Exam 6: The Trigonometric Functions

Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature is 19 , and the low temperature of -19 occurs at 4 A.M.

A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

Find the exact value.

An airplane flying at a speed of 300 mi/hr flies from a point A in the direction for 30 minutes and then flies in the direction for 15 minutes. Approximate, to the nearest mile, the distance from the airplane to A.

Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points.

Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , .

Find the period of the equation.

Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , .

A ladder 24 feet long leans against the side of a building, and the angle between the ladder and the building is . A. Approximate the distance from the bottom of the ladder to the building. Give the answer to one decimal place. B. If the distance from the bottom of the ladder to the building is increased by 2 feet, approximately how far does the top of the ladder move down the building? Give the answer to one decimal place.

Find the exact value.

A forester, 180 feet from the base of a redwood tree, observes that the angle between the ground and the top of the tree is . Estimate the height of the tree.

Find the exact values of the six trigonometric functions of the angle, whenever possible.

The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R ₀ is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 ^o when the sun is in the east and = 0 ^o when the sun is in the south. What percentage of R ₀ is striking the wall when is equal to 60 ^o and the sun is in the southeast? Round the answer to the nearest integer.

A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.

Find the exact value.

The thickness of the ozone layer can be estimated using the formula where I ₀ is the intensity of a particular wavelength of light from the sun before it reaches the atmosphere, I is the intensity of the same wavelength after passing through a layer of ozone x centimeters thick, k is the absorption constant of ozone for that wavelength, and is the acute angle that the sunlight makes with the vertical. Suppose that for a wavelength of 3,055 10 ^{- 8} centimeter with k 1.93, I _o /I is measured as 1.72 and . Approximate the thickness of the ozone layer to the nearest 0.01 centimeter.

Find the exact value.

A ship leaves port at 1:00 P.M. and sails in the direction at a rate of 20 mi/hr. Another ship leaves port at 1:30 P.M. and sails in the direction at a rate of 18 mi/hr. Approximately how far apart are the ships at 4:00 P.M.?

Find the period of the equation.

Estimate the horizontal asymptote.