Question

use reference angles to find the exact value of the following expression. do not use a calculator. sec 1050°. determine the reference angle for 1050°. the reference angle is □°.

Explanation:

Step1: Find equivalent angle within 360°

Subtract multiples of 360° from 1050°. $1050 - 2\times360=1050 - 720 = 330^{\circ}$.

Step2: Determine reference angle

Since 330° is in the fourth - quadrant, the reference angle $\theta_{r}=360 - 330=30^{\circ}$.

Step3: Recall secant identity

$\sec\theta=\frac{1}{\cos\theta}$, and $\cos330^{\circ}=\cos(360 - 30)^{\circ}=\cos30^{\circ}=\frac{\sqrt{3}}{2}$.

Step4: Calculate secant value

$\sec330^{\circ}=\frac{1}{\cos330^{\circ}}=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2\sqrt{3}}{3}$.

Answer:

The reference angle is $30^{\circ}$ and $\sec1050^{\circ}=\frac{2\sqrt{3}}{3}$