Question

Find the value:
Sin5π/12
Tan7π/12.

Answer 1

we have to find the value of sin(5π/12) and tan(7π/12)

in degree, 5π/12 => 75° and 7π/12 = 105°

means, we have to find sin75° and tan105°

sin(75°) = sin(30° + 45°)

use formula, sin(A + B) = sinA.cosB + cosA.sinB

then, sin(30° + 45°) = sin30°.cos45° + cos30°.sin45°

= 1/2 × 1/√2 + √3/2 × 1/√2

= (√3 + 1)/2√2

hence, sin(75°) = sin(5π/12) = (√3 + 1)/2√2

again, tan(7π/12) = tan(105°)

= tan(180° - 75°)

= -tan(75°)

= - tan(30° + 45°)

use formula, tan(A + B) = (tanA + tanB)/(1 - tanA.tanB)

so, -tan(30° + 45°) = -(tan30° + tan45°)/(1 - tan30° . tan45°)

= -(1/√3 + 1)/(1 - 1/√3 × 1)

= -(1 + √3)/(√3 - 1)

= -(1 + √3)²/(√3² - 1²)

= -(1 + 3 + 2√3)/2

= -(2 + √3)

hence, tan(105°) = tan(7π/12) = -(2 + √3)