Exercise 104 Page 599

a To find the angles that makes the equation true, we can use the following diagram.

The cosine of a trig expression is given by the unit circle's horizontal axis. Examining the diagram, we notice that two angles result in a cosine value of $- \frac{1}{2} .$

Our two solutions are $θ = \frac{2 π}{3}$ and $θ = \frac{4 π}{3} .$

b The tangent value of an angle is the ratio of the angle's sine value to its cosine value.

tan θ = \frac{sin θ}{cos θ}

Therefore, we have to identify an angle of rotation where the ratio of the sine value to the cosine value equals

\frac{3}{3} .

Notice that this is the same ratio as

\frac{1}{3} .

We have two such angles.

If we calculate the ratio of the angle's sine value to their cosine value, we can see that the ratio equals

\frac{1}{3} .

tan \frac{π}{6} tan \frac{7 π}{6} = \frac{1 / 2}{3 / 2} = \frac{1}{3} = \frac{- 1 / 2}{- 3 / 2} = \frac{1}{3}

Our two solutions are

x = \frac{π}{6}

and

x = \frac{7 π}{6} .

c The sine of a trig expression is given by the unit circle's vertical axis. Examining the diagram, we notice that two angles result in a sine value of