Question

If we have an expression as \[\sin \theta. \cos \theta = \dfrac{1}{2}\] . then find the value of \[\sin \theta + \cos \theta\]

Answer 1

Hint: In this question, we need to find the value of \[\sin \theta + \cos \theta\] and also given \[\sin \theta. \cos \theta = \dfrac{1}{2}\] . Sine , cosine and tangent are the basic trigonometric functions . Sine is nothing but a ratio of the opposite side of a right angle to the hypotenuse of the right angle. Similarly, cosine is nothing but a ratio of the adjacent side of a right angle to the hypotenuse of the right angle . Here we need to find the value of \[\sin \theta + \cos \theta\] . With the help of the Trigonometric functions , we can find the value of \[\sin \theta + \cos \theta\] .
Identity used :
\[sin^{2}\theta + cos^{2}\theta = 1\]
Algebraic formula used :
\[\left( a + b \right)^{2} = a^{2} + b^{2} + 2ab\]

Complete step by step solution:
Given,
\[\Rightarrow \sin \theta + \cos \theta\]
By squaring ,
We get,
\[\Rightarrow \left( \sin \theta + \cos \theta \right)^{2}\]
We know that \[\left( a + b \right)^{2} = a^{2} + b^{2} + 2ab\]
By expanding,
We get,
\[\left( \sin \theta + \cos \theta \right)^{2} = \sin^{2}\theta + \cos^{2} \theta + 2\sin \theta \cos \theta\]
From the trigonometry identity , \[\sin^{2} \theta + \cos^{2} \theta = 1\]
By substituting \[\sin^{2}\theta + \cos^{2}\theta = 1\]
We get,
\[\left( \sin \theta + \cos \theta \right)^{2} = 1 + 2\sin \theta \cos \theta\]
In question, given that \[\sin \theta. \cos \theta = \dfrac{1}{2}\]
By substituting \[\sin \theta. \cos \theta = \dfrac{1}{2}\]
We get,
\[\Rightarrow \left( \sin \theta + \cos \theta \right)^{2} = 1 + 2 \times \dfrac{1}{2}\]
By simplifying,
We get,
\[\Rightarrow \left( \sin \theta + \cos \theta \right)^{2} = 1 + 1\]
By adding,
We get,
\[\Rightarrow \left( \sin \theta + \cos \theta \right)^{2} = 2\]
On taking square root on both sides,
We get,
\[\Rightarrow \sqrt{\left( \sin \theta + \cos \theta \right)^{2}} = \pm \sqrt{2}\]
Thus we get, \[\sin \theta + \cos \theta = \pm \sqrt{2}\]
Final answer :
The value of \[\sin \theta + \cos \theta \] is equal to \[\pm \sqrt{2}\]

Note: The concept used in this problem is trigonometric identities and ratios. Trigonometric identities are nothing but they involve trigonometric functions including variables and constants. The common technique used in this problem is the algebraic formula with the use of trigonometric functions . Trigonometric functions are also known as circular functions or geometrical functions.