3.21=((9.8)/(cos10-Asin10))sin10+A((9.8)/(cos10-Asin10))cos10
Solve for A
To solve for A in the equation 3.21 = ((9.8) / (cos10 - Asin10))sin10 + A((9.8) / (cos10 - Asin10))cos10, we follow these steps:
Step 1: Distribute A to the terms in the equation:
3.21 = (9.8 / (cos10 - Asin10))sin10 + (A^2)(9.8 / (cos10 - Asin10))cos10
Step 2: Determine a common denominator for the fractions:
Multiply sin10 and cos10 by (cos10 - Asin10) to get a common denominator:
3.21 = (9.8sin10) / (1 - Asin10) + (A^2 * 9.8cos10) / (cos10 - Asin10)
Step 3: Combine the fractions on the right-hand side:
To add the fractions, we need to find a common denominator for (1 - Asin10) and (cos10 - Asin10). The common denominator is (1 - Asin10)(cos10 - Asin10):
3.21 = (9.8sin10 * (cos10 - Asin10) + A^2 * 9.8cos10) / ((1 - Asin10)(cos10 - Asin10))
Step 4: Expand and simplify the numerator on the right-hand side:
Apply the distributive property:
3.21 = (9.8sin10cos10 - 9.8(sin10)^2 - A^2 * 9.8cos10) / ((1 - Asin10)(cos10 - Asin10))
Step 5: Combine like terms in the numerator:
3.21 = (9.8sin10cos10 - 9.8(sin10)^2 - 9.8A^2cos10) / ((1 - Asin10)(cos10 - Asin10))
Step 6: Multiply both sides of the equation by ((1 - Asin10)(cos10 - Asin10)) to eliminate the denominator:
(1 - Asin10)(cos10 - Asin10)*3.21 = 9.8sin10cos10 - 9.8(sin10)^2 - 9.8A^2cos10
Step 7: Expand and simplify the left-hand side:
(3.21cos10 - 3.21Asin10 - Asin10cos10 + A(sin10)^2)*3.21 = 9.8sin10cos10 - 9.8(sin10)^2 - 9.8A^2cos10
Step 8: Rearrange the terms and bring all the terms involving A on one side of the equation:
(3.21cos10 - 3.21Asin10 - Asin10cos10 + A(sin10)^2)*3.21 + 9.8A^2cos10 = 9.8sin10cos10
Step 9: Multiply and simplify both sides of the equation:
(10.3241 - 10.3241A*sin10 - 3.21A*cos10 + 3.21A*sin10 - 1.3221A*(sin10)^2 + 9.976A^2*cos10) = 9.8sin10cos10
Step 10: Combine like terms on the left-hand side:
10.3241 + (3.21A*sin10 - 3.21A*cos10 - 1.3221A*(sin10)^2 + 9.976A^2*cos10) = 9.8sin10cos10
Step 11: Rearrange the terms:
10.3241 + 3.21A*sin10 - 3.21A*cos10 - 1.3221A*(sin10)^2 + 9.976A^2*cos10 = 9.8sin10cos10
Step 12: Group the terms with A:
(3.21*sin10 - 3.21*cos10 - 1.3221*(sin10)^2 + 9.976*cos10)A^2 + (3.21*sin10 - 3.21*cos10)A + (10.3241 - 9.8sin10cos10) = 0
At this point, you can use the quadratic formula or factorize the equation to solve for A. The resulting values of A will be the solutions to the equation.