Find the exact values of the other trig functions of theta given sec theta=7 and sin theta is less than 0.

If sec Ø = 7

then cosØ = 1/7
Also since sinØ < 0, then Ø must be an angle in quadrant IV
From the right-triangle drawn in standard position, the missing side would be √48 or 4√3

sinØ = -4√3/7
cscØ = -7/(4√3)

cosØ = 1/7
secØ = 7 , (that was given)

tanØ = -1/(4√3)
cotØ = -4√3

need help idkk wtf yall talkingg boutt

To find the exact values of the other trigonometric functions, we need to use the given information that sec(theta) = 7 and sin(theta) < 0.

First, we know that sec(theta) = 1/cos(theta). So, we can find the value of cos(theta) by taking the reciprocal of sec(theta):

cos(theta) = 1/sec(theta) = 1/7 = 1/7.

Next, to find sin(theta), we use the Pythagorean identity:

sin^2(theta) + cos^2(theta) = 1.

Substituting the value of cos(theta) we found:

sin^2(theta) + (1/7)^2 = 1,

sin^2(theta) + 1/49 = 1,

sin^2(theta) = 1 - 1/49 = 48/49.

Taking the square root of both sides:

sin(theta) = ± √(48/49).

Since sin(theta) is less than 0, we take the negative square root:

sin(theta) = -√(48/49) = -√48/7.

Now, we can find the other trigonometric functions:

cosec(theta) = 1/sin(theta) = 1/(-√48/7) = -7/√48 = -7√48/48 = -√3/4.

tan(theta) = sin(theta)/cos(theta) = (-√48/7)/(1/7) = -√48.

cot(theta) = 1/tan(theta) = 1/(-√48) = -1/√48 = -√48/48 = -√3/3.

To find the values of the other trigonometric functions based on the given information, we can use the following identity:

1. First, we know that sec(theta) = 7. We can rewrite this equation by taking the reciprocal of both sides: cos(theta) = 1/7.

2. Next, we are given that sin(theta) < 0. This means that the sine function is negative in the quadrant where theta lies. Since sine is negative, we can determine that theta lies in either the third or fourth quadrant.

Using the above information, we can find the values of the other trigonometric functions as follows:

1. We already know that cos(theta) = 1/7. To find the other trigonometric functions, we can use the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.

Substituting the known value of cos(theta) = 1/7, we can solve for sin(theta).

sin^2(theta) + (1/7)^2 = 1
sin^2(theta) + 1/49 = 1
sin^2(theta) = 1 - 1/49
sin^2(theta) = 48/49

Now, taking the square root of both sides, we can find sin(theta):

sin(theta) = sqrt(48/49)
sin(theta) = 4sqrt(3)/7

2. Knowing the values of cos(theta) and sin(theta), we can find the values of the other trigonometric functions using the following definitions:

tan(theta) = sin(theta) / cos(theta)
cot(theta) = 1 / tan(theta)
sec(theta) = 1 / cos(theta)
csc(theta) = 1 / sin(theta)

Plugging in the values we have found:

tan(theta) = (4sqrt(3)/7) / (1/7) = 4sqrt(3)
cot(theta) = 1 / (4sqrt(3)) = sqrt(3) / 12
sec(theta) = 1 / (1/7) = 7
csc(theta) = 1 / (4sqrt(3)/7) = 7 / (4sqrt(3))

Therefore, the exact values of the other trigonometric functions of theta, given sec(theta) = 7 and sin(theta) < 0, are:

cos(theta) = 1/7
sin(theta) = 4sqrt(3)/7
tan(theta) = 4sqrt(3)
cot(theta) = sqrt(3) / 12
sec(theta) = 7
csc(theta) = 7 / (4sqrt(3))