sin ⁡ x + cos ⁡ x = 1 2 What is the value of tan ⁡ x? I tried using sin ⁡ 2 x = 2 tan ⁡ x 1 + tan 2 ⁡ x and cos ⁡ 2 x = 1 − tan 2 ⁡ x 1 + tan 2 ⁡ x but we get a quadratic for tan ⁡ ( x 2 ) . So any better approach would be much appreciated.

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sin  ⁡  x  +  cos  ⁡  x  =      1    2  What is the value of   tan  ⁡  x? I tried using  sin  ⁡  2  x  =            2      tan      ⁡      x              1      +              tan        2            ⁡      x      and  cos  ⁡  2  x  =            1      −              tan        2            ⁡      x              1      +              tan        2            ⁡      x      but we get a quadratic for   tan  ⁡      (          x      2        )   . So any better approach would be much appreciated.

Braxton Gallagher · Accepted Answer

Given  sin  ⁡  x  +  cos  ⁡  x  =      1    2  squaring both the sides,  (  sin  ⁡  x  +  cos  ⁡  x      )    2    =            (              1        2            )        2        sin    2    ⁡  x  +      cos    2    ⁡  x  +  2  sin  ⁡  x  cos  ⁡  x  =      1    4    1  +  sin  ⁡  2  x  =      1    4    sin  ⁡  2  x  =  −      3    4              2      tan      ⁡      x              1      +              tan        2            ⁡      x        =  −      3    4    3      tan    2    ⁡  x  +  8  tan  ⁡  x  +  3  =  0let   tan  ⁡  x  =  t  3      t    2    +  8  t  +  3  =  0I hope you can take it from here.

Annabel Sullivan · Accepted Answer

HINT....Write the LHS as      2    sin  ⁡  (  x  +      π    4    )You can find x then find   tan  ⁡  xUsually the best way to proceed when you have an equation of the form  a  sin  ⁡  x  +  b  cos  ⁡  x  =  cis to rewrite the LHS as  R  cos  ⁡  (  x  +  α  )or  R  sin  ⁡  (  x  +  α  )This is known as a compound angle transformation.

sin ⁡ x + cos ⁡ x = 1 2

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