Finding $sin$, $cos$, and $ an$ Values on the Unit Circle

Consider the angle $45^\circ$ (or $\frac{\pi}{4}$ radians) on the unit circle. Find the sine, cosine, and tangent values for this angle.

Step 1: Identify the coordinates on the unit circle for the angle $45^\circ$. The coordinates are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$.

Step 2: Using these coordinates, we can determine:

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

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

Step 3: Tangent is the ratio of sine to cosine:

$\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$

Therefore,

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

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

$\tan 45^\circ = 1$

Let’s find the sine, cosine, and tangent values for the angle $30^circ$ (or $frac{pi}{6}$ radians) on the unit circle.

Step 1: Identify the coordinates on the unit circle for the angle $30^circ$. The coordinates are $(frac{sqrt{3}}{2}, frac{1}{2})$.

Step 2: Using these coordinates, we can determine:

$sin 30^circ = frac{1}{2}$

$cos 30^circ = frac{sqrt{3}}{2}$

Step 3: Tangent is the ratio of sine to cosine:

$ an 30^circ = frac{sin 30^circ}{cos 30^circ} = frac{frac{1}{2}}{frac{sqrt{3}}{2}} = frac{1}{sqrt{3}} = frac{sqrt{3}}{3}$

Therefore,

$sin 30^circ = frac{1}{2}$

$cos 30^circ = frac{sqrt{3}}{2}$

$ an 30^circ = frac{sqrt{3}}{3}$

Find the sine, cosine, and tangent values for the angle $60^circ$ (or $frac{pi}{3}$ radians) on the unit circle.

Coordinates for $60^circ$: $(frac{1}{2}, frac{sqrt{3}}{2})$

$sin 60^circ = frac{sqrt{3}}{2}$

$cos 60^circ = frac{1}{2}$

$ an 60^circ = frac{sqrt{3}}{2} div frac{1}{2} = sqrt{3}$

Therefore,

$sin 60^circ = frac{sqrt{3}}{2}$

$cos 60^circ = frac{1}{2}$

$ an 60^circ = sqrt{3}$

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