How to find tan &#x2061;<!-- ⁡ --> ( &#x2212;<!-- − --> 5 &#x03

Kaeden Woodard

Kaeden Woodard

Answered question

2022-05-18

How to find tan ( 5 π 16 ) with half-angle formulas?

Answer & Explanation

Melina Glover

Melina Glover

Beginner2022-05-19Added 11 answers

I assume you know (or can figure out)
cos ( 5 π / 4 ) = 2 2 ,   sin ( 5 π / 4 ) = + 2 2
Applying the half-angle formulas to that, and noting that π < 5 π / 8 < π / 2
cos ( 5 π / 8 ) = 1 + cos ( 5 π / 4 ) 2 = 2 2 2 sin ( 5 π / 8 ) = 1 cos ( 5 π / 4 ) 2 = 2 + 2 2
Similarly, since π / 2 < 5 π / 16 < 0,
cos ( 5 π / 16 ) = + 1 + cos ( 5 π / 8 ) 2 = 2 2 2 2 sin ( 5 π / 16 ) = 1 cos ( 5 π / 8 ) 2 = 2 + 2 2 2
so that
tan ( 5 π / 16 ) = sin ( 5 π / 16 ) cos ( 5 π / 16 ) = 2 + 2 2 2 2 2
It turns out (but this is somewhat harder) that you can do some simplification here: you can write it as
tan ( 5 π / 16 ) = 1 2 4 2 2

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