Answer
Verified
478.2k+ views
Hint: To find the minimum value of the given trigonometric expression, we need to rearrange the form of the given expression into a trigonometric sine function. To calculate the minimum value, we have to minimise the trigonometric sine function.
Complete step-by-step answer:
The given trigonometric expression = \[\sin x+\cos x\]
Let the expression be equal to \[f(x)\]
Multiply and divide the function \[f(x)\] with the value of \[\dfrac{1}{\sqrt{2}}\], we get
$\Rightarrow$ \[f(x)=\dfrac{\sqrt{2}\sin x+\sqrt{2}\cos x}{\sqrt{2}}\]
By the numerator common value \[\sqrt{2}\], we get
$\Rightarrow$ \[f(x)=\sqrt{2}\left( \dfrac{\sin x+\cos x}{\sqrt{2}} \right)\]
By reframing the arrangement, we get
$\Rightarrow$ \[f(x)=\sqrt{2}\left( \dfrac{\sin x}{\sqrt{2}}+\dfrac{\cos x}{\sqrt{2}} \right)\]
We know that the trigonometric values of \[\cos \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}},\sin \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}\]
By substituting the above values in the given trigonometric expression, we get
\[f(x)=\sqrt{2}\left( \cos \left( \dfrac{\pi }{4} \right)\sin x+\cos x\sin \left( \dfrac{\pi }{4} \right) \right)\]
We know that the formula for sine angle of summation of two angles is given by \[\sin \left( \alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \]
By comparing the above formula and the trigonometric expression, we get
The values of \[\alpha ,\beta \] are equal to the value of \[x,\dfrac{\pi }{4}\] respectively.
Thus, we obtain the above expression as
\[f(x)=\sqrt{2}\left( \sin \left( x+\dfrac{\pi }{4} \right) \right)\]
To calculate the minimum value of the expression we have to minimise the trigonometric sine function.
We know that the value of the trigonometric function \[\sin \theta \] lies between -1 and 1 and \[\theta \] is an angle.
Thus, we can say that the minimum of any angle of sine is equal to -1 which implies that the minimum value of \[\sin \left( x+\dfrac{\pi }{4} \right)\] is equal to -1.
By substituting the above value in the given trigonometric expression, we get
\[f(x)=\sqrt{2}\left( -1 \right)\]
Thus, the minimum value of the given trigonometric expression is equal to \[-\sqrt{2}\].
Note: The possibility of mistake can be adding the minimum value of both terms of the expression, which is wrong since both the functions are not minimum at the same point. The alternative procedure of solving the question is to use the formula \[\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \] and then substitute the minimum value of \[\cos \theta \] which is equal to -1, to calculate the minimum value.
Complete step-by-step answer:
The given trigonometric expression = \[\sin x+\cos x\]
Let the expression be equal to \[f(x)\]
Multiply and divide the function \[f(x)\] with the value of \[\dfrac{1}{\sqrt{2}}\], we get
$\Rightarrow$ \[f(x)=\dfrac{\sqrt{2}\sin x+\sqrt{2}\cos x}{\sqrt{2}}\]
By the numerator common value \[\sqrt{2}\], we get
$\Rightarrow$ \[f(x)=\sqrt{2}\left( \dfrac{\sin x+\cos x}{\sqrt{2}} \right)\]
By reframing the arrangement, we get
$\Rightarrow$ \[f(x)=\sqrt{2}\left( \dfrac{\sin x}{\sqrt{2}}+\dfrac{\cos x}{\sqrt{2}} \right)\]
We know that the trigonometric values of \[\cos \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}},\sin \left( \dfrac{\pi }{4} \right)=\dfrac{1}{\sqrt{2}}\]
By substituting the above values in the given trigonometric expression, we get
\[f(x)=\sqrt{2}\left( \cos \left( \dfrac{\pi }{4} \right)\sin x+\cos x\sin \left( \dfrac{\pi }{4} \right) \right)\]
We know that the formula for sine angle of summation of two angles is given by \[\sin \left( \alpha +\beta \right)=\sin \alpha \cos \beta +\cos \alpha \sin \beta \]
By comparing the above formula and the trigonometric expression, we get
The values of \[\alpha ,\beta \] are equal to the value of \[x,\dfrac{\pi }{4}\] respectively.
Thus, we obtain the above expression as
\[f(x)=\sqrt{2}\left( \sin \left( x+\dfrac{\pi }{4} \right) \right)\]
To calculate the minimum value of the expression we have to minimise the trigonometric sine function.
We know that the value of the trigonometric function \[\sin \theta \] lies between -1 and 1 and \[\theta \] is an angle.
Thus, we can say that the minimum of any angle of sine is equal to -1 which implies that the minimum value of \[\sin \left( x+\dfrac{\pi }{4} \right)\] is equal to -1.
By substituting the above value in the given trigonometric expression, we get
\[f(x)=\sqrt{2}\left( -1 \right)\]
Thus, the minimum value of the given trigonometric expression is equal to \[-\sqrt{2}\].
Note: The possibility of mistake can be adding the minimum value of both terms of the expression, which is wrong since both the functions are not minimum at the same point. The alternative procedure of solving the question is to use the formula \[\cos \left( \alpha -\beta \right)=\cos \alpha \cos \beta +\sin \alpha \sin \beta \] and then substitute the minimum value of \[\cos \theta \] which is equal to -1, to calculate the minimum value.
Recently Updated Pages
Change the following sentences into negative and interrogative class 10 english CBSE
A Paragraph on Pollution in about 100-150 Words
One cusec is equal to how many liters class 8 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What were the social economic and political conditions class 10 social science CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE
Trending doubts
Change the following sentences into negative and interrogative class 10 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
A Paragraph on Pollution in about 100-150 Words
Write an application to the principal requesting five class 10 english CBSE
Complete the following word chain of verbs Write eat class 10 english CBSE
Discuss the main reasons for poverty in India