use reference angles to find the exact value of the following expression. do not use a calculator. tan 420°…

Explanation:

Step1: Find the coterminal angle

Since angles are coterminal if they differ by a multiple of 360°, we find a coterminal angle of 420° in the range of 0° to 360°. Subtract 360° from 420°: $420 - 360=60^{\circ}$.

Step2: Determine the reference - angle

For an angle $\theta$ in the first quadrant (0° < $\theta$ < 90°), the reference angle is the angle itself. Since 60° is in the first quadrant, the reference angle for 420° (which is coterminal with 60°) is 60°.

Step3: Find the value of the tangent function

We know that $\tan\theta$ has the same value for an angle and its reference angle when considering the sign based on the quadrant. Since 420° is in the first - quadrant (coterminal with 60°) and tangent is positive in the first quadrant, and $\tan60^{\circ}=\sqrt{3}$, so $\tan420^{\circ}=\sqrt{3}$.

Answer:

The reference angle is 60°. $\tan420^{\circ}=\sqrt{3}$