Q : Identify the x-values that are solutions of each multiple choice equation:
1. 8cos x +4 = 0
A. x= 2π/3 , 4π/3
B. x=4π/3, 5π/3
C. x = π/3 , 5π/3
D. x =π/3 , 4π/3
2. 10cos x – 5 √(3) = 0
A. x =5π/6 , 7π/6
B. x = π/6 , 11π/6
C. x = 7π/6 , 11π/6
D. x = π/6 , 7π/6
3. 18cos x – 9 = 0
A. x = 4π/3 , 5π/3
B . x = 2π/3 , 4π/3
C . x = π/3 , 5π/3
D. x = π/3 , 4π/3
Answer:
In each of these problems, we will end up refering back to our exact values chart to find the values for x… So, be prepared!
1. 8cos x +4 = 0
Solve:
8cos x = -4
cos x = -4/8
cos x = -1/2
OK… First we ask ourselves: where does cos x = 1/2 (look at the chart).
The answer is when x = 60°.
However, what we really want to know is where cos x = -1/2 not +1/2…. So, we ask ourselves:
In what quadrants in cos negative? Quadrant II and Quadrant III
So, what angles in Q II and Q III have reference angles of 60°?
Q II angle: 180° – 60° = 120°
Q III angle: 180° + 60° = 240°
Our answers are x = 120°, 240° (convert this to radians): x = 2π/3, 4π/3
Correct Answer: A
2. 10cos x – 5 √(3) = 0
Start solving:
10cos x = 5 √(3)
cos x = 5 √(3) / 10
cos x = √(3) / 2
Where does cos x = √(3) / 2 ? At 30°. And, in what other quadrant is cos positive? Quadrant IV.
So, what angle in Q IV has a reference angle of 30°? 360° – 30° = 330°.
x = 30° or 330°… convert to radians… x = π/6 or 11π/6
Correct Answer: B
3. 18cos x – 9 = 0
Start solving:
18cos x = 9
cos x = 9/18
cos x = 1/2
Very simlar to problem (1)! Where does cos x = 1/2? 60°…. And, it what other quadrant is cos positive? Quadrant IV.
What angle in Q IV has a reference angle of 60°? 360° – 60° = 300°.
So, x = 60°, 300°… convert to radians… x = π/3 , 5π/3
Correct Answer: C