Expert Answer to "How to solve 12sin⁡50∘+2sin⁡10∘"

High School
Geometry
Trigonometry
Trigonometric equations and identities

zagonek34

Answered question

2021-12-31

How to solve

\frac{1}{2 {\sin 50}^{\circ}} + 2 {\sin 10}^{\circ}

Answer & Explanation

Paul Mitchell

Beginner2022-01-01Added 40 answers

\frac{1}{2 {\sin 50}^{\circ}} + 2 {\sin 10}^{\circ} = \frac{1 + 4 {\sin 50}^{\circ} {\sin 10}^{\circ}}{2 {\sin 50}^{\circ}}

= \frac{1 + 2 ({\cos 40}^{\circ} - {\cos 60}^{\circ})}{2 {\cos 40}^{\circ}}

David Clayton

Beginner2022-01-02Added 36 answers

\frac{1}{2 {\sin 50}^{\circ}} + 2 {\sin 10}^{\circ} = \frac{1}{2 {\sin 50}^{\circ}} + 2 \sin (60^{\circ} - 50^{\circ}) = \frac{1}{2 {\sin 50}^{\circ}} + \sqrt{3} {\cos 50}^{\circ} - {\sin 50}^{\circ} =

= \frac{1 - 2 \sin^{2} 50^{\circ}}{2 {\sin 50}^{\circ}} + \frac{2 \sqrt{3} {\cos 50}^{\circ} {\sin 50}^{\circ}}{2 {\sin 50}^{\circ}} = \frac{{\cos 100}^{\circ}}{2 {\sin 50}^{\circ}} + \frac{\sqrt{3} {\sin 100}^{\circ}}{2 {\sin 50}^{\circ}} =

= \frac{{\frac{12}{\cos 100}}^{\circ} + \frac{\sqrt{3}}{2} {\sin 100}^{\circ}}{{\sin 50}^{\circ}} = \frac{\sin (30^{\circ} + 100^{\circ})}{{\sin 50}^{\circ}} = 1

karton

Expert2022-01-08Added 613 answers

Always remember : "never simplify denominator unless you know the values."
Onto the answer :
just cross multiply to get
$\frac{1 + 4 \sin 50 \sin 10}{2 \sin 50}$
which would further simplify as
$\frac{1 + 2 (\cos 40 - \cos 60)}{2 \sin 50}$
which is easy to solve as we get
$\frac{\cos 40}{\sin 50} = 1$
so the answer is 1

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