How do I find the value of sin(a-b) if tana=4/3, cotb=5/12, 0<a<90, and 0<b<90?
so both a and b are in the first quadrant.
Draw two right-angled triangles in the first quadrant with
1. angle a and opposite of 4 and adjacent of 3 (tan a=4/3)
then the hypotenuse is 5 and
sin a = 4/5, cos a = 3/5
2. angle b and opposite 12 and adjacent of 5 (cot b = 5/12)
then the hypotenuse is 13 and
sin b = 12/13, cos b = 5/13
we know
sin(a - b)
= sinacosb - cosasinb
= (4/5)(5/13) - (3/5)(12/13)
= -16/65
I was only two steps from being right!
To find the value of sin(a-b), we can use the trigonometric identity for the sine of the difference of two angles:
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
Given that tana = 4/3 and cotb = 5/12, we can find the values of sin(a) and cos(b) using the following trigonometric identities:
sin(a) = tana / √(1 + tana^2)
cos(b) = 1 / √(1 + cotb^2)
Substituting the given values into these identities, we have:
sin(a) = (4/3) / √(1 + (4/3)^2)
cos(b) = 1 / √(1 + (5/12)^2)
Let's calculate these values step-by-step:
1. Calculate sin(a):
sin(a) = (4/3) / √(1 + (4/3)^2)
= (4/3) / √(1 + 16/9)
= (4/3) / √(25/9)
= (4/3) / (5/3)
= 4/5
2. Calculate cos(b):
cos(b) = 1 / √(1 + (5/12)^2)
= 1 / √(1 + 25/144)
= 1 / √(169/144)
= 1 / (13/12)
= 12/13
Now that we have the values for sin(a) and cos(b), we can substitute them back into the sine difference formula:
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
= (4/5)(12/13) - sin(a)sin(b)
However, we still need to find sin(b). We can use the identity sin^2(b) = 1 - cos^2(b) to find sin(b):
sin^2(b) = 1 - cos^2(b)
sin(b) = √(1 - cos^2(b))
3. Calculate sin(b):
sin(b) = √(1 - cos^2(b))
= √(1 - (12/13)^2)
= √(1 - 144/169)
= √(25/169)
= 5/13
Now we can substitute the values for sin(a), cos(b), and sin(b) back into the sine difference formula:
sin(a-b) = (4/5)(12/13) - sin(a)sin(b)
= (4/5)(12/13) - (4/5)(5/13)
= (48/65) - (20/65)
= 28/65
So, the value of sin(a-b) is 28/65.
To find the value of sin(a-b), we can use trigonometric identities and the given information. Here's how you can solve it step by step:
1. Start by drawing a right triangle to represent the given information. In this case, since 0<a<90 and 0<b<90, both angles lie in the first quadrant.
2. Since tana = 4/3, we can use this information to determine the lengths of the sides of the triangle. Recall that tan is the ratio of the opposite side to the adjacent side.
3. Let's assume that the opposite side of angle a is 4x (since the ratio is 4/3) and the adjacent side is 3x.
4. Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the hypotenuse of the triangle. In this case, using the values we assumed earlier, we have (3x)^2 + (4x)^2 = c^2.
5. Simplifying the equation, we get 9x^2 + 16x^2 = c^2, which leads to 25x^2 = c^2, and taking the square root of both sides, we have 5x = c.
6. Now, let's focus on cotb = 5/12. Cotangent is the ratio of the adjacent side to the opposite side. So, in this case, let's assume the adjacent side of angle b is 12y and the opposite side is 5y.
7. Using the Pythagorean theorem again, we have (12y)^2 + (5y)^2 = c^2. Simplifying, we get 144y^2 + 25y^2 = c^2, which leads to 169y^2 = c^2, and taking the square root of both sides, we have 13y = c.
8. Since we assumed the hypotenuse to be the same in both triangles (c = 5x = 13y), we can set 5x = 13y and solve for y in terms of x. Dividing both sides by 13, we get y = (5/13)x.
9. Now, we can calculate sin(a - b) using the difference of angles identity for sine: sin(a - b) = sin(a)cos(b) - cos(a)sin(b).
10. Since we know the values of tan(a) and cot(b), we can find sin(a) and cos(b) using the definitions of tangent and cotangent.
11. tan(a) = opposite/adjacent = (4x)/(3x) = 4/3. Rearranging, we have 4x = 3tan(a), so x = (3/4)tan(a).
12. From step 8, we know that y = (5/13)x, so substituting x into the equation, we have y = (5/13)(3/4)tan(a).
13. Now, we can find sin(a) and cos(b):
- sin(a) = opposite/hypotenuse = (4x)/(5x) = 4/5.
- cos(b) = adjacent/hypotenuse = (12y)/(13y) = 12/13.
14. Finally, we can substitute the values of sin(a), cos(b), cos(a), and sin(b) into the formula from step 9:
- sin(a - b) = sin(a)cos(b) - cos(a)sin(b)
= (4/5)(12/13) - cos(a)sin(b).
15. At this point, we need more information about either cos(a) or sin(b) in order to find sin(a - b). If there are additional given values or equations relating to cos(a) or sin(b), those can be used to solve for the remaining unknown variable(s).
So, to find the value of sin(a - b), we would need additional information about either cos(a) or sin(b) in order to complete the calculation.