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Question: How do you evaluate \(\sin \left( \dfrac{\pi }{12} \right).\cos \left( \dfrac{3\pi }{4} \right)-\cos...

How do you evaluate sin⁡(π12).cos⁡(3π4)−cos⁡(π12).sin⁡(3π4)\sin \left( \dfrac{\pi }{12} \right).\cos \left( \dfrac{3\pi }{4} \right)-\cos \left( \dfrac{\pi }{12} \right).\sin \left( \dfrac{3\pi }{4} \right) ?
(a) Using trigonometric angle identities
(b) Using linear formulas
(c) a and b both
(d) none of the above

Explanation

Solution

In the given problem we are to find the value of sin⁡(π12).cos⁡(3π4)−cos⁡(π12).sin⁡(3π4)\sin \left( \dfrac{\pi }{12} \right).\cos \left( \dfrac{3\pi }{4} \right)-\cos \left( \dfrac{\pi }{12} \right).\sin \left( \dfrac{3\pi }{4} \right). To find that we are going to use a trigonometric angle identity known as sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A-B)=\sin A\cos B-\cos A\sin B. Thus we can decrease the given term in a single term of sin function. Then using the quadrant rule and table of trigonometric values we get our desired value.

Complete step by step solution:
According to the question, we are to evaluate the value of sin⁡(π12).cos⁡(3π4)−cos⁡(π12).sin⁡(3π4)\sin \left( \dfrac{\pi }{12} \right).\cos \left( \dfrac{3\pi }{4} \right)-\cos \left( \dfrac{\pi }{12} \right).\sin \left( \dfrac{3\pi }{4} \right)in this given problem.
We will start with, given from the trigonometric identities, sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A-B)=\sin A\cos B-\cos A\sin B ,
So clearly, here A=π12A=\dfrac{\pi }{12} and B=3π4B=\dfrac{3\pi }{4} ,
Then we can write,
sin⁡(π12).cos⁡(3π4)−cos⁡(π12).sin⁡(3π4)=sin⁡(π12−3π4)\sin \left( \dfrac{\pi }{12} \right).\cos \left( \dfrac{3\pi }{4} \right)-\cos \left( \dfrac{\pi }{12} \right).\sin \left( \dfrac{3\pi }{4} \right)=\sin \left( \dfrac{\pi }{12}-\dfrac{3\pi }{4} \right)
Trigonometric Identities are useful whenever trigonometric functions are involved in an expression or an equation. Identity inequalities which are true for every value occurring on both sides of an equation. Geometrically, these identities involve certain functions of one or more angles. There are various distinct identities involving the side length as well as the angle of a triangle. The trigonometric identities hold true only for the right-angle triangle.
Now simplifying,
sin⁡(π12−3π4)\sin \left( \dfrac{\pi }{12}-\dfrac{3\pi }{4} \right)
Again, 3π4\dfrac{3\pi }{4} can be written as 9π12\dfrac{9\pi }{12} ,
⇒sin⁡(π12−9π12)\Rightarrow \sin \left( \dfrac{\pi }{12}-\dfrac{9\pi }{12} \right)
Simplifying, we get,
⇒sin⁡(−8π12)\Rightarrow \sin \left( -\dfrac{8\pi }{12} \right)
Dividing denominator and numerator by 4, we are getting,
⇒sin⁡(−2π3)\Rightarrow \sin \left( -\dfrac{2\pi }{3} \right)
Again, we are familiar with the fact that, with the help of trigonometric identities, sin⁡(−x)=−sin⁡x\sin \left( -x \right)=-\sin x ,
So, we can write,
sin⁡(−2π3)=−sin⁡2π3\sin \left( -\dfrac{2\pi }{3} \right)=-\sin \dfrac{2\pi }{3}
Putting the value of sin⁡2π3\sin \dfrac{2\pi }{3}from the trigonometric table, we have,
⇒−32\Rightarrow -\dfrac{\sqrt{3}}{2}
Thus we can conclude that, sin⁡(π12).cos⁡(3π4)−cos⁡(π12).sin⁡(3π4)=−32\sin \left( \dfrac{\pi }{12} \right).\cos \left( \dfrac{3\pi }{4} \right)-\cos \left( \dfrac{\pi }{12} \right).\sin \left( \dfrac{3\pi }{4} \right)=-\dfrac{\sqrt{3}}{2}

So, the correct answer is “Option a”.

Note: The sin of angle difference identity is a trigonometric identity. It’s used to expand the sin of subtraction of two angles functions such as sin⁡(A−B)\sin \left( A-B \right) and so on. You know how to expand the sin of difference of two angles and it’s essential to learn how it is derived in mathematical form in trigonometry. The sin of difference of two angles formula is derived in trigonometry by the geometrical approach. Geometrically, a right triangle is constructed and it contains a difference of two angles. Then, the angle difference identity is derived in mathematical form on the basis of this triangle. This formula can be also proved with the help of the Euler’s theorem.