Given 12 ( 2 sin ⁡ ( α ) − cos ⁡ ( α ) ) ≤ ± 5how can I solve for the value of α in degrees?

Question

Given  12  (  2  sin  ⁡  (  α  )  −  cos  ⁡  (  α  )  )  ≤  ±  5how can I solve for the value of   α in degrees?

cokeman206 · Accepted Answer

A right triangle with legs 2 and 1 has hypotenuse

\sqrt{5}

. Let the angle opposite the leg of length 1 be t. Then

\sin t = 1 / \sqrt{5}

and

\cos t = 2 / \sqrt{5}

Factor

\sqrt{5}

out of your left-hand side to get

12 \sqrt{5} (\frac{2}{\sqrt{5}} \sin α - \frac{1}{\sqrt{5}} \cos α)

= 12 \sqrt{5} (\cos t \sin α - \sin t \cos α)

= 12 \sqrt{5} \sin (α - t)

where

t = \tan^{- 1} \frac{1}{2} .

Now it should be do-able

Iris Vaughn · Accepted Answer

HintIf you use the tangent half-angle substitution   α  =  2      tan          −      1        ⁡  (  t  ), the problem becomes  −  (  12  +  5  k  )  +  48  t  +  (  12  −  5  k  )      t    2    =  0    with    k  =  ±  1Solve each quadratic and select the proper roots to obtain the value of   tan  ⁡      (          α      2        )

Solving 12(2 sin alpha−cos alpha) <= +- 5 for alpha

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