Download Math 112: Review for Chapter 6 Exam

Math 112: Review for Chapter 6 Exam
You should know:
• The commonly used angles (in degrees and radians) and their corresponding coordinates on the unit circle
• The definitions for the sine, cosine, tangent, secant, cosecant, and cotangent functions.
• The Pythagorean Identities involving the trig functions above
You should be able to:
• Identify the period, amplitude, midline, horizontal and vertical phase shifts of a periodic function from its graph or equation
• Find a formula for a sinusoidal function based on its graph
• Convert an angle from radian to degree measure or from degree to radian measure
• Convert an angle from decimal angle form to DMS form and back
• Solve applications involving arc length along a circle given the angle and radius of the circle.
• Solve trigonometric equations using inverse trigonometric functions
• Evaluate sin(θ), cos(θ), tan(θ), sec(θ), csc(θ), cot(θ) given either the angle, θ, or the coordinates of a point on the unit circle
Here are some practice problems for the exam.
1. a) If θ = 37˚, what is the measure of θ in radians? (Round to 4 decimal places)
b) The minute hand on a watch is 0.8 inches long. How far does the tip of the minute hand travel as the hand turns through 37˚?
c) How fast (in inches/minute) is the tip of the minute hand moving in part b?
2. Convert the angle to an exact radian measure:
a) 245˚
b) 720˚
c) 240˚
3. Convert the angle to a degree measure (round to the 2 decimal places when needed): a)
4. a) Convert the angle to decimal form: 50˚ 26’ 32”
3π
4
b) 5
c)
11π
5
b) Convert the angle to DMS form: 23.6541˚
5. Find the value of θ in degrees (0˚ < θ < 360˚) and radians (0 < θ < 2π) in exact form:
a)
6. Given
cos(θ ) =
1
2
b)
tan(θ ) = 3
c)
csc(θ ) = 2
f (t ) = −4 cos(2t + π4 ) − 6 and g (t ) = 3 sin(π t + π4 ) + 2
a) For each function state the period, amplitude, midline, horizontal and vertical shifts, reflections, and vertical stretches or
compressions.
b) Sketch the graph with the information above.
7. Suppose you are on a Ferris wheel (that turns in a counter-clockwise direction) and that your height in meters, above the ground at
time, t, in minutes is given by h(t ) = 15 sin( π2 t ) + 15 .
a)
b)
c)
d)
e)
Sketch the graph for one full revolution.
How high above the ground are you at time t = 0?
What is your clock position on the wheel at t = 0?
What is the radius of the wheel?
How long does one revolution take?
8. For each graph below, identify the midline, amplitude, period, vertical and horizontal shifts, and phase shift. Then find a sine and
cosine formula for each function.
9. The position (in inches), S, of a piston in a 6-inch stroke in an engine is given as a function of time, t, in seconds by the formula
S = 3 sin(200π t ) .
a) What is the period of the function?
b) Sketch a graph of one period of this function.
c) Why is this engine said to have a 6-inch stroke?
d) How many revolutions per minute (rmp) is this engine performing?
10. A caribou population dropped from a high of 200,000 in 1943 to a low of 76,000 in 1989, and has risen since then. Scientists
hypothesize that this population may follow a sinusoidal cycle affected by predation and other environmental conditions, and that
the population will again reach its previous high.
a) Let t be the number of years since 1943. Create a sinusoidal formula to describe the caribou population as a function of time.
b) Sketch the graph.
c) When does your model predict that the caribou population will next reach 200,000 again?
d) Use your model to approximate when (from 1943 to 2050) the population is 175,000?
11. Sketch θ in standard position and find the reference angle, θ’.
a) θ= 2.4
b) θ = 895˚
c) θ = 5π/6
12. If sin(θ) = -3/5 and θ is in the fourth quadrant, find the exact values for cos(θ), tan(θ), sec(θ), and cot(θ).
13. What quadrant is θ in if csc(θ) < 0 and tan(θ) > 0?
14. Solve symbolically for all solutions in the interval, 0 ≤ x ≤ 2π, of the following equation. (Show your work)
a)
2 cos( x) = 1
b)
sin(2 x) + 3 = 4
c)
2 tan( x) = sec 2 ( x)
d)
2 cos 2 ( x) + cos( x) = 1
15. Solve symbolically for all solutions in the interval, 0˚ ≤ x ≤ 360˚, of the following equations. (Show your work)
a)
16. Is
4 tan( x) = −1
b)
3 sin 2 ( x) + 4 = 5
sin −1 (sin( x)) = x ? Why or why not? (Hint: Calculate sin −1 (sin(1)) and sin −1 (sin(4)) . What is happening and why?)
17. The Big Ben in London is 320 feet high. How long will the new London Ferris Wheel rider be above Big Ben during a ride of
π
one revolution if the height of the rider on the Wheel is h = 250 − 250 cos( 10
t ) after t minutes?
Give your answer in three ways:
a) Approximated to the nearest minute
b) Approximated to the nearest second
c) As an exact expression
18. Use the figure below on the left to find sin θ , cosθ , cos φ , tan φ , and tan θ . (Keep values exact.)
19. Use the figure above on the right to find x, Α, and B. (Round to the tenth of a degree if needed.)
20. A ladder 10 feet tall leans against a house, making an angle of 75 degrees to the ground. How high up the wall of the house does
the ladder reach? (Round to the tenth of a foot if needed)