Exact values & multiple quadrant answers

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

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