# To solve this problem, use the Pythagorean Theorem to find the third side. Next, find the other five trigonometric functions of theta (sin, tan, csc, sec and cot).

Cos=5/7

## so you are given sides 5, and 7. The other side is sqrt(7^2-5^2)=sqrt 24

draw the triangle. I will do tan

tanTheta=sqrt24/5

## To solve this problem, we have given the value of cos(theta) = 5/7. We will use the Pythagorean Theorem to find the third side and then calculate the other trigonometric functions.

Here's how you can do it step by step:

Step 1: Use the Pythagorean Theorem to find the third side:

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, let's assume the adjacent side (the side adjacent to the angle theta) is represented by a, the opposite side (the side opposite to the angle theta) is represented by b, and the hypotenuse (the third side) is represented by c.

Since cos(theta) = adjacent side / hypotenuse, we know that a = 5 and c = 7.

Now we can use the Pythagorean Theorem to solve for b:

a^2 + b^2 = c^2

5^2 + b^2 = 7^2

25 + b^2 = 49

b^2 = 49 - 25

b^2 = 24

b = √24

b ≈ 4.899

So, the length of the third side is approximately 4.899.

Step 2: Calculate the other trigonometric functions (sin, tan, csc, sec, and cot):

Using the values of a, b, and c that we just found, we can calculate the other trigonometric functions:

- sin(theta) = opposite side / hypotenuse

sin(theta) = b / c

sin(theta) = 4.899 / 7

sin(theta) ≈ 0.7

- tan(theta) = opposite side / adjacent side

tan(theta) = b / a

tan(theta) = 4.899 / 5

tan(theta) ≈ 0.979

- csc(theta) = 1 / sin(theta)

csc(theta) = 1 / 0.7

csc(theta) ≈ 1.43

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

sec(theta) = 1 / (5/7)

sec(theta) ≈ 1.4

- cot(theta) = 1 / tan(theta)

cot(theta) = 1 / 0.979

cot(theta) ≈ 1.022

So, the values of the other trigonometric functions for the given angle theta are approximately:

sin(theta) ≈ 0.7,

tan(theta) ≈ 0.979,

csc(theta) ≈ 1.43,

sec(theta) ≈ 1.4, and

cot(theta) ≈ 1.022.