Question

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

Answer 1

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.

Answer 2

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.