If theta is an acute angle of a right triangle and cos theta equals 3/8, what is the value of csc theta?

recall that sin^2 Ø + cos^2 Ø = 1

so
sin^2 Ø + 9/64= 1
sin^2Ø = 55/64
sinØ = √55/8

then cscØ = 8/√55
or 8√55/55

To find the value of csc theta, we first need to find the value of sin theta.

Since theta is an acute angle of a right triangle, we can use the Pythagorean identity: sin^2 theta + cos^2 theta = 1

Given that cos theta equals 3/8, we can substitute this value into the equation:

sin^2 theta + (3/8)^2 = 1

Simplifying, we have:

sin^2 theta + 9/64 = 1

Subtracting 9/64 from both sides:

sin^2 theta = 1 - 9/64

sin^2 theta = 64/64 - 9/64

sin^2 theta = 55/64

Taking the square root of both sides:

sin theta = √(55/64)

Now, to find the value of csc theta, we can take the reciprocal of sin theta:

csc theta = 1/sin theta

csc theta = 1/√(55/64)

To simplify further, we can rationalize the denominator:

csc theta = √(64/55)

csc theta = √(64)/√(55)

csc theta = 8/√(55)

Therefore, the value of csc theta is 8/√(55).

To find the value of csc theta, we need to find the value of the sine of theta first. Since theta is an acute angle of a right triangle and cos theta equals 3/8, we can use the Pythagorean identity to find the value of sin theta.

The Pythagorean identity states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). Using this identity, we have:

cos^2 theta + sin^2 theta = 1

Substituting cos theta = 3/8, we can solve for sin theta:

(3/8)^2 + sin^2 theta = 1
9/64 + sin^2 theta = 1
sin^2 theta = 1 - 9/64
sin^2 theta = 55/64

Taking the square root of both sides, we get:

sin theta = sqrt(55/64)

Now, to find the value of csc theta, we can take the reciprocal of sin theta:

csc theta = 1/sin theta = 1/(sqrt(55/64))

Simplifying the expression, we get:

csc theta = sqrt(64/55) = 8/√55

Therefore, the value of csc theta is 8/√55.