in triangle cat a=4 c=5 and cos t =(1/8). what is the length of t?

t^2 = 4^2 + 5^2 - 2(4)(5)(1/8)

t = 6

How did u get to 6 i can't follow what was done

To find the length of side b in triangle CAT, we can use the Law of Cosines. The Law of Cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of the lengths of those sides multiplied by the cosine of the included angle.

In our case, we are given side lengths a = 4, c = 5 and the cosine of angle T, cos(T) = 1/8.

Using the Law of Cosines, we can write:

b^2 = a^2 + c^2 - 2ac * cos(T)

Substituting the given values:

b^2 = 4^2 + 5^2 - 2 * 4 * 5 * (1/8)

Simplifying:

b^2 = 16 + 25 - 40/8
b^2 = 16 + 25 - 5
b^2 = 36

Taking the square root of both sides:

b = √36
b = 6

Therefore, the length of side b is 6.

To find the length of side b in triangle CAT, we can use the law of cosines, which states that in a triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice their product multiplied by the cosine of the angle between them.

In this case, we are given side a (AC) = 4, side c (CT) = 5, and the cosine of angle T (cos(t)) = 1/8. We want to find side b (AT).

The law of cosines formula is:
b^2 = a^2 + c^2 - 2*a*c*cos(t)

Plugging in the given values, we have:
b^2 = 4^2 + 5^2 - 2*4*5*(1/8)

Simplifying:
b^2 = 16 + 25 - 10*(1/8)
b^2 = 41 - 10/8
b^2 = 41 - 5/4
b^2 = (164 - 5) / 4
b^2 = 159 / 4

To find the length of side b, we need to take the square root of both sides of the equation:
b = √(159/4)

Calculating the square root:
b ≈ √39.75

Therefore, the length of side b (AT) is approximately equal to √39.75.