If theta =( -7pi )/( 3) , then sec(theta) equals...?

sec(-7pi/3) = sec(7pi/3) = sec(7pi/3 - 2pi) = sec(pi/3) = 2

Can you visualize angles better in degrees?

most students can, so
7π/3 = 7(6-°) = 210°

210° is in quadr III
the reference angle is 30° ( we are 30° from the x-axis)
I know cos 30° = √3/2
in III, the cosine, and thus the secant, is negative
sec (7π/3)
= sec 210°
= 1/cos 210°
= 1/(-√3/2)
= -2/√3

scrap my solution

had another "senior moment"

To find the value of sec(theta), which represents the secant of theta, we'll use the formula:

sec(theta) = 1 / cos(theta)

We know that theta = (-7pi) / 3.

To find cos(theta), we'll use the unit circle, where cos(theta) is represented by the x-coordinate of the corresponding point on the unit circle.

Since theta = (-7pi) / 3, we need to find the point on the unit circle that corresponds to that angle.

To do that, we'll convert theta to degrees by using the fact that 180 degrees = pi radians:

theta_degrees = theta * (180 / pi)
= (-7pi / 3) * (180 / pi)
= -420 degrees

Now we'll find the point on the unit circle that corresponds to -420 degrees.

Since the unit circle is periodic, we'll find an equivalent angle in the range of 0 to 360 degrees.

To determine the equivalent angle, we'll add or subtract a multiple of 360 degrees until we get a value within that range:

-420 degrees + 360 degrees = -60 degrees

So, an equivalent angle within the range of 0 to 360 degrees is -60 degrees.

Now we'll find the point on the unit circle that corresponds to -60 degrees.

At -60 degrees, or in quadrant IV, the x-coordinate (cosine) is positive, whereas the y-coordinate (sine) is negative. The distance from the origin to the point is 1 unit, as it lies on the unit circle.

Therefore, cos(-60 degrees) = cos(theta) = 1.

Now we can calculate sec(theta):

sec(theta) = 1 / cos(theta)
= 1 / 1
= 1

Therefore, sec(theta) is equal to 1.