Question

How do you simplify: $\sin \left( {x + \dfrac{\pi }{4}} \right)$ ?

Answer 1

Hint: The given question deals with basic simplification of trigonometric functions by using some of the simple trigonometric formulae such as the compound angle formulae for sine function $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$. Basic algebraic rules and trigonometric identities are to be kept in mind while doing simplification in the given problem.

Complete answer:
In the given problem, we have to simplify the trigonometric expression given to us as: $\sin \left( {x + \dfrac{\pi }{4}} \right)$
So, we will use the compound angle formula for sine function as $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ where $A = x$ and $B = \left( {\dfrac{\pi }{4}} \right)$. Hence, we get the expanded form as,
$ \Rightarrow \sin \left( {x + \dfrac{\pi }{4}} \right) = \sin x\cos \left( {\dfrac{\pi }{4}} \right) + \cos x\sin \left( {\dfrac{\pi }{4}} \right)$
Now, we have to substitute in the values of cosine and sine function for the angle $\left( {\dfrac{\pi }{4}} \right)$ and then get the simplified form of the trigonometric expression.
We know that the value of cosine function for the standard angle $\left( {\dfrac{\pi }{4}} \right)$ is $\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$ and value of sine function for the same angle is $\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$.
So, we get,
$ \Rightarrow \sin \left( {x + \dfrac{\pi }{4}} \right) = \sin x\left( {\dfrac{1}{{\sqrt 2 }}} \right) + \cos x\left( {\dfrac{1}{{\sqrt 2 }}} \right)$
Now, taking the common terms outside the brackets, we get,
$ \Rightarrow \sin \left( {x + \dfrac{\pi }{4}} \right) = \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\sin x + \cos x} \right)$
Hence, the simplified form of the trigonometric expression $\sin \left( {x + \dfrac{\pi }{4}} \right)$ can be simplified as $\left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\sin x + \cos x} \right)$ by the use of basic algebraic rules and simple trigonometric formulae.

Note:
Trigonometric functions are also called Circular functions. Trigonometric functions are the functions that relate an angle of a right angled triangle to the ratio of two side lengths. There are $6$trigonometric functions, namely: $\sin (x)$,$\cos (x)$,$\tan (x)$,$\cos ec(x)$,$\sec (x)$and \[\cot \left( x \right)\] . We must know the compound angle formulae of sine as $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ and cosine as $\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B$.