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Which of the following best explains why tan 5pi/6 does not equal to tan 5pi/3?

AThe angles do not have the same reference angle.
BTangent is positive in the second quadrant and negative in the fourth quadrant.
CTangent is negative in the second quadrant and positive in the fourth quadrant.
DThe angles do not have the same reference angle or the same sign.

Verified Answer :

Edufirst

Option A. The angles do not have the same reference angle.

Explanation:

1) Angle 5π / 3 radians:

  • Convert radians to degrees: 5π/3 × 180° / π = 300°
  • 300° is in the fourth quadrant
  • The reference angle for angles in the fourth quadrant is 360° - angle ⇒ 360° - 300° = 60°.
  • ∴ The reference angle for this angle is 60°.

2) Angle 5π / 6 radians:

  • Convert radians to degrees: 5π/6 × 180° / π = 150°
  • 150° is in the second quadrant
  • The reference angle for angles in the second quadrant is 180° - angle ⇒ 180° - 150° = 30°.
  • ∴ The reference angle for this angle is 30°.

3) Conclusion:

  • Since the reference angles are different, the tangent ratios have different values.
  • tan (5π/3) = - tan(60°) = - √3
  • tan (5π/6) = - tan(30°) = - (√3)/3

Note that the tangent is negative in both second and fourth quadrants.

Answer:

The correct option is A.

Step-by-step explanation:

The first expression is,

\(\tan (\frac{5\pi}{6})\)

\(\tan (\frac{6\pi-\pi}{6})\)

\(\tan (\pi-\frac{\pi}{6})\)

The reference angle is \(\frac{\pi}{6}\) . In second quadrant the sign of tangent is negative.

\(-\tan (\frac{\pi}{6})\)

\(-\frac{1}{\sqrt{3}}\)

The second expression is,

\(\tan (\frac{5\pi}{3})\)

\(\tan (\frac{6\pi-\pi}{3})\)

\(\tan (2\pi-\frac{\pi}{3})\)

The reference angle is \(\frac{\pi}{3}\) . In fourth quadrant the sign of tangent is negative.

\(-\tan (\frac{\pi}{3})\)

\(-\sqrt{3}\)

Since the sign are same but the reference angle is are different, therefore the correct option is A.

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