1. ## Solving trigonometric equations

Solve the trigonometric equation analytically for values of $\displaystyle x$ for $\displaystyle 0\le x <2\pi$

$\displaystyle sin\left(x - \frac{\pi}{4}\right) = cos\left(x - \frac{\pi}{4}\right)$

$\displaystyle sin\;x\;cos\;\frac{\pi}{4} - cos\;x\;sin\;\frac{\pi}{4} = cos\;x\;cos\;\frac{\pi}{4} + sin\;x\;sin\;\frac{\pi}{4}$

$\displaystyle sin\;x\;\left(\frac{\sqrt{2}}{2}\right) - cos\;x\;\left(\frac{\sqrt{2}}{2}\right) = cos\;x\;\left(\frac{\sqrt{2}}{2}\right) + sin\;x\;\left(\frac{\sqrt{2}}{2}\right)$

$\displaystyle 2\;cos\;x\;\left(\frac{\sqrt{2}}{2}\right)$

If what I did so far is right. I'm stuck at this point.

2. Originally Posted by TI-84
Solve the trigonometric equation analytically for values of $\displaystyle x$ for $\displaystyle 0\le x <2\pi$

$\displaystyle sin\left(x - \frac{\pi}{4}\right) = cos\left(x - \frac{\pi}{4}\right)$

$\displaystyle sin\;x\;cos\;\frac{\pi}{4} - cos\;x\;sin\;\frac{\pi}{4} = cos\;x\;cos\;\frac{\pi}{4} + sin\;x\;sin\;\frac{\pi}{4}$

$\displaystyle sin\;x\;\left(\frac{\sqrt{2}}{2}\right) - cos\;x\;\left(\frac{\sqrt{2}}{2}\right) = cos\;x\;\left(\frac{\sqrt{2}}{2}\right) + sin\;x\;\left(\frac{\sqrt{2}}{2}\right)$

$\displaystyle 2\;cos\;x\;\left(\frac{\sqrt{2}}{2}\right)$

If what I did so far is right. I'm stuck at this point.

$\displaystyle 2\;cos\;x\;\left(\frac{\sqrt{2}}{2}\right)$ = 0,

that is

$\displaystyle \sqrt{2}\;\cos x$ = 0,

3. But when I bring everything over to the right side to isolate x doesnt it just become x=0? the books answer is $\displaystyle \frac{\pi}{2}\;and\;\frac{3\pi}{2}$

4. Originally Posted by mr fantastic

$\displaystyle 2\;cos\;x\;\left(\frac{\sqrt{2}}{2}\right)$ = 0,

that is

$\displaystyle \sqrt{2}\;\cos x$ = 0,
Next step: Divide both sides by $\displaystyle \sqrt{2}$:

$\displaystyle \cos x = 0$

Therefore x = ......

5. Originally Posted by mr fantastic
Next step: Divide both sides by $\displaystyle \sqrt{2}$:

$\displaystyle \cos x = 0$

Therefore x = ......
Perhaps I should write it as $\displaystyle \cos (x) = 0$.

Hmmmmmm ....... You do understand that $\displaystyle \cos (x) = 0$ means cos of x, NOT $\displaystyle cos \times x$ (which has no meaning.

In the same way that $\displaystyle \sqrt{2}$ means the square root of 2, not square root times 2 ......

So you're looking for values of x such that the cos of those values is equal to 0 ......

6. $\displaystyle 2\;cos\;x\;\left(\frac{\sqrt{2}}{2}\right) = 0$

$\displaystyle cos\;x\;\left(\frac{\sqrt{2}}{2}\right) = 0$

$\displaystyle cos\;x = 0$

$\displaystyle x = Cos^{-1}\;(0)$

$\displaystyle x=90$ and $\displaystyle x=270$

Is that correct?

7. Originally Posted by TI-84
$\displaystyle 2\;cos\;x\;\left(\frac{\sqrt{2}}{2}\right) = 0$

$\displaystyle cos\;x\;\left(\frac{\sqrt{2}}{2}\right) = 0$

$\displaystyle cos\;x = 0$

$\displaystyle x = Cos^{-1}\;(0)$

$\displaystyle x=90$ and $\displaystyle x=270$

Is that correct?
Yes. You can substitute into the original equation and confirm this.

8. Hello, TI-84

Solve for $\displaystyle x,\;\;0 \le x <2\pi$

. . $\displaystyle \sin\left(x - \frac{\pi}{4}\right) \:= \:\cos\left(x - \frac{\pi}{4}\right)$
Divide both sides by $\displaystyle \cos\left(x-\frac{\pi}{4}\right)$

. . $\displaystyle \frac{\sin\left(x-\frac{\pi}{4}\right)}{\cos\left(x-\frac{\pi}{4}\right)} \;=\;1 \quad\Rightarrow\quad \tan\left(x-\frac{\pi}{4}\right) \;=\;1$

Hence: .$\displaystyle x - \frac{\pi}{4} \;=\;\left\{\frac{\pi}{4},\:\frac{5\pi}{4}\right\} \quad\Rightarrow\quad x \;=\;\left\{\frac{\pi}{2},\:\frac{3\pi}{2}\right\}$