Download Trig Review

Name
January 24/25, 2017
Math 4 problem set
sections 4.1-4.3 page 1
Trigonometry: further concepts and practice
Quadrants
Objective: Find angles and trigonometric values using knowledge of quadrants.
The four quadrants of the plane are numbered as follows:
•
•
Quadrant I has the angles 0 < θ < π2 or 0° < θ < 90°. It’s the upper-right quadrant.
Quadrant II has the angles π2 < θ < π or 90° < θ < 180°. It’s the upper-left quadrant.
•
•
Quadrant III has the angles π < θ < 32π or 180° < θ < 270°. It’s the lower-left quadrant.
Quadrant IV has the angles 32π < θ < 2π or 270° < θ < 360°. It’s the lower-right quadrant.
Often a key step in a problem is to determine in which quadrant(s) an angle may fall based on
given information. If the given information involves trigonometric functions, it may help to use
the x-y-r definitions of the functions and consider that r is always positive but x and y may be
either positive or negative, and each quadrant has these signs in a different combination.
1. For each of the following, based on the given information, identify the quadrant(s) in which
angle θ could possibly fall.
a. cos(θ) > 0
b. sin(θ) < 0
c. tan(θ) = 2
d. sec(θ) = –3
e. cos(θ) < 0 and tan(θ) < 0
f. sin(θ) = − 53 and cos(θ) =
4
5
g. csc(θ) = 1.3 and cot(θ) > 0
2. Suppose that tan(θ) = –2.4 =
−12
5
.
a. Identify the two quadrants in which angle θ could fall.
b. Choose one of the quadrant possibilities. Draw a diagram showing what angle θ could
look like. Then draw a reference triangle and use it to find the values of cos(θ) and sin(θ).
c. Take the other quadrant possibility and do the same as in part b.
d. How do the cos(θ) and sin(θ) values in part b compare to those in part c? Explain why
this happened.
3. Suppose that sec(θ) =
2
3
. Find two possible radian angle measures in different quadrants
for angle θ.
4. Find sin(θ) and tan(θ) given that cos(θ) =
1
3
and csc(θ) < 0.
Name
January 24/25, 2017
Math 4 problem set
sections 4.1-4.3 page 2
Practice problems
Objective: Practice using trigonometry in preparation for a first quiz tomorrow.
5. From the top of a 150-foot building, Flora observes a car moving toward her. If the angle of
depression of the car changes from 18° to 42° during the observation (see illustration above),
how far does the car travel?
6. A supermarket sells a round 20"-diameter pizza that is cut into 16 equal sectors as slices.
Try to answer the following questions without looking up any formulas (if there’s a formula
you don’t recall, find a way to work around it using something you do remember).
a. Find the central angle measure in radians for one slice of this pizza.
b. Find the perimeter and the area for one slice of this pizza.
7. Here is a trigonometric function value that is much less well-known than those for the special
angles we have studied:
sin(36°) =
5− 5
≈ 0.588.
8
Use this information to answer the following questions:
a. Find an angle in one of the other quadrants whose sine approximately equals 0.588.
b. Find an angle greater than 360° whose sine approximately equals 0.588.
c. Find two angles in different quadrants for which the sine approximately equals –0.588.
d. Without using the cos and tan keys on your calculator, find the approximate values of
cos(36°) and tan(36°). Hint: Try using a right triangle with a hypotenuse of 1.
e. Without any further calculator use, find sin(54°) and cos(54°).
f. Challenge: Find an exact value (involving
’s) for cos(36°).
Review problems from the textbook: Page 400 problems 11-33 and 39-52 review the core skills
needed for tomorrow’s quiz. These problems are very similar to past homework, so you may
decide how much of this review you personally need to do. For many of you, doing the
odd-numbered problems from this set would be an appropriate amount of practice.
Name
January 24/25, 2017
Math 4 problem set
sections 4.1-4.3 page 3
Answers
1. a. I or IV, b. III or IV, c. I or III, d. II or III, e. II, f. IV, g. I
2. a. II or IV
b/c. Quadrant II: reference triangle involves point (–5, 12); cos(θ) = –5/13, sin(θ) = 12/13.
Quadrant IV: reference triangle involves point (5, –12); cos(θ) = 5/13, sin(θ) = –12/13.
d. The cosines and sines are opposites because the x- and y-coordinates are opposites.
3. θ could be in Quadrant I or IV. Use a reference triangle with x = 3 and r = 2. It’s a
30°-60°-90° so answers are 30° and 330° in degrees, π/6 and 11π/6 in radians.
4. θ must be in Quadrant IV. From x = 1 and r = 3, get y = – 8 . sin(θ) =
− 8
3
and tan(θ) =
− 8
1
5. Let F = Flora, G = ground point below Flora, E = end, S = start.
Use right triangle FGE to get GE = 150/tan(42°) ≈ 166.59.
Use right triangle FGS to get GS = 150 tan(18°) ≈ 461.65.
Finally ES = GE – GS = 295.06. So the car traveled about 295 feet.
6. a. 2π/16 ≈ 0.393
b. perimeter = 10 + 10 +
area =
7. a. 144°
1
16
1
16
(2π·10) ≈ 23.93 inches.
2
π·10 ≈ 19.63 sq. in.
b. 396°
c. 216°, 324°
d. opp ≈ 0.588, hyp = 1; use Pythagorean Theorem to get adj = 0.809.
cos(36°) ≈ 0.809, tan(36°) ≈ 0.588/0.809 ≈ 0.727.
e. Hint: In a right triangle with a 36° angle, the other acute angle is 54°.
sin(54°) ≈ 0.809, cos(54°) ≈ 0.588
f. Use the Pythagorean Theorem but use exact expressions throughout. Answer:
3+ 5
.
8
.