Question

What is \[\sqrt {1 + \sin 2\theta } \] equal to \[?\]

Answer 1

Hint: First we have to know the formula \[\sin 2x = 2\sin x\cos x\] to convert the equation from compound angle to an angle. Using the trigonometric identities rewrite the given equation and some algebra formulas simplify it as much as possible.

Complete step-by-step answer:
The word trigonometry comes from the Greek words trigono (“triangle”) and metry (“to measure”). Until about the 16th century, trigonometry was mainly concerned with computing the numerical values of the missing parts of a triangle (or any shape that can be dissected into triangles) when the values of other parts were given.
Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).
Given \[\sqrt {1 + \sin 2\theta } \]--(1)
 Since, we know that \[\sin 2\theta = 2\sin \theta \cos \theta \] and \[{\sin ^2}\theta + {\cos ^2}\theta = 1\] for every angle \[\theta \]. Then the equation (1) becomes
\[\sqrt {1 + \sin 2\theta } = \sqrt {{{\sin }^2}\theta + {{\cos }^2}\theta + 2\sin \theta \cos \theta } \]---(2)
We know that \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\], In the equation (2) we have \[a = \sin \theta \] and \[b = \cos \theta \]. Then the equation (2) can be written as
\[\sqrt {1 + \sin 2\theta } = \sqrt {{{\left( {\sin \theta + \cos \theta } \right)}^2}} \]
Hence simplifying the above equation, we get
\[\sqrt {1 + \sin 2\theta } = \sin \theta + \cos \theta \].
So, the correct answer is “\[\sin \theta + \cos \theta \]”.

Note: Note that Trigonometry developed from a need to compute angles and distances in such fields as astronomy, map making, surveying and artillery range finding. Problems involving angles and distances in one plane are covered in plane trigonometry. Applications to similar problems in more than one plane of three-dimensional space are considered in spherical trigonometry.