4-6 Inverse Trigonometric Functions

Transcription

1 Find the exact value of each expression, if it exists 1 sin 1 0 with a y-coordinate of 0 3 arcsin When t = 0, sin t = 0 Therefore, sin 1 0 = 0 2 arcsin When t =, sin t = Therefore, arcsin = 4 sin 1 When t =, sin t = Therefore, arcsin = When t =, sin t = Therefore, sin 1 = esolutions Manual - Powered by Cognero Page 1

2 5 7 cos 1 with an x-coordinate of When t =, sin t = Therefore, sin 1 6 arccos 0 = with an x-coordinate of 0 When t =, cos t = Therefore, cos 1 = 8 arccos ( 1) with an x-coordinate of 1 When t =, cos t = 0 Therefore, arccos 0 = When t =, cos t = 1 Therefore, arccos ( 1)= esolutions Manual - Powered by Cognero Page 2

3 16 RESCUE A cruise ship sailed due west 24 miles before turning south When the cruise ship became disabled and the crew radioed for help, the rescue boat found that the fastest route covered a distance of 48 miles Find the angle θ at which the rescue boat should travel to aid the cruise ship a Write a function expressing θ in terms of distance d b Use a graphing calculator to determine the distance for the maximum projecting angle Use inverse trigonometric functions and the unit circle to solve a Make a diagram of the situation with an x-coordinate of We are asked to find θ in terms of d However, we do not know the value of the angle α either Using right triangle trigonometry, we can determine that tan α = and tan (θ + α) = Now we have two equations, but still three variables We need to find a way to eliminate α If we can get α and θ + α isolated in each equation, we can eliminate α When t =, cos t = Therefore, cos 1 = 28 SPORTS Steve and Ravi want to project a pro soccer game on the side of their apartment building They have placed a projector on a table that stands 5 feet above the ground and have hung a 12-foot-tall screen 10 feet above the ground esolutions Manual - Powered by Cognero Page 3

4 b The goal is to maximize θ, so if we graph, we can identify the maximum value of θ The d-value that corresponds with this maximum is the distance 38 sin (tan 1 1 sin 1 1) First, find tan 1 1 To do this, find a point on the unit circle on the interval [0, 2π] with an x-coordinate equal to the y-coordinate When t =, cos t = sin t = Therefore, tan 1 1 = The d-value that corresponds with this maximum is about 92 feet Next, find sin 1 1 To do this, find a point on the unit circle on the interval [0, 2π] with a y-coordinate of 1 When t =, sin t = 1 Therefore, sin 1 1 = Find the exact value of each expression, if it exists 35 cos (tan 1 1) First, find tan 1 1 The inverse property applies, because 1 is on the interval Therefore, tan 1 1 = Next, find cos On the unit circle, Find On the unit circle, corresponds to So, = corresponds to So, cos = sin (tan 1 1 sin 1 1) = cos (tan 1 1) = 55 SAND When piling sand, the angle formed between the pile and the ground remains fairly consistent and is called the angle of repose Suppose Jade creates a pile of sand at the beach that is 3 feet in diameter and 11 feet high a What is the angle of repose? b If the angle of repose remains constant, how many feet in diameter would a pile need to be to reach a height of 4 feet? a Draw a diagram to model this situation esolutions Manual - Powered by Cognero Page 4

5 Draw a diagram to model this situation Use the tangent function to find θ Therefore, the angle of repose is about 36º b Draw a diagram to model this situation, where the height of the triangle is 4 ft and angle of repose is 36º Use the tangent function to find x The pile would reach 4 feet if the diameter was about 2(55) or 11 feet esolutions Manual - Powered by Cognero Page 5