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All Textbook Solutions for Trigonometry (MindTap Course List)
Simplifying a Trigonometric Expression In Exercises 33-40, use the fundamental identities to simplify the expression. (There is more than one correct form of each answer.) tanxcosx35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59ETrigonometric Substitution In Exercises 59-62, use the trigonometric substitution to write the algebraic equation as a trigonometric equation of , where /2 /2 . Then find sin and cos. 22=164x2,x=2cos61E62E63E64E65E66E67E68E69ETrue or False? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer. A cofunction identity can transform a tangent function into a cosecant function.71E72E73E74E75E76ERewriting a Trigonometric Expression Rewrite the expression below in terms of sin andcos. sec1+tansec+csc1ECP2ECP3ECP4ECPVerify the identity cscx+cotx=sinx1cosx.Verify the identity tan21+sec=1coscosVerify the identity a. tan3x=tanx sec2xtanx b. sin3xcos4x=cos4xcos6xsinx1E2E3E4E5E6EIn Exercises 38, fill in the blank to complete the fundamental trigonometric identity. cscu=8E9E10E11E12E13E14E15E16E17EVerifying a Trigonometric Identity In Exercises 918, verify the identity. sec2ycot22y=119E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48EDetermining Trigonometric Identities In Exercises 4550, (a) use a graphing utility to graph each side of the equation to determine whether the equation is an identity, (b) use the table feature of the graphing utility to determine whether the equation is an identity, and (c) confirm the results of parts (a) and (b) algebraically. 1+cosxsinx=sinx1cosx50E51E52E53E54E55E56E57E58E59E60E61EShadow length The length s of a shadow cast by a vertical gnomon (a device used to tell time) of height h when the angle of the sun above the horizon is can be modeled by the equation s=hsin90sin 090 (a) Verify that the expression for s is equal to h cot . (b) Use a graphing utility to create a table of the lengths s for different values of . Let h=5 feet. (c) Use your table from part (b) to determine the angle of the sun that results in the minimum length of the shadow. (d) Based on your results from part (c), what time of day do you think it is when the angle of the sun above the horizon is 90?63E64E65E66E67E68E69E70ESolve sinx2=sinx.2ECP3ECP4ECP5ECP6ECP7ECP8ECPSolve 4tan2x+5tanx6=0.10ECP11ECP1E2EFill in the blanks. The equation 2tan2x3tanx+1=0 is a trigonometric equation of type.4E5E6EVerifying Solutions In Exercises 5-10, verify that each x-value is a solution of the equation. 3tan22x1=0 (a) x=12 (b) x=512Verifying Solutions In Exercises 5-10, verify that each x -value is a solution of the equation. 2cos24x1=0 (a) x=16 (b) x=3169E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31ESolving a Trigonometric Equation In Exercises 2938, find all solutions of the equation in the interval 0,2. tan2x=secx1Solving a Trigonometric Equation In Exercises 2938, find all solutions of the equation in the interval 0,2. sin2x=3cos2xSolving a Trigonometric Equation In Exercises 29-38, find all solutions of the equation in the interval 0,2. 2sec2x+tan2x3=0Solving a Trigonometric Equation In Exercises 29-38, find all solutions of the equation in the interval 0,2. 2sinx+cscx=0Solving a Trigonometric Equation In Exercises 29-38, find all solutions of the equation in the interval 0,2. 3secx4cosx=0Solving a Trigonometric Equation In Exercises 2938, find all solutions of the equation in the interval 0,2. cscx+cotx=1Solving a Trigonometric Equation In Exercises 2938, find all solutions of the equation in the interval 0,2. secx+tanx=1Solving a Multiple-Angle Equation In Exercises 3946, solve the multiple-angle equation. 2cos2x1=0Solving a Multiple-Angle Equation In Exercises 3946, solve the multiple-angle equation. 2sin2x+3=041E42E43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59EUsing Inverse Functions In Exercises 59-70, solve the equation. tan2xtanx2=061E62E63E64E65E66EUsing Inverse Functions In Exercises 5970, solve the equation. sec2x4secx=0Using Inverse Functions In Exercises 5970, solve the equation. sec2x+2secx8=0Using Inverse Functions In Exercises 5970, solve the equation. csc2x+3cscx4=0Using Inverse Functions In Exercises 5970, solve the equation. csc2x5cscx=071E72E73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90EEquipment Sales The monthly sales S (in hundreds of units) of skiing equipment at a sports store are approximated by S=58.3+32.5cost6 where t is the time (in months), with t=1 corresponding to January. Determine the months in which sales exceed 7500 units.92E93EFerris Wheel The height h (in feet) above ground of a seat on a Ferris wheel at time t (in minutes) can be modeled by ht=53+50sin16t2. The wheel makes one revolution every 32 seconds. The ride begins when t=0. (a) During the first 30 seconds of the ride, when will a person’s seat on the Ferris wheel be 53 feet above ground? (b) When will a person’s seat be at the top the Ferris wheel for the first time during the ride? For a ride that lasts 160 seconds, how many times will a person’s seat be at the top of the ride, and at what times?95E96E97E98E99ETrue or False? In Exercises 99 and 100, determine whether the statement is true or false. Justify your answer. The trigonometric equation sinx=3.4 can be solved using an inverse trigonometric function.Think About it Explain what happens when you divide each side of the equation cotxcos2x=2cotx by cotx. Is this a correct method to use when solving equations?HOW DO YOU SEE IT? Explain how to use the figure to solve the equation 2cosx1=0Graphical Reasoning Use a graphing utility to confirm the solutions found in Example 6 in two different ways. (a) Graph both sides of the equation and find the x -coordinates of the points at which the graphs intersect. Left side : y=cosx+1 Right side : y=sinx (b) Graph the equation y=cosx+1sinx find the x-intercepts of the graph. (c) Do both methods produce the same x -values? Which method do you prefer? Explain.Find the exact value of cos12.2ECPFind the exact value of cosu+v given sinu=1213, where 0u2 and cosv=35, where 2v.4ECP5ECP6ECP7ECP8ECP1E2EFill in the blank. tanu+v=4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30ERewriting a Trigonometric Expression In Exercises 2734, write the expression as the sine, cosine, or tangent of an angle. tan/15+tan2/51tan/15tan2/532E33E34E35E36E37E38E39E40E41E42EEvaluating a Trigonometric Expression In Exercises 4146, find the exact value of the trigonometric expression given that sinu=35, where 3/2u2, and cosv=1517, where 0v/2. tanu+v44E45E46E47E48E49E50E51E52E53E54E55E56E57E