In a triangle ABC, if $a = 18, b = 24, c = 30$ find the value of sin(A/2) and cos(A/2).

Step 1: Calculate the semi-perimeter (s)

To use the half-angle formulas for sine and cosine, we first need to calculate the semi-perimeter of the triangle, denoted by $s$. The semi-perimeter is half the sum of the lengths of the three sides of the triangle.

Step 2: Substitute the side lengths

We substitute the given values of the side lengths $a=18$, $b=24$, and $c=30$ into the formula for $s$ and perform the addition and division to find its value.

Step 3: Apply the half-angle formula for sin(A/2)

Now we use the half-angle formula for $\sin(A/2)$, which relates the sine of half an angle to the side lengths and the semi-perimeter of the triangle. This formula is derived using the Law of Cosines and trigonometric identities.

Step 4: Substitute values into sin(A/2) formula

Substitute the calculated semi-perimeter $s=36$ and the given side lengths $b=24$, $c=30$ into the formula. Then, perform the subtractions and multiplications within the square root.

Step 5: Simplify to find sin(A/2)

Simplify the fraction inside the square root. Since $720/72 = 10$, the fraction becomes $1/10$. Finally, take the square root and rationalize the denominator by multiplying the numerator and denominator by $\sqrt{10}$.

Step 6: Apply the half-angle formula for cos(A/2)

Next, we use the half-angle formula for $\cos(A/2)$. This formula, similar to the sine half-angle formula, connects the cosine of half an angle to the side lengths and the semi-perimeter of the triangle.

Step 7: Substitute values into cos(A/2) formula

Substitute the calculated semi-perimeter $s=36$ and the given side lengths $a=18$, $b=24$, $c=30$ into the formula. Then, perform the subtraction and multiplications within the square root.

Step 8: Simplify to find cos(A/2)

Simplify the fraction inside the square root by finding common factors. Both 648 and 720 are divisible by 72 ($648 = 9 \times 72$ and $720 = 10 \times 72$). After simplification, take the square root of the numerator and denominator, and rationalize the denominator.

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