Find the values of $sin$, $cos$, and $ an$ at $45^circ$ on the unit circle

To find the values of $\sin$, $\cos$, and $\tan$ at $45^\circ$ on the unit circle, we start by noting that $45^\circ$ is the same as $\frac{\pi}{4}$ radians.

The coordinates of the point on the unit circle at $\frac{\pi}{4}$ radians are $\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$. Thus,

$\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$

$\tan \frac{\pi}{4} = \frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

Therefore, the values are:

$\sin 45^\circ = \frac{\sqrt{2}}{2}$

$\cos 45^\circ = \frac{\sqrt{2}}{2}$

$\tan 45^\circ = 1$

We know that $sin$, $cos$, and $ an$ values for $45^circ$ (or $frac{pi}{4}$ radians) on the unit circle can be derived from the coordinate point $left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$.

Thus, using the definitions of trigonometric functions:

$sin left( 45^circ
ight) = sin left( frac{pi}{4}
ight) = frac{sqrt{2}}{2}$

$cos left( 45^circ
ight) = cos left( frac{pi}{4}
ight) = frac{sqrt{2}}{2}$

$ an left( 45^circ
ight) = an left( frac{pi}{4}
ight) = frac{sin left( frac{pi}{4}
ight)}{cos left( frac{pi}{4}
ight)} = 1$

So, the values are:

$sin 45^circ = frac{sqrt{2}}{2}$

$cos 45^circ = frac{sqrt{2}}{2}$

$ an 45^circ = 1$

At $45^circ$, which is $frac{pi}{4}$ radians, the point on the unit circle is $left( frac{sqrt{2}}{2}, frac{sqrt{2}}{2}
ight)$. This gives:

$sin 45^circ = frac{sqrt{2}}{2}$

$cos 45^circ = frac{sqrt{2}}{2}$

$ an 45^circ = 1$

Start Using PopAi Today