What is the length of AB given that CA is 30 cm and CB is 72 cm? Simplify fractions. The sine of angle B is equal

To find the length of segment AB, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and angle \(C\) opposite side \(c\):

\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]

In our case, we have side CA with length 30 cm and side CB with length 72 cm. We need to find the length of segment AB. Let"s denote the length of segment AB as \(x\). Angle C is angle CBA.

Plugging in the values into the formula, we have:

\[x^2 = 30^2 + 72^2 - 2 \cdot 30 \cdot 72 \cos(CBA)\]

Now, we need to find the value of \(\cos(CBA)\). Since the sine of angle B is equal to the cosine of angle CBA, we can use the identity \(\sin(B) = \cos(CBA)\).

We can find the value of \(\sin(B)\) using the given information. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, the side opposite angle B is AC (length 30 cm) and the hypotenuse is AB (which we are trying to find). So:

\[\sin(B) = \frac{AC}{AB} = \frac{30}{x}\]

Since the sine of angle B is equal to the cosine of angle CBA, we have:

\[\cos(CBA) = \sin(B) = \frac{30}{x}\]

Plugging this value back into the Law of Cosines formula, we have:

\[x^2 = 30^2 + 72^2 - 2 \cdot 30 \cdot 72 \cdot \frac{30}{x}\]

We can simplify the equation by simplifying the fractions and combining like terms:

\[x^2 = 900 + 5184 - 43200 \cdot \frac{1}{x}\]

\[x^2 = 6084 - \frac{43200}{x}\]

To simplify further and solve for \(x\), let"s multiply both sides of the equation by \(x\) to eliminate the fractions:

\[x^3 = 6084x - 43200\]

Now we have a cubic equation. To find the value of \(x\), we can use numerical methods like graphing or approximations. However, since simplifying fractions was mentioned in the question, let"s try to simplify the expression further.

We can rewrite the expression as:

\[x^2 + 43200 = 6084x\]

This gives us a quadratic equation. We can solve it by applying the quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Comparing it to our equation:

\[x^2 - 6084x + 43200 = 0\]

We got \(a = 1\), \(b = -6084\), \(c = 43200\). Applying the quadratic formula, we get:

\[x = \frac{-(-6084) \pm \sqrt{(-6084)^2 - 4 \cdot 1 \cdot 43200}}{2 \cdot 1}\]

Simplifying further, we have:

\[x = \frac{6084 \pm \sqrt{37035876 - 172800}}{2}\]

\[x = \frac{6084 \pm \sqrt{37018676}}{2}\]

\[x = \frac{6084 \pm 1924}{2}\]

Taking the positive square root, since lengths cannot be negative, we have:

\[x = \frac{6084 + 1924}{2}\]

\[x = \frac{8008}{2}\]

\[x = 4004\]

Therefore, the length of segment AB is 4004 cm.