Question

If \[sin\left( {A - B} \right) = sinAcosB-cosAsinB\] and \[\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B\]. Find the values of $\sin {15^ \circ }$ and $\cos {15^ \circ }$.

Answer 1

Hint: Start by assuming two trigonometric angles whose difference would be 15 and we already know the corresponding values of their cos and sin . Substitute all these values into the given formula and find the required value.

Complete step-by-step answer:
Given,
$sin\left( {A - B} \right) = sinAcosB-cosAsinB\] and \[\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B$.
Let the angle $A = {45^ \circ }$ and $B = {30^ \circ }$
Now putting these angles in the equation, we get
$sin\left( {A - B} \right) = sinAcosB—cosAsinB$
$ \Rightarrow sin\left( {{{45}^ \circ } - {{30}^ \circ }} \right) = sin{45^ \circ }cos{30^ \circ }-cos{45^ \circ }sin{30^ \circ }$
$\because \sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\;and\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$,$\cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\;and\sin {30^ \circ } = \dfrac{1}{2}$
Substituting these values , we get
$\Rightarrow \sin {15^ \circ } = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2} - \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{2}$
On further simplification , we get
$\sin {15^ \circ } = \dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}$
Similarly , we will try to find $\cos {15^ \circ }$.
So ,Now putting the same angles in the equation \[\cos \left( {A - B} \right) = \cos A\cos B + \sin A\sin B\] , we get
$\cos \left( {{{45}^ \circ } - {{30}^ \circ }} \right) = \cos {45^ \circ }cos{30^ \circ } + \sin {45^ \circ }sin{30^ \circ }$
$\because \sin {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\;and\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$,$\cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\;and\sin {30^ \circ } = \dfrac{1}{2}$
Substituting these values, we get
$ \Rightarrow \cos {15^ \circ } = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{2}$
On further simplification, we get
$\cos {15^ \circ } = \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$
Therefore, The values of $\sin {15^ \circ }$ and $\cos {15^ \circ }$are $\dfrac{{\sqrt 3 - 1}}{{2\sqrt 2 }}$and $\dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$ respectively.

Note: Similar question can be asked for different other angles such as ${22.5^ \circ }$ etc. Students must also know all the double angles and half angles formula which facilitates finding some extraordinary angles. All the values must be substituted correctly otherwise would lead to wrong answers only.