Question

find the exact value of cos open parentheses fraction numerator 7 straight pi over denominator 6 end fraction close parentheses by using the unit circle.

Answer 1

To find the exact value of cos(7π/6) using the unit circle, divide 7π/6 by the period 2π to get 7/12. Locate this angle in the unit circle and find its x-coordinate, which is -√3/2.

Step-by-step explanation:

To find the exact value of cos(7π/6) using the unit circle, we need to determine which angle in the unit circle corresponds to 7π/6. To do this, we divide the angle 7π/6 by the period of 2π, which gives us 7π/6 ÷ 2π = 7/6 ÷ 2/1 = 7/12. Since the cosine function represents the x-coordinate of a point on the unit circle, we need to find the x-coordinate of the angle 7π/12 on the unit circle.

To do this, we draw a unit circle and divide it into 12 equal parts (each representing π/6). Starting from the positive x-axis, we move counterclockwise to the 7/12 position, which is in the third quadrant. The x-coordinate of this position is -√3/2.

Therefore, the exact value of cos(7π/6) is -√3/2.

Answer 2

The exact value of cos(7π/6) is negative since it is in the third quadrant of the unit circle, and the reference angle of π/6 gives a cosine value of √3/2. Accordingly, the exact value is -√3/2.

Step-by-step explanation:

To find the exact value of cos(7π/6) using the unit circle, we need to consider that 7π/6 is an angle located in the third quadrant where the cosine values are negative. The reference angle for 7π/6 is π/6, which corresponds to 30°. Since the cosine of 30° is √3/2, and considering that in the third quadrant cosine values are negative, the exact value for cos(7π/6) will be -√3/2.

To ensure the answer is reasonable, we check if the sign and magnitude align with the properties of the unit circle. Indeed, in the third quadrant, all cosine values are negative, and √3/2 is the expected magnitude for a 30° reference angle. Thus, the answer of -√3/2 does make sense.