Question

What sign has $\sin {\text{A + cosA}}$for the following value of ${\text{A}}$?
$ - {1125^\circ }$

Answer 1

Hint: Since they have told to find the value of $\sin {\text{A + cosA}}$ where ${\text{A = - 112}}{{\text{5}}^\circ }$ first try to solve the trigonometric equation by bringing in in some basic trigonometric identity and after solving this equation try write the angle as a multiple of 360 as it will help us get our solution more accurately and quickly.

Formula used:
Trigonometric identities used in this sum are
(1) $\sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b$
(2) $\sin ( - \theta ) = - \sin (\theta )$

Complete step by step answer:
We have been asked to find the sign of $\sin {\text{A + cosA}}$ when ${\text{A = - 112}}{{\text{5}}^\circ }$
Since the given equation is in trigonometric form let us try to bring the equation in basic trigonometric identity if possible so that it becomes easier for us to further solve the equation.
Let us first start with multiplying and dividing the equation by $\sqrt 2 $.
We get
$
  \sin {\text{A + cosA = }}\dfrac{{\sqrt 2 }}{{\sqrt 2 }}\left( {\sin {\text{A + cosA}}} \right) \\
   \Rightarrow \sin {\text{A + cosA}} = \sqrt 2 \left( {\dfrac{1}{{\sqrt 2 }}\sin {\text{A + }}\dfrac{1}{{\sqrt 2 }}\cos {\text{A}}} \right) \\
 $
We know that$\cos 45 = \sin 45 = \dfrac{1}{2}$
Therefore substituting this value above we get,
$
  \sin {\text{A + cosA = }}\sqrt 2 \left[ {\sin {\text{Acos45 + cosAsin45}}} \right] \\
   \Rightarrow \sin {\text{A + cosA = }}\sqrt 2 \left[ {\sin \left( {{\text{A + 45}}} \right)} \right] - - - \left( 1 \right) \\
 $
Since $\sin \left( {a + b} \right) = \sin a\cos b + \cos a\sin b$
Now since we have been given the value of ${\text{A = - 112}}{{\text{5}}^\circ }$ substituting it in equation (1) we get
$
  \sin {\text{A + cosA = }}\sqrt 2 \sin ( - 1125 + 45) \\
   \Rightarrow \sin {\text{A + cosA = }}\sqrt 2 \sin ( - 1080) \\
   \Rightarrow \sin {\text{A + cosA = }}\sqrt 2 \sin ( - 3 \times 360) \\
 $
Since we know that $\sin ( - \theta ) = - \sin (\theta )$ , we get
$\sin {\text{A + cosA = - }}\sqrt 2 \sin (3 \times 360)$
Here we know that value of $\sin (360) = 0$ that implies that the value of $\sin (3 \times 360) = 0$
$
  \sin {\text{A + cosA = - }}\sqrt 2 \times 0 \\
   \Rightarrow \sin {\text{A + cosA = }}0 \\
 $

Therefore we can conclude from the above calculations that the value of $\sin {\text{A + cosA}}$ is 0

Note: While trying to solve this kind of trigonometric equation and asking for signs of the equation try to use identities and bring the solution in the basic form. Also the angle given also determines the sign of the final equation so writing these huge angles as multiples of 360 gives us a better view to find the sign more accurately.