find the value of tan π/8
Answers
→ tan π/8
→ let π = 180°
→ tan 180° / 8
→ tan 45° / 2
[ Use tan 2x formula ]
→ tan 2x = 2 tan x / 1 - tan² x
Put x = 45° / 2
tan ( 2 × 45°/2 ) = 2 tan 45°/2 / 1 - tan² 45°/2
tan 45° = 2 tan 45°/2 / 1 - tan² 45°/2
tan 45° = 2 tan 45°/2 / 1 - tan²45°/2
1 = 2 tab 45°/2 / 1 - tan² 45°/2 [ tan 45° = 1 ]
1 - tan² 45°/2 = 2 tan 45°/2
let tan 45°/2 = x
1 - x² = 2x
0 = 2x + x² - 1
x² + 2x - 1 = 0
The above equation is in the form of
ax² + bx + c = 0
where
a = 1
b = 2
c = -1
x = -b ± √b² - 4ac / 2a
x = -2 ± √ (-2)²-4.1.(-1) / 2.1
x = -2 ± √4 + 4 / 2
x = -2 ± √8 / 2
x = -2 ± √ 2.2.2. / 2
x = -2 ± 2√2 / 2
x = 2 ( -1 ± √2 ) / 2
x = -1 ± √2
tan 45° / 2 = -1 - √2 is not possible because
45° / 2 = 22.5° lies in first quadrant and
we know that tan value is positive
tan 45° / 2 = -1 ± √2
:. tan π/8 = √2 - 1
π=22/7
A.T.Q.
10×22/7/8
=3.92(approx)