পিথাগোরিয় অভেদসমূহ

The Pythagorean identities state that $$\begin{align*} \sin^2 \theta+\cos^2 \theta &= 1 \\ \end{align*}$$ Simple, right?

The proof isn't that difficult either. Construct a right triangle with angle $\theta$. Sine of $\theta$ is defined as $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ and $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$ Squaring them and adding gives $$\begin{align*} \sin^2 \theta+\cos^2 \theta &= \left( \frac{\text{opposite}}{\text{hypotenuse}} \right)^2+\left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)^2 \\ &= \dfrac{\text{opposite}^2+\text{adjacent}^2}{\text{hypotenuse}^2} \\ &= 1 \end{align*}$$ The third equation follows form the second by the Pythagorean identity, hence the name.

Got it?

These identities are also useful because they can be used to derive other ones such as $$\begin{align*} \tan^2 \theta + 1 &= \sec^2 \theta \\ 1 + \cot^2 \theta &= \csc^2 \theta\\ \end{align*}$$

These follow from $\sin^2\theta+\cos^2\theta=1$ via division by $\cos^2\theta$ and $\sin^2\theta$, respectively.

For the right Triangle $XYZ$, with angles $x, y,$ and $z$, If $sin (x)= 4/5$ what is $cos (x)$?

Here are some practice problems with pythagorean identities! Keep in mind that they mostly use the ones involving secants and cosecants. Remember that you can derive them from the normal pythagorean identity by dividing by either $\sin^2x$ or $\cos^2x$. Best of luck!