Exam 6: Trigonometric Functions

Solve the right triangle. If two sides are given, give angles in degrees and minutes. -In 1838 , the German mathematician and astronomer Friedrich Wilhelm Bessel was the first person to calculate the distance to a star other than the Sun. He accomplished this by first determining the parallax of the star, 61 Cygni, at $0.314$ arc seconds (Parallax is the change in position of the star measured against background stars as Earth orbits the Sun. See illustration.) If the distance from Earth to the Sun is about $150,000,000 \mathrm {~km}$ and $\theta = 0.314$ seconds $= \frac { 0.314 } { 60 }$ minutes $= \frac { 0.314 } { 60 \cdot 60 }$ degrees, determine the distance $\mathrm { d }$ from Earth to 61 Cygni using Bessel's figures. Express the answer in scienti notation.

Use the appropriate identity to find the indicated function value. Rationalize the denominator, if applicable. If the given value is a decimal, round your answer to three decimal places. - $\cos \theta$ , given that $\sec \theta = - 4$

Find the exact value of the expression. - $\cos \theta = \frac { \sqrt { 3 } } { 2 }$

Sketch an angle $\theta$ in standard position such that $\theta$ has the least positive measure and the given point is on the terminal side of $\theta$ (3, 6) - $( 0 , - 5 )$

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize the denominator if applicable. -Find $\tan \mathrm { B }$ when $\mathrm { a } = 96$ and $\mathrm { c } = 100$ .

Suppose that $\theta$ is in standard position and the given point is on the terminal side of $\theta$ . Give the exact value of the indicated trig function for $\theta$ - $( 21,28 )$ ; Find $\sin \theta$ .

Without using a calculator, give the exact trigonometric function value with rational denominator. -Find a formula for the area of the figure in terms of s.

Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length using the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. Rationalize the denominator if applicable. -Find $\csc \mathrm { A }$ when $\mathrm { b } = 32$ and $c = 68$

Solve the right triangle. If two sides are given, give angles in degrees and minutes. -A contractor needs to know the height of a building to estimate the cost of a job. From a point 96 feet away from the base of the building, the angle of elevation to the top of the building is found to be $48 ^ { \circ } 53 ^ { \prime }$ . Find the height of the building. Round your answer to the hundredths place when necessary.

The number represents an approximate measurement. State the range represented by the measurement. -24.5 m

Solve the problem. -Radio direction finders are set up at points $\mathrm { A }$ and $\mathrm { B } , 8.68 \mathrm { mi }$ apart on an east-west line. From $\mathrm { A }$ it is found that the bearing of a signal from a transmitter is $\mathrm { N } 54.3 ^ { \circ } \mathrm { E }$ , while from $\mathrm { B }$ it is $\mathrm { N } 35.7 ^ { \circ } \mathrm { W }$ . Find the distance of the transmitter from $\mathrm { B }$ , to the nearest hundredth of a mile.

Solve the problem. -A person is watching a car from the top of a building. The car is traveling on a straight road away from the building. When first noticed, the angle of depression to the car is $48 ^ { \circ } 21 ^ { \prime }$ . When the car stops, the angle of depression is $21 ^ { \circ } 43 ^ { \prime }$ . The building is 250 feet tall. How far did the car travel from when it was first noticed until it stopped? Round to the nearest foot.

Provide an appropriate response. -Find the supplement of an angle whose measure is $74 ^ { \circ } 51 ^ { \prime }$ .

An equation of the terminal side of an angle $\theta$ in standard position is given along with a restriction on x. Find the indicated trigonometric function value of $\theta$ . Do not use a calculator. - $- 5 x + y = 0 , x \geq 0 ; \quad$ Find $\sin \theta$

Solve the problem. -A person is watching a boat from the top of a lighthouse. The boat is approaching the lighthouse directly. When first noticed, the angle of depression to the boat is $15 ^ { \circ } 30 ^ { \prime }$ . When the boat stops, the angle of depression is $46 ^ { \circ } 19 ^ { \prime }$ . The lighthouse is 200 feet tall. How far did the boat travel from when it was first noticed until it stopped? Round to the nearest foot.

Solve the right triangle. If two sides are given, give angles in degrees and minutes. -A conservation officer needs to know the width of a river in order to set instruments correctly for a study of pollutants in the river. From point A, the conservation officer walks 105 feet downstream And sights point B on the opposite bank to determine that $\theta = 50 ^ { \circ }$ (see figure). How wide is the river (round to the nearest foot)?

Without using a calculator, give the exact trigonometric function value with rational denominator. -Find the exact value of $\mathrm { x }$ in the figure.

Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary. - $\tan \theta = 1.7009401$

Without using a calculator, give the exact trigonometric function value with rational denominator. - $\sin 60 ^ { \circ }$

Write the function in terms of its cofunction. Assume that any angle in which an unknown appears is an acute angle. - $\cos 46.7 ^ { \circ }$