Explain how special right triangles, reference angles, and quadrants of a coordinate grid help us find the exact answer to the following: cos 210°.

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  cos(210°) = -(√3)/2

Step-by-step explanation:

The terminal ray of a 210° angle is in the third quadrant. The angle it makes with the -x axis is (210° -110°) = 30°. This is the "reference angle". In the 3rd quadrant, the x-coordinate of the terminal ray's intersection with the unit circle has a negative sign. This will be the sign of the cosine of the angle.

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The reference angle of 30° tells you that the trig functions of the angle can be found from the side ratios of the "special right triangle" with angles of 30°, 60°, and 90°. The side ratios, shortest to longest, in that triangle are 1 : √3 : 2.

The cosine of the angle is the ratio ...

  Cos = Adjacent/Hypotenuse

In the above special triangle, the side adjacent to the 30° angle is the one that is √3 ratio unis. The hypotenuse is 2 ratio units. So, the cosine of 30° is ...

  cos(30°) = (√3)/2

As we said above, the sign of the adjacent side of the reference angle for 210° has a negative value. (The hypotenuse is always considered to be positive.) Then the desired cosine is ...

  cos(210°) = -cos(30°)

  cos(210°) = -(√3)/2

step-by-step explanation: you take each of the numbers and divide by the amount then you take the range