If sin A=(20)/(29) and cos B=(28)/(53) and angles A and B are in Quadrant I, find the value of tan(A-B). Answer:

Question

If  sin A=(20)/(29) and  cos B=(28)/(53) and angles A and B are in Quadrant I, find the value of  tan(A-B). Answer:

Bytelearn · Accepted Answer

Find Side Lengths: Use the sine and cosine values to find the lengths of the opposite and adjacent sides of the right triangles for angles A and B. For angle A, sin A = opposite/hypotenuse = 20/29, which means the opposite side is 20 and the hypotenuse is 29. For angle B, cos B = adjacent/hypotenuse = 28/53, which means the adjacent side is 28 and the hypotenuse is 53.Calculate Missing Side: Calculate the missing side (adjacent for angle A and opposite for angle B) using the Pythagorean theorem. For angle A, adjacent side = √(hypotenuse^2 - opposite^2) = √(29^2 - 20^2) = √(841 - 400) = √441 = 21. For angle B, opposite side = √(hypotenuse^2 - adjacent^2) = √(53^2 - 28^2) = √(2809 - 784) = √2025 = 45.Find Tangents: Now that we have the lengths of all sides of the right triangles for angles A and B, we can find tan A and tan B. tan A = opposite/adjacent for angle A = 20/21. tan B = opposite/adjacent for angle B = 45/28.Use Angle Difference Identity: Use the angle difference identity for tangent to find tan(A-B): tan(A-B) = (tan A - tan B) / (1 + tan A * tan B). Substitute the values of tan A and tan B into the formula: tan(A-B) = (20/21 - 45/28) / (1 + (20/21 * 45/28)).Simplify Expression: Simplify the expression for tan(A-B): tan(A-B) = ((20*28 - 45*21) / (21*28)) / (1 + (20*45) / (21*28)). tan(A-B) = ((560 - 945) / (588)) / (1 + (900) / (588)). tan(A-B) = (-385 / 588) / (1 + 900 / 588).Find Common Denominator: Further simplify the expression by finding a common denominator for the terms in the denominator of the fraction: tan(A-B) = (-385 / 588) / ((588 + 900) / 588). tan(A-B) = (-385 / 588) / (1488 / 588).Simplify Complex Fraction: Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: tan(A-B) = (-385 / 588) * (588 / 1488).Cancel Common Factors: Cancel out the common factors in the numerator and the denominator: tan(A-B) = (-385 / 1488).Reduce Fraction: Reduce the fraction to its simplest form: tan(A-B) = (-55 / 212).

If $\sin A=\frac{20}{29}$ and $\cos B=\frac{28}{53}$ and angles A and B are in Quadrant I, find the value of $\tan (A-B)$ . $\newline$ Answer:

Full solution

More problems from Find trigonometric ratios using multiple identities

Related topics

If sin⁡A=2029 \sin A=\frac{20}{29} sinA=2920​ and cos⁡B=2853 \cos B=\frac{28}{53} cosB=5328​ and angles A and B are in Quadrant I, find the value of tan⁡(A−B) \tan (A-B) tan(A−B).\newlineAnswer:

If $\sin A=\frac{20}{29}$ and $\cos B=\frac{28}{53}$ and angles A and B are in Quadrant I, find the value of $\tan (A-B)$ . $\newline$ Answer: