If sinA=3/5 when pi/2 < A < pi and cosB=5/13 when 3pi/2 < B < 2pi, find the exact value of the function cos(5pi/6+B).

Draw your angles in the proper quadrants.

Recall again that
sinθ = y/r
cosθ = x/r
Given the points for A and B, and recall your basic Pythagorean triples,
sinA = 3/5
cosA = -4/5
sinB = -12/13
cosB = 5/13

cos(5π/6 + B) = cos(5π/6) cosB - sin(5π/6) sinB
Now just plug and chug. Did angle A get lost?

To find the exact value of the function cos(5π/6 + B), we'll use the following trigonometric identities:

1. cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b)
2. cos(a) = sin(π/2 - a)

Let's break down the problem step by step:

Step 1: Identify the given information:
- sin(A) = 3/5, where π/2 < A < π
- cos(B) = 5/13, where 3π/2 < B < 2π

Step 2: Use the cosine identity cos(a) = sin(π/2 - a) to find cos(A):
Since sin(A) = 3/5, we can use Pythagorean identity to find cos(A):

cos(A) = √(1 - sin^2(A))
cos(A) = √(1 - (3/5)^2)
cos(A) = √(1 - 9/25)
cos(A) = √(16/25)
cos(A) = 4/5

Step 3: Plug the values into the cosine identity cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b):
cos(5π/6 + B) = cos(5π/6) * cos(B) - sin(5π/6) * sin(B)

Step 4: Evaluate cos(5π/6) and sin(5π/6):
5π/6 is an angle in the second quadrant, where cos is negative and sin is positive.

cos(5π/6) = -cos(π/6) = -√3/2
sin(5π/6) = sin(π/6) = 1/2

Step 5: Substitute the values obtained into the equation from Step 3:
cos(5π/6 + B) = (-√3/2) * (5/13) - (1/2) * (5/13)
cos(5π/6 + B) = -5√3/26 - 5/26
cos(5π/6 + B) = (-5√3 - 5) / 26

Therefore, the exact value of the function cos(5π/6 + B) is (-5√3 - 5) / 26.

To find the value of cos(5pi/6 + B), we need to use the trigonometric identity for the sum of angles:

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

First, let's find the values of sinA and cosB:

Given that sinA = 3/5 when pi/2 < A < pi, we can use the Pythagorean identity to find cosA:

cosA = sqrt(1 - sin^2A) = sqrt(1 - (3/5)^2) = sqrt(1 - 9/25) = sqrt(16/25) = 4/5

Given that cosB = 5/13 when 3pi/2 < B < 2pi, we can use the Pythagorean identity to find sinB:

sinB = sqrt(1 - cos^2B) = sqrt(1 - (5/13)^2) = sqrt(1 - 25/169) = sqrt(144/169) = 12/13

Now, let's substitute the values of sinA, cosA, sinB, and cosB into the trigonometric identity:

cos(5pi/6 + B) = cos(5pi/6) * cosB - sin(5pi/6) * sinB

Since cos(5pi/6) = cos(pi/6) = sqrt(3)/2 and sin(5pi/6) = sin(pi/6) = 1/2, we can simplify the expression:

cos(5pi/6 + B) = (sqrt(3)/2) * (5/13) - (1/2) * (12/13)
= (5sqrt(3) - 6)/26

Therefore, the exact value of the function cos(5pi/6 + B) is (5sqrt(3) - 6)/26.