Question

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

A. The angles do not have the same reference angle.
B. Tangent is positive in the second quadrant and negative in the fourth quadrant.
C. Tangent is negative in the second quadrant and positive in the fourth quadrant.
D. The angles do not have the same reference angle or the same sign.

Answer 1

The correct option is A.

Step-by-step explanation:

The first expression is,

`\tan ((5\pi)/(6))`

`\tan ((6\pi-\pi)/(6))`

`\tan (\pi-(\pi)/(6))`

The reference angle is
`(\pi)/(6)` . In second quadrant the sign of tangent is negative.

`-\tan ((\pi)/(6))`

`-(1)/(√(3))`

The second expression is,

`\tan ((5\pi)/(3))`

`\tan ((6\pi-\pi)/(3))`

`\tan (2\pi-(\pi)/(3))`

The reference angle is
`(\pi)/(3)` . In fourth quadrant the sign of tangent is negative.

`-\tan ((\pi)/(3))`

`-√(3)`

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

Answer 2

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

Step-by-step 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.