In a triangle ABC, if a=18,b=24,c=30 find the value of sin(A/2) and cos(A/2).
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Step-by-Step Solution
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 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=9×72 and 720=10×72). After simplification, take the square root of the numerator and denominator, and rationalize the denominator.