find the 5 other trig functions if cos(theta) = square root of 2/2 and cotangent is less than 0

if the cotØ < 0 then tanØ < 0

then Ø is in either II or IV
and since the cosØ = √2/2 (positive)
Ø is in I or IV
so Ø is in IV
the angle in standard position is 45°
(since cos45° = √2/2)

you should know the trig ratios of the 45° by heart.
so
sinØ = -√2/2
cscØ = -2/√2

cosØ = √2/2 --- the given value
secØ = 2/√2

tanØ = -1
cotØ = -1

To find the values of the other trigonometric functions, we need to utilize the given information about the cosine and cotangent.

Given:
cos(theta) = √2/2
cot(theta) < 0

We can start by finding the value of sin(theta).

We know that sin^2(theta) + cos^2(theta) = 1, which is a fundamental property of trigonometric functions. Let's substitute the given value of cos(theta) into this equation:

sin^2(theta) + (√2/2)^2 = 1
sin^2(theta) + 2/4 = 1
sin^2(theta) + 1/2 = 1
sin^2(theta) = 1 - 1/2
sin^2(theta) = 1/2
sin(theta) = ±√(1/2) = ±√2/2

Since cot(theta) = cos(theta) / sin(theta), we can evaluate the value of cot(theta) using the given information:

cot(theta) = cos(theta) / sin(theta)
cot(theta) = (√2/2) / (±√2/2)

Since cot(theta) is less than 0, we know that the signs of sin(theta) and cos(theta) must differ.

Let's analyze the two possible cases:

Case 1: sin(theta) = √2/2 and cos(theta) = -√2/2

In this case:
cot(theta) = (-√2/2) / (√2/2)
cot(theta) = -1

Now, let's find the values of the remaining trigonometric functions based on these values of sin(theta) and cos(theta):

1. tan(theta) = sin(theta) / cos(theta)
tan(theta) = (√2/2) / (-√2/2)
tan(theta) = -1

2. sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (-√2/2)
sec(theta) = -√2

3. csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (√2/2)
csc(theta) = √2

4. cot(theta) = cos(theta) / sin(theta) = -1 (Already calculated)

Therefore, for this case, the values of the five other trigonometric functions are:
tan(theta) = -1
sec(theta) = -√2
csc(theta) = √2
cot(theta) = -1

Case 2: sin(theta) = -√2/2 and cos(theta) = √2/2

In this case:
cot(theta) = (√2/2) / (-√2/2)
cot(theta) = -1

Now, let's find the values of the remaining trigonometric functions based on these values of sin(theta) and cos(theta):

1. tan(theta) = sin(theta) / cos(theta)
tan(theta) = (-√2/2) / (√2/2)
tan(theta) = -1

2. sec(theta) = 1 / cos(theta)
sec(theta) = 1 / (√2/2)
sec(theta) = √2

3. csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (-√2/2)
csc(theta) = -√2

4. cot(theta) = cos(theta) / sin(theta) = -1 (Already calculated)

Therefore, for this case, the values of the five other trigonometric functions are:
tan(theta) = -1
sec(theta) = √2
csc(theta) = -√2
cot(theta) = -1

In both cases, the values of the other trigonometric functions are the same due to the symmetry of the unit circle.

To summarize, the values of the five other trigonometric functions for the given conditions (cos(theta) = √2/2 and cot(theta) < 0) are:

tan(theta) = -1
sec(theta) = -√2
csc(theta) = √2
cot(theta) = -1