If α = tan^-1(tan 5π/4) and β = tan^-1(-tan 2π/3), then A. 4α = 3β B. 3α = 4β C. α- β = 7π/12 D. none of these

Question

If α = tan^-1(tan \(\frac{5\pi}4\)) and β = tan^-1(-tan \(\frac{2\pi}3\)), then

A. 4α = 3β

B. 3α = 4β

C. α - β = \(\frac{7\pi}{12}\)

D. none of these

Answer 1

Correct option is A. 4α = 3β

We are given that,

[\(\because\), tan(π + \(\frac{\pi}4\)) lies in III Quadrant and tangent is positive in III Quadrant]

⇒ α = tan^-1(tan\(\frac{\pi}4\))

Using the property of inverse trigonometry, that is, tan ^-1(tan A) = A.

⇒ α = \(\frac{\pi}4\)

Now, take

[\(\because\), tan(π - \(\frac{\pi}3\)) lies in II Quadrant and tangent is negative in II Quadrant]

Using the property of inverse trigonometry, that is, tan^-1(tan A) = A.

Since, the values of 4α and 3β are same, that is,

4α = 3β = π

Therefore, 4α = 3β