How do I find the value of sin ( A-B)
When tan A= 4/3
Cot B= 5/12
If all the angles are in QI, then
sinA = 4/5
cosA = 3/5
sinB = 12/13
cosB = 5/13
now just use the difference formula
To find the value of sin(A-B) when tan A = 4/3 and cot B = 5/12, we can use the trigonometric identities and properties to calculate it step by step.
Step 1: Find sin A and cos A using the given value of tan A.
Since tan A = opposite/adjacent = 4/3, we can use the Pythagorean theorem to find the hypotenuse. Let's assume that the opposite side of angle A is 4, and the adjacent side is 3. The hypotenuse, according to the Pythagorean theorem, can be calculated as:
hypotenuse = √(opposite^2 + adjacent^2) = √(4^2 + 3^2) = √(16 + 9) = √25 = 5
Now, we can find sin A and cos A using the values we calculated:
sin A = opposite/hypotenuse = 4/5
cos A = adjacent/hypotenuse = 3/5
Step 2: Find sin B and cos B using the given value of cot B.
Since cot B = adjacent/opposite = 5/12, we can assume that the adjacent side of angle B is 5, and the opposite side is 12. Using the Pythagorean theorem, we can calculate the hypotenuse as:
hypotenuse = √(adjacent^2 + opposite^2) = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Now, we can find sin B and cos B using the values we calculated:
sin B = opposite/hypotenuse = 12/13
cos B = adjacent/hypotenuse = 5/13
Step 3: Use the trigonometric identity sin(A-B) = sin A * cos B - cos A * sin B.
Plugging in the values we calculated, we get:
sin(A-B) = (sin A * cos B) - (cos A * sin B)
= (4/5 * 5/13) - (3/5 * 12/13)
= 20/65 - 36/65
= -16/65
Therefore, the value of sin(A-B) is -16/65.