = π π. arcsec sec = π. arccsc csc π = π 7. arccsc csc π = π π 9. arcsec sec = π. sin arccos =. sin arctan =. sin arccsc = π. arccsc csc. arccsc csc 8. arccsc csc 0. arcsec sec = π π = 7π 9π = 9π 8 8 π = π. arcsec sec π is undefined. π. arccsc csc = π π. arccsc csc = π π 8. arccsc csc = π 9π 0. arccsc csc = π 8 8. sin arccos =. sin arccot =. cos arcsin = 7. cos arctan 7 = 8. cos arccot = cos arcsec = 0. tan arcsin =. tan arccos =. tan arcsec =. tan arccot =. cot arcsin =

14 0. The Inverse Trigonometric Functions 89. cot arccos =. cot arccsc = 7. cot arctan 0. = 8. sec arccos 9. sec arcsin = 0. sec arctan 0 = 0 0. sec arccot = 0. csc arccot 9 = 8. csc arcsin =. sin arcsin 7. tan arctan + arccos 9. sin arccsc. cos arcsin + π = 7 = 0 9 = 7. cos arccot = =. sin arccos x = x for x. cos arctan x = 7. tan arcsin x = + x x x for all x for < x < 8. sec arctan x = + x for all x 9. csc arccos x = x 70. sin arctan x = x x + for < x < for all x 7. sin arccos x = x x for x =. csc arctan = 0. cos arcsec + arctan = 8. sin arcsin = 0. sin arctan =. cos arcsec = 7 7 arctan. sin = 0

15 89 Foundations of Trigonometry 7. cos arctan x = x for all x + x 7. sinarccosx = x for x x x 7. sin arccos = for x x x 7. cos arcsin = for x 7. cos arctan x = + 9x for all x 77. sin arcsin7x = x 9x for 7 x 7 x 78. sin arcsin = x x for x 79. cos arcsinx = x for x 80. secarctanx tanarctanx = x + x for all x 8. sin arcsinx + arccosx = for x x 8. cos arcsinx + arctanx = x for x + x 8. 0 tan arcsinx = x x x for x in,, 8. sin arctanx = x + x + x + x + for x 0 for x < 0 8. If sinθ = x for π < θ < π, then θ + sinθ = arcsin x 8. If tanθ = x 7 for π < θ < π, then θ sinθ = arctan x 7 + x x, 7x x The equivalence for x = ± can be verified independently of the derivation of the formula, but Calculus is required to fully understand what is happening at those x values. You ll see what we mean when you work through the details of the identity for tant. For now, we exclude x = ± from our answer.

16 0. The Inverse Trigonometric Functions If secθ = x for 0 < θ < π, then tanθ θ = x x arcsec x = arcsin 89. x = arccos πk or x = π arcsin + πk or x = arccos 9 + πk, in [0, π, x 0.898,.8 + πk, in [0, π, x.799, x = π + arcsin0.9 + πk or x = π arcsin0.9 + πk, in [0, π, x.79, x = arccos0.7 + πk or x = π arccos0.7 + πk, in [0, π, x., x = arcsin πk or x = π arcsin πk, in [0, π, x , x = arccos + πk or x = π arccos + πk, in [0, π, x 0.07, x = arctan7 + πk, in [0, π, x., x = arctan + πk, in [0, π, x.08, x = arccos 97. x = π + arcsin + πk or x = π arccos 7 + πk or x = π arcsin x = arctan 0 + πk, in [0, π, x.877, x = arcsin + πk or x = π arcsin x = arccos 7 + πk or x = arccos 7 0. x = arctan0.0 + πk, in [0, π, x 0.000,.7 + πk, in [0, π, x 0.8,. 7 + πk, in [0, π, x., πk, in [0, π, x 0.8,.77 + πk, in [0, π, x.0,.9 0. x = arcsin0.0 + πk or x = π arcsin0.0 + πk, in [0, π, x 0.78, x = π + arcsin0.7 + πk or x = π arcsin0.7 + πk, in [0, π, x.98, x = arccos πk or x = π arccos πk, in [0, π, x 0.879, x = arccos πk or x = arccos πk, in [0, π, x.97,. 0. x = arctan7 + πk, in [0, π, x., x = arctan πk, in [0, π, x.9,.78

17 89 Foundations of Trigonometry and and and fx = sinx + cosx = sin x + arcsin sinx fx = cosx + sinx = sin x + arcsin sinx fx = cosx sinx = 0 sin x + arccos 0 0 sinx fx = 7 sin0x cos0x = sin 0x + arcsin sin0x fx = cosx sinx = sin x + π + arcsin sinx +.8. fx = sinx cosx = sin x + arcsin sinx 0... [, ] [, ]. [, ]., ] [, ] [,., 7.,,, [ 8., 9., 0., ] [,., ] [,., ] [,. [0,