Determine if the value of each trig function is positive, negative, zero, or undefined.

a) tan 310 degrees

b) cos 7pi/6

c) sin -245 degrees

d) sec( -7/2)

Determine in which quadrant you are in,

then use the CAST rule

e.g.
sin(-245°)
= sin( 115°) , (going 115 counterclockwise is the same as going 245° clockwise )

115° is in quad II
in quad II , sine is positive, so
sin(-245°) is positive

you can always check with your calculator.

in sec(-7/2) , remember that if no units are stated, radians are assumed.
so -7/2 is coterminal with 2π - 7/2 which is appr 2.78

a) To determine if the value of tan 310 degrees is positive, negative, zero, or undefined, we can refer to the unit circle.

First, let's convert 310 degrees to its equivalent angle within one full circle. Since one full circle is 360 degrees, we can subtract 360 from 310 to obtain the reference angle:

310 - 360 = -50 degrees.

Next, we look at the sign of the tangent function in each quadrant of the unit circle.

In the first quadrant (0 to 90 degrees), tan is positive.
In the second quadrant (90 to 180 degrees), tan is negative.
In the third quadrant (180 to 270 degrees), tan is positive.
In the fourth quadrant (270 to 360 degrees), tan is negative.

Since -50 degrees falls in the fourth quadrant, the value of tan 310 degrees is negative.

b) To determine if the value of cos (7pi/6) is positive, negative, zero, or undefined, we can also refer to the unit circle.

The angle 7pi/6 is equivalent to the angle -pi/6 in radians.

Again, we look at the sign of the cosine function in each quadrant of the unit circle.

In the first quadrant (0 to pi/2), cos is positive.
In the second quadrant (pi/2 to pi), cos is negative.
In the third quadrant (pi to 3pi/2), cos is negative.
In the fourth quadrant (3pi/2 to 2pi), cos is positive.

Since -pi/6 falls in the fourth quadrant, the value of cos (7pi/6) is positive.

c) To determine if the value of sin -245 degrees is positive, negative, zero, or undefined, we again refer to the unit circle.

Since -245 degrees is equivalent to the angle -65 degrees, we can look at the sign of the sine function in each quadrant of the unit circle.

In the first quadrant (0 to 90 degrees), sin is positive.
In the second quadrant (90 to 180 degrees), sin is positive.
In the third quadrant (180 to 270 degrees), sin is negative.
In the fourth quadrant (270 to 360 degrees), sin is negative.

Since -65 degrees falls in the third quadrant, the value of sin -245 degrees is negative.

d) To determine if the value of sec(-7/2) is positive, negative, zero, or undefined, we need to remember the relationship between secant and cosine.

Recall that secant is the reciprocal of cosine. So, if the value of cosine is positive, then secant will also be positive. Conversely, if cosine is negative, secant will be negative.

Since we determined in part b that the value of cos(7pi/6) is positive, sec(-7/2) will also be positive.

To determine if the value of each trig function is positive, negative, zero, or undefined, we can use the unit circle and the properties of each trigonometric function.

a) tan 310 degrees:
To determine the value of tan 310 degrees, we first need to locate the angle 310 degrees on the unit circle. Since 310 degrees is greater than 360 degrees, we can subtract 360 degrees to get an equivalent angle. Thus, we have 310 degrees - 360 degrees = -50 degrees.

Now, we need to determine the quadrant in which the angle -50 degrees lies. In the unit circle, angles in the second and fourth quadrants have a positive tangent value, while angles in the first and third quadrants have a negative tangent value.

In this case, -50 degrees is in the fourth quadrant, so the value of tan 310 degrees is negative.

b) cos 7pi/6:
To determine the value of cos 7pi/6, we can recognize that 7pi/6 is in the third quadrant. In the third quadrant, the cosine value is negative. Therefore, the value of cos 7pi/6 is negative.

c) sin -245 degrees:
To determine the value of sin -245 degrees, we first need to convert -245 degrees to an angle within 0 and 360 degrees. We can do this by adding 360 degrees. Thus, we have -245 degrees + 360 degrees = 115 degrees.

Now, we need to determine the quadrant in which the angle 115 degrees lies. In the unit circle, angles in the first and second quadrants have a positive sine value, while angles in the third and fourth quadrants have a negative sine value.

In this case, 115 degrees is in the second quadrant, so the value of sin -245 degrees is positive.

d) sec(-7/2):
To determine the value of sec(-7/2), we need to evaluate the reciprocal of the cosine of -7/2 (since sec(x) is equivalent to 1/cos(x)).

To find the value of cosine -7/2, we first need to convert -7/2 to radians. Since there are 2π radians in a full circle, we can multiply -7/2 by 2π/360 to convert it to radians. Thus, we have -7/2 * (2π/360) = -7π/360 radians.

Now, we need to determine the quadrant in which the angle -7π/360 radians lies. In the unit circle, angles in the second and third quadrants have a negative cosine value, while angles in the first and fourth quadrants have a positive cosine value.

In this case, -7π/360 radians lies in the second quadrant. Therefore, the cosine value is negative, and the value of sec(-7/2) is negative.