Test 1 September 15, 1998
SOLUTIONS
1. Find the values of the other five trig functions of θ exactly if 0o < θ < 90o and csc θ = 5/2.
cos θ =
/5
sin θ = 2/5
tan θ = 2/
csc θ = 5/2
sec θ = 5/
cot θ =
/2
2. Use a calculator to find the missing parts of the following right triangle accurate to the thousandths place.
angle B = 90° - 22.3° = 67.7°
side a = 7 sin 22.3° = 2.656
side b = 7cos22.3° = 6.476
3. Which quadrant does the terminal side of the following angles lie in?
a. -936o __II____
-936°/360° = -2.6
b. 9R __II___
9/(2p ) = 1.43
4. Fill in the missing parts for each of the following circular sectors accurate to the thousandths place.
10/r =.8 so r = 10/.8 =12.5 cm
S = 70° (π/180°)(8cm) = 9.774 cm
5. Find the exact values of the following trigonometry functions:
cos 5π = -1
sin (5π/3) = -
/2
tan (-5π/6) = 1/
cot(3π/4) = -1
6. Convert 7° 18′ 20″ to decimal degrees accurate to the ten-thousandths place.
7° + 18°/60 + 20°/3600 = 7.3056°
7. From a point 70 yards from the base of a tower, the angle of elevation is 40° . Find the height of the tower.
h = 70tan 40° = 58.74 yds
8. If a roulette wheel with a diameter of 3 feet is spinning at a rate of 18 rev./min, what is its angular speed in
radians per minute accurate to the hundredths place, and what is the speed of a fixed point on its edge in inches per
minute accurate to the hundredths place. (Recall: there are 12 inches per foot.)
angular speed = (18 rev/min)(2πR/rev) = 113.10 R/min
speed of a fixed point = (18 rev/min)(2π (18)in/rev) = 2035.75in
9. Given that sin α= 0.3, cos α = 0.9539 find:
sin(90° - α) = .9539
cos(360° - α) = .9539
sin(180° + α) = -.3
10. Find sin α and cos α if the terminal side of α is on the line 12x +35y = 0 in quadrant IV and α is in standard
position.
y = -12x/35
sin α = -12/37
cos α = 35/37
11. Suppose csc θ = -3 in quadrant IV. Find:
cos θ = 2
/3
sin θ = -1/3
tan θ = -1/2
12 Find all values of x in [0, 2π] such that sin x < 0.7 . Put your answer in interval notation accurate to the
thousandths place.
I used the Thinking on the Unit Circles Program to create this image, which could also be done by hand.
Notice that I used a horizontal line at y=0.7 since we want all the angle values that are associated with points
below that line. Of course the program doesn't give the level of accuracy desired, so you still need to use your
calculator to evaluate the answer: [0,sin-1(0.7)) ∪ (2π-sin-1(0.7), 2π] ≈ [0, 0.775) ∪ (2.366, 2π]
13. Let cot α = 5.47 where π < α < 3π/2. Evaluate α.
tan α = 1/5.47, so we need to evaluate tan-1 (1/5.47). You will notice that the angle you get is in
quadrant I instead of quadrant III so we will need to add π in order to get the correct answer. Thus α =
tan-1(1/5.47) + π ≈ 3.322