How do I do this trig problem?

First, draw (or imagine) the unit circle on an x-y plot.

It is a circle of radius 1, centred on the origin (0, 0)

cos(t) = 0.4

Angle t is the angle measured at the origin, between the x-axis and the vector that goes from the origin to the unit circle.

cosine is the value of x at the end of the vector.

The end of the vector (on the unit circle) is always

(x, y) = ( cos(t), sin(t) )

Draw a vertical line at x = +0.4

There are two places where the vertical line

x = +0.4

cuts the circumference of the circle:

once in the first quadrant (up and to the right), and

once in the fourth quadrant (down and to the right).

The interval for t asks you to begin with a full turn (-2π) backwards.

So start at the x axis (1, 0) on the unit circle, and follow the circumference, counter-clockwise (angle t will increase).

On the circumference, near the x-axis, draw a tiny arrow going counterclockwise (in the first quadrant, the arrow would point upwards).

After a full circle, t will have reached 0. Keep going; after another turn, it will reach +2π. Another turn 4π and another turn still, and you will have reached the end of the interval at 6π.

In total, you will have gone 4 times around the circle.

During every turn, you will have passed twice over the vertical line (x = +0.4).

Therefore, you are expected to provide 8 answers.

The third one (when t is between 0 and +2π) will be the principal angle where cos(t) = 0.4, which is t = 1.1592795 radians (approx.)

draw a line x = 0.4 which intersects with the unit circle at 2 points A and B

measure the angle between AO and x-axis and between BO and x-axis

you get two angles +/- 66 degrees