A Trigo Sum

Geometry Level 3

If $\cos x + \sin x = c$ , find $\sin x - \cos x$ .

\pm \sqrt{1-c^2}

c

\pm c

\sqrt{c}

\pm \sqrt{c}

\pm \sqrt{2-c^2}

3 solutions

Chew-Seong Cheong
Jul 21, 2016

$\begin{aligned} \cos x + \sin x & = c & \small \color{#3D99F6}{\text{Squaring both sides}} \\ \cos^2 x + 2 \sin x \cos x + \sin^2 x & = c^2 \\ 1 + 2 \sin x \cos x & = c^2 \\ \implies 2 \sin x \cos x & = c^2 - 1 \end{aligned}$

Now, we have:

$\begin{aligned} (\sin x - \cos x)^2 & = \sin^2 x - 2 \sin x \cos x + \cos^2 x \\ & = 1 - (c^2 - 1) \\ \implies \sin x - \cos x & = \boxed{\pm \sqrt{2-c^2}} \end{aligned}$

what about the negative solution?

Milton Shimabukuro - 4 years, 10 months ago

It should be $\pm \sqrt{ 2 - c^2 }$ right?

Calvin Lin Staff - 4 years, 10 months ago

Clément Berthelot
Jul 22, 2016

For x = pi/4, c=sqrt(2). The difference being null, only 6) can be the good answer

Md Zuhair
Jul 20, 2016

Squaring we get $cos^2x+ sin^2x +2 coxsinx= c^2$ ... $1-sin^2x+1-cos^2x+2cosx sinx =c^2$ ... that is $2-c^2 = (sinx - cosx)^2$ that is $\sqrt(2-c^2) = |sinx -cosx|$

@Md Zuhair , you need to learn up LaTex, so that members will do you problem. You can use Toggle LaTex on the top right pull-down menu to see the LaTex codes. You can also put your mouse cursor over the formulas to see the code. I have changed the problem phrasing for you.

Chew-Seong Cheong - 4 years, 10 months ago

A Trigo Sum

3 solutions

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