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Question: How do you write the expression \[\sin {140^ \circ }\cos {50^ \circ } + \cos {140^ \circ }\sin {50^0...

How do you write the expression sin⁡140∘cos⁡50∘+cos⁡140∘sin⁡500\sin {140^ \circ }\cos {50^ \circ } + \cos {140^ \circ }\sin {50^0}as the sine cosine or tangent of an angle?

Explanation

Solution

Here the question is related to the trigonometry, we use the trigonometry functions of sum and difference of two angles and we can solve this question. In this question we have to simplify the given trigonometric ratios to its simplest form. By using the trigonometry ratios and trigonometry formulas we simplify the given trigonometric function.

Complete step-by-step solution:
The question is related to trigonometry and it includes the trigonometry ratios. The trigonometry ratios are sine, cosine, tangent, cosecant, secant and cotangent. The tangent trigonometry ratio is defined as tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}} , The cosecant trigonometry ratio is defined as csc⁡x=1sin⁡x\csc x = \dfrac{1}{{\sin x}}, The secant trigonometry ratio is defined as sec⁡x=1cos⁡x\sec x = \dfrac{1}{{\cos x}} and The tangent trigonometry ratio is defined as cot⁡x=cos⁡xsin⁡x\cot x = \dfrac{{\cos x}}{{\sin x}}.
By the trigonometry functions of sum and difference of two angles formula we have a sum and difference formula for the sine trigonometry ratio also.
The trigonometry function of sum of two angles is given by
sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin (a + b) = \sin a\cos b + \cos a\sin b
The trigonometry function of difference of two angles is given by
sin⁡(a−b)=sin⁡acos⁡b−cos⁡asin⁡b\sin (a - b) = \sin a\cos b - \cos a\sin b
Now consider the given expression which is given in the question sin⁡140∘cos⁡50∘+cos⁡140∘sin⁡500\sin {140^ \circ }\cos {50^ \circ } + \cos {140^ \circ }\sin {50^0}, when we compare the given expression to the trigonometry function of sum or difference of two angles. the given expression is similar to the trigonometry function of sum of two angles.
Here the value of a is 140∘{140^ \circ }and the value of b is 50∘{50^ \circ }
Then by the trigonometry function of sum of two angles we have
⇒sin⁡(140∘+50∘)=sin⁡140∘cos⁡50∘+cos⁡140∘sin⁡50∘\Rightarrow \sin ({140^ \circ } + {50^ \circ }) = \sin {140^ \circ }\cos {50^ \circ } + \cos {140^ \circ }\sin {50^ \circ }
On simplification we have
⇒sin⁡140∘cos⁡50∘+cos⁡140∘sin⁡50∘=sin⁡(190∘)\Rightarrow \sin {140^ \circ }\cos {50^ \circ } + \cos {140^ \circ }\sin {50^ \circ } = \sin ({190^ \circ })
hence the given expression is written in the trigonometry ratio.

Note: In the trigonometry we have six trigonometry ratios and 3 trigonometry standard identities. The trigonometry ratios are sine, cosine, tangent, cosecant, secant and cotangent. We must know the trigonometry function of sum and difference of two angles. For determining the trigonometry ratio of an angle we must follow the table of trigonometry ratio for the standard angles or Clark’s table.