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A V-gauge is used to find the diameters of pipes. The measure of angle AVB is 54°. A pipe is placed in the V-shaped slot and the distance VP is used to predict the diameter. a. b. c. Suppose that the diameter of a pipe is 2 cm. What is the distance VP? Suppose that the distance VP is 3.93 cm. What is the diameter of the pipe? Find the formula for d in terms of VP. Find a formula for VP in terms of d. The gauge setup is like this: VB is tangential to the pipe, Therefore the angle VPO is a right angle, since a tangent is always perpendicular to the radius. The angle is half of the angle AVB. Therefore the ratio between the radius and VP yields: R tan VP R VP tan The radius is half the diameter, therefore: d 2 R 2VP tan d VP 2 tan Plugging in the numbers for part a: 54 27 2 d 2cm VP d 2 1.96cm 2 tan 2 0.51 And for part b: 54 27 2 VP 3.93cm d 2VP tan 2 3.93 0.51 4cm 1. Two cars with new tires are driven at an average speed of 60 mph for a test drive of 2000 miles. The diameter of the wheels of one car is 15 inches. The diameter of the wheels of the other car is 16 inches. If the tires are equally durable and differ only by diameter, which car will probably need new tires first? Why? If the tires are equally durable, the tire that completes more revolutions wears sooner. At each revolution the car advances the circumference of the wheel. The wheel with the larger diameter has a larger circumference (c=d). The larger wheel will need to complete fewer revolutions in order to advance the car the same distance as the car with the smaller wheel. Hence the car with the smaller wheel will probably need new tires sooner. 2. Explain why tan(x + 450 degrees) cannot be simplified using the tangent sum formulas but can be simplified by using the sine and cosine formulas. If we use the tangent sum formula tan tan tan 1 tan tan Then we get: tan 450 tan tan 450 1 tan tan 450 The problem arises at tan(450). We know that every trigonometric function has a periodicity of 360. That is, if wee add an integer multiplier of 360 to the function’s argument; the function will still be evaluated to the same result: f f 360n For example: sin 15 sin 15 360 sin 15 720 cos22 cos22 360 cos22 720 cos22 360 3 In our case we have: tan 450 tan 90 360 tan 90 But tan(90) is not defined (it approaches infinity), therefore tan(450) is not defined. Thus the expression tan 450 tan tan 450 1 tan tan 450 Is not defined. However, this expression can be manipulated using the sine and cosine functions because they are always limited to values between -1 and 1: tan sin sin cos sin cos cos cos cos sin sin Thus: tan 450 sin cos450 sin 450 cos cos cos450 sin 450sin Using the periodicity property of trigonometric functions: sin 450 sin 90 360 sin 90 1 cos450 cos90 360 cos90 0 So: tan 450 3. sin cos450 sin 450 cos cos cot cos cos450 sin 450sin sin What is the difference between a trig equation that is an identity and a trig equation that is not an identity? Provide an example to clarify. A trigonometric identity is an expression which is true for any angle. For example, we already know that the periodicity function holds for any angle: sin x 360n sin x n ,,3,2,1,0,1,2,3 This statement is true for any x. Other examples are the expressions: sin 2 x cos 2 x 1 sin 180 x sin x sin x cos90 x Which is again, true for any x. A trigonometric equation is an expression that is true only for some specific values of x. For example cosx sin x Is true for x=45 or x=225, but obviously it is wrong for any other value (excluding the periodic values).