If sin = 4/5 with A in QII, and sin B = -3/5 with B in QIV find cos(A-B)

cos(A-B)= cosAcosB+sinAsinB

If sinA = 4/5 in QII
draw you triangle in QII, and find the third side.
The third side is 3. (Do the same thing for sin B= -3/5
Plug in:
cosAcosB-sinAsinB
=(-3/5)*(4/5)+(4/5)*(-3/5)
=(-12/25) -(-12/25)
=-24/25

53. the longest side is 55 cm. find the length of the shortest side

To find cos(A-B), we can use the formula:

cos(A-B) = cosA * cosB + sinA * sinB

Given that sinA = 4/5 (in QII) and sinB = -3/5 (in QIV), we need to find cosA and cosB.

To find cosA, we can use the Pythagorean identity:

sin^2A + cos^2A = 1

Since sinA = 4/5, we can substitute this in the equation:

(4/5)^2 + cos^2A = 1

16/25 + cos^2A = 1

cos^2A = 1 - 16/25

cos^2A = 9/25

Taking the square root of both sides:

cosA = ±√(9/25)

Note that cosA is negative in QII and QIII.

Since sinB = -3/5, we can use the Pythagorean identity again:

sin^2B + cos^2B = 1

(-3/5)^2 + cos^2B = 1

9/25 + cos^2B = 1

cos^2B = 1 - 9/25

cos^2B = 16/25

Taking the square root of both sides:

cosB = ±√(16/25)

Note that cosB is positive in QIV and QI.

Now, we can substitute the values of cosA (-√(9/25)) and cosB (√(16/25)) into the formula:

cos(A-B) = cosA * cosB + sinA * sinB

cos(A-B) = (-√(9/25)) * (√(16/25)) + (4/5) * (-3/5)

cos(A-B) = -12/25 + (-12/25)

cos(A-B) = -24/25

Therefore, cos(A-B) = -24/25.

To find cos(A-B), we first need to find the values of cos(A) and cos(B).

Given sin(A) = 4/5 and A is in Quadrant II, we can use the Pythagorean identity to find cos(A). The Pythagorean identity states that sin^2(A) + cos^2(A) = 1.

Since sin(A) = 4/5, we can square both sides:
(4/5)^2 + cos^2(A) = 1
16/25 + cos^2(A) = 1
cos^2(A) = 1 - 16/25
cos^2(A) = 9/25
cos(A) = ±√(9/25)
cos(A) = ±3/5

Since A is in Quadrant II, cos(A) is negative:
cos(A) = -3/5

Given sin(B) = -3/5 and B is in Quadrant IV, we can use the Pythagorean identity to find cos(B).

Using the same steps mentioned above, we find that cos(B) = 4/5.

Now that we have the values of cos(A) and cos(B), we can use the cosine of the difference formula to find cos(A-B).

The cosine of the difference formula states that cos(A-B) = cos(A) * cos(B) + sin(A) * sin(B).

Substituting the values we found earlier, we get:
cos(A-B) = (-3/5) * (4/5) + (4/5) * (-3/5)
cos(A-B) = -12/25 - 12/25
cos(A-B) = -24/25

Therefore, cos(A-B) is equal to -24/25.