kerrum75
2021-12-13
Answered

How do you solve ${\mathrm{sin}}^{2}x-{\mathrm{cos}}^{2}x=0$ for x in the interval $[0,2\pi )$ ?

You can still ask an expert for help

usumbiix

Answered 2021-12-14
Author has **33** answers

Applying the identity ${\mathrm{cos}}^{2}\left(x\right)-{\mathrm{sin}}^{2}\left(x\right)=\mathrm{cos}\left(2x\right)$ we have

${\mathrm{sin}}^{2}\left(x\right)-{\mathrm{cos}}^{2}\left(x\right)=-\mathrm{cos}\left(2x\right)$

In general,

$\mathrm{cos}\left(u\right)=0\to u=\frac{n\pi}{2}$ for some $n\in \mathbb{Z}$

Thus we have

${\mathrm{sin}}^{2}\left(x\right)-{\mathrm{cos}}^{2}\left(x\right)=0$

$\to 2x=\frac{n\pi}{2}$ for $n\in \mathbb{Z}$

$\Rightarrow x=\frac{n\pi}{4}$ for $n\in \mathbb{Z}$

Restricting our values to the interval$[0,2\pi ]$ gives our final result:

$x\in (\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4})$

In general,

Thus we have

Restricting our values to the interval

lovagwb

Answered 2021-12-15
Author has **50** answers

Yes! Of course!

asked 2021-05-13

Transform $\mathrm{cot}B+\mathrm{tan}B\to \mathrm{csc}B\mathrm{sec}B$

asked 2021-12-26

How do you calculate the slope, x intercept, and y intercept of the following equation: $f\left(x\right)=-6x-3$ ?

asked 2021-02-12

The question asks for the exact value of the trigonometric function at the given real number:

$\mathrm{sin}\left(\frac{3\pi}{4}\right)$

asked 2021-02-02

(a) Compute

(b) Verify the Cauchy-Schwarz inequality for the given pair of vector.

asked 2021-12-09

Tell, what is the derivative of $\mathrm{sec}\left(x\right)$ ?

asked 2021-05-30

asked 2021-12-20

What is $\frac{\pi}{6}$ radians in degrees?