Solve trigonometric inequality sin ⁡ x + 2 cos ⁡ x &lt; 2 2 t 1 + t 2 + 2 1 − t 2 1 + t 2 &lt; 2 4 t 2 − 2 t &gt; 0 2 t ( 2 t − 1 ) &gt; 0 t ( 2 t − 1 ) &gt; 0 ( t &gt; 0 ∧ t &gt; 1 2 ) ∨ ( t &lt; 0 ∧ t &lt; 1 2 )From this, I can only find x &lt; 2 π + 2 k π, and, x &lt; 2 k π, these are good (I think), but I should find another two solutions.

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Solve trigonometric inequality   sin  ⁡  x  +  2  cos  ⁡  x  &amp;lt;  2                    2        t                    1        +                  t          2                      +  2                    1        −                  t          2                            1        +                  t          2                      &amp;lt;  2  4      t    2    −  2  t  &amp;gt;  0  2  t  (  2  t  −  1  )  &amp;gt;  0  t  (  2  t  −  1  )  &amp;gt;  0  (  t  &amp;gt;  0  ∧  t  &amp;gt;            1      2        )  ∨  (  t  &amp;lt;  0  ∧  t  &amp;lt;            1      2        )From this, I can only find   x  &amp;lt;  2  π  +  2  k  π, and,   x  &amp;lt;  2  k  π, these are good (I think), but I should find another two solutions.

Morisio64moc · Accepted Answer

You have used the substitution:  t  =  tan  ⁡      (          x      2        )  and the solution of the inequality is   t  &amp;lt;  0  ∨  t  &amp;gt;            1      2      so you have:  tan  ⁡      (          x      2        )    &amp;lt;  0    ∨    tan  ⁡      (          x      2        )    &amp;gt;            1      2      the solution of the first inequality is:  −            π      2        +  k  π  &amp;lt;            x      2        &amp;lt;  0  +  k  π    ⟺    (  2  k  −  1  )  π  &amp;lt;  x  &amp;lt;  2  k  πand the solution of the second inequality is:      tan          −      1        ⁡      (                  1        2              )    +  k  π  &amp;lt;            x      2        &amp;lt;            π      2        +  k  π    ⟺    2      tan          −      1        ⁡      (                  1        2              )    +  2  k  π  &amp;lt;  x  &amp;lt;  π  +  2  k  π

Kade Ibarra · Accepted Answer

sin  ⁡  (  x  +      cos          −      1        ⁡            1              5              )  =            2              5              ,    o  r  ,  sin  ⁡  (  x  +      tan          −      1        ⁡  2  )  =      2          5        .which you can further.

Solve trigonometric inequality sin ⁡ x + 2 cos ⁡

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