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Question: How do you rewrite \[\sin 37\cos 22+\cos 37\sin 22\] as a function of a single angle and then evalua...

How do you rewrite sin⁡37cos⁡22+cos⁡37sin⁡22\sin 37\cos 22+\cos 37\sin 22 as a function of a single angle and then evaluate ?

Explanation

Solution

In the given question, we have been asked to write an equation as a function of a single angle. In order to solve the question, we need to apply the trigonometric formula of sum and difference of angles. The formula we will use in the given question issin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B. Here in this formula we need to put the value of ‘A’ equals to 37 and the value of ‘B’ equals to 22. Then we will find the sum of the angles and further evaluate.

Formula used:
The formulae of sum and difference of angles for sin ratio, i.e.
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B

Complete step by step answer:
We have given that, sin⁡37cos⁡22+cos⁡37sin⁡22\sin 37\cos 22+\cos 37\sin 22.
Using the formulae of sum and difference of angles for sin ratio, i.e.
sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B
Applying this formula for the given question, we obtain
Here,
A = 37 and B = 22
Therefore,
sin⁡37cos⁡22+cos⁡37sin⁡22=sin⁡(37+22)\sin 37\cos 22+\cos 37\sin 22=\sin \left( 37+22 \right)
Now, we get
sin⁡(59)0\sin {{\left( 59 \right)}^{0}}
Using the trigonometric identity, i.e. sin⁡a=cos⁡(90−a)\sin a=\cos \left( 90-a \right)
Here, putting the value of ‘a’ equals to 59. So,
sin⁡(59)0=cos⁡(90−59)=cos⁡(31)0\sin {{\left( 59 \right)}^{0}}=\cos \left( 90-59 \right)=\cos {{\left( 31 \right)}^{0}}
⇒sin⁡37cos⁡22+cos⁡37sin⁡22 =sin⁡590=cos⁡310\Rightarrow \sin 37\cos 22+\cos 37\sin 22~=\sin {{59}^{0}}=\cos {{31}^{0}}
To evaluate the sin⁡(59)0\sin {{\left( 59 \right)}^{0}}, we need to use the calculator. Thus,
∴sin⁡(59)0=0.8571\therefore \sin {{\left( 59 \right)}^{0}}=0.8571

Therefore, the value of sin⁡37cos⁡22+cos⁡37sin⁡22\sin 37\cos 22+\cos 37\sin 22 is sin⁡(59)0\sin {{\left( 59 \right)}^{0}} or cos⁡(31)0\cos {{\left( 31 \right)}^{0}}.

Note: While solving these types of questions, students need to know the basic concepts of trigonometry. In order to solve the above given question, we can convert the sin ratio into cos ratio and make the equation in such a way that we can apply formulae of sum and differences of cos angle to find the value of the answer. In both ways we will get the same answer.