Unit Circle, important angles

Unit Circle, important angles

Now that we have seen some of the trigonometric functions, it is time to start putting elements together to get ready for solving trigonometric problems.

In this part of our trigonometry adventure, we will see the only table we need to learn by heart and use that as a base to generate more formulas needed in our future lessons.

We have already seen that when we know the sine and the cosine of an angle, we are likely to find it.

Radians and degrees

We may have seen this in previous lessons. However, it is crucial to grasp the conversion procedure from one to the other.

\pi\; radians\;=180 ^{\circ}

This yields the following statements:

-To convert an angle from degrees to radians, we multiply it by \pi and divide the result by 180.

Example:

Convert 30^{\circ} to radians

30\cdot \frac{\pi}{180}=\frac{\pi}{6}. We usually leave it in that form whenever possible.

Another Example:

Convert \frac{\pi}{4} to degrees

\frac{\pi}{4}\cdot\frac{180}{\pi}=\frac{180}{4}=45^{\circ}

Table of main angles in unit circle:

{supertable table transpose}

{highlight rows}

{active 3}

Degrees

{headrow}{headcol}

0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360
Radians 0 \frac{\pi}{6} \frac{\pi}{4} \frac{\pi}{3} \frac{\pi}{2} \frac{2\pi}{3} \frac{3\pi}{4} \frac{5\pi}{6} \pi \frac{7\pi}{6} \frac{5\pi}{4} \frac{4\pi}{3} \frac{3\pi}{2} \frac{5\pi}{3} \frac{7\pi}{4} \frac{11\pi}{6} 2\pi
Sine 0 \frac{1}{2} \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} 1 \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2} \frac{1}{2} 0 -\frac{1}{2} -\frac{\sqrt{2}}{2} -\frac{\sqrt{3}}{2} -1 -\frac{\sqrt{3}}{2} -\frac{\sqrt{2}}{2} -\frac{1}{2} 0
Cosine 1 \frac{\sqrt{3}}{2} \frac{\sqrt{2}}{2} \frac{1}{2} 0 -\frac{1}{2} -\frac{\sqrt{2}}{2} -\frac{\sqrt{3}}{2} -1 -\frac{\sqrt{3}}{2} -\frac{\sqrt{2}}{2} -\frac{1}{2} 0 \frac{1}{2} \frac{\sqrt{2}}{2} \frac{\sqrt{3}}{2} 1

{/supertable}

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