Help Needed with this Trigonometric Proof

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February 3rd 2011, 09:58 PM
sammysparrow

Help Needed with this Trigonometric Proof

Hi, can someone show me how to prove that

sin A + cos A = (the square root of 2) sin ( A + pi over 4) ?

Thanks, any help would be very much appreciated!
February 3rd 2011, 10:06 PM
harish21

$RHS = \sqrt{2}\;\bigg[ \sin\bigg(A+\dfrac{\pi}{4}\bigg)\bigg]$

now use $\sin(A+B)=\sin(A)\cos(B)+\cos(A)\sin(B)$ where $B = \dfrac{\pi}{4}$
February 4th 2011, 07:13 AM
Soroban

Hello, sammysparrow!

Quote:

$\text{Prove that: }\:\sin A + \cos A \:=\:\sqrt{2}\sin\left(A + \frac{\pi}{4}\right)$

Multiply the left side by $\frac{\sqrt{2}}{\sqrt{2}}$

. . $\frac{\sqrt{2}}{\sqrt{2}}\left(\sin A + \cos A\right) \;=\;\sqrt{2}\left(\sin A\cdot\frac{1}{\sqrt{2}}+ \cos A\cdot\frac{1}{\sqrt{2}}\right)$ .[1]

Note that: . $\sin\frac{\pi}{4} \,=\,\cos\frac{\pi}{4} \,=\,\frac{1}{\sqrt{2}}$

Then [1] becomes: . $\sqrt{2}\left(\sin A\cos\frac{\pi}{4} + \cos A\sin\frac{\pi}{4}\right) \;=\;\sqrt{2}\sin(A +\frac{\pi}{4})$
February 4th 2011, 12:55 PM
Archie Meade

Quote:

Originally Posted by sammysparrow

Hi, can someone show me how to prove that

sin A + cos A = (the square root of 2) sin ( A + pi over 4) ?

Thanks, any help would be very much appreciated!

If you had to begin from the LHS, another way is

$2SinACosB=Sin(A+B)+Sin(A-B)$

$2CosASinB=Sin(A+B)-Sin(A-B)$

$\Rightarrow\ 2(SinACosB+CosASinB)=2Sin(A+B)$

$\Rightarrow\ SinACosB+CosASinB=Sin(A+B)$

$SinACosB+CosASinB=k(SinA+CosA)$

$\Rightarrow\ SinB=CosB\Rightarrow\ B=\frac{\pi}{4}$

$\displaystyle\ Sin\frac{\pi}{4}=Cos\frac{\pi}{4}=\frac{1}{\sqrt{2 }}$

$\Rightarrow\ SinA+CosA=\sqrt{2}\left[SinA+\frac{\pi}{4}\right]$
February 6th 2011, 06:05 PM
sammysparrow

(Hi) Thanks everyone!