tan A=-4/5;A in Quadrant 2 and cos B=-5/13: B in Quadrant 3.
What is the exact value of the expression cos(A+B)?
draw your triangles in standard position. It will be clear that
sinA = 4/√41
cosA = -5/√41
sinB = -12/13
cosB = -5/13
Now just plug that into the formula for cos(A+B)
Although I suspect a typo...
73*41^1/2/533
To find the exact value of the expression cos(A+B), we can use the trigonometric identity:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Let's find the values of sin A and sin B first:
Given that tan A = -4/5 in Quadrant 2, we can find sin A by using the Pythagorean identity: sin^2 A = 1 - cos^2 A.
Since tan A = -4/5, we have sin A = -4/5 and cos A = -3/5 (which is negative because it's in Quadrant 2). Plugging this into the Pythagorean identity, we get: 1 - (9/25) = 16/25 => sin A = -4/5.
Similarly, given that cos B = -5/13 in Quadrant 3, we can find sin B using the Pythagorean identity sin^2 B = 1 - cos^2 B.
So, sin B = sqrt(1 - cos^2 B) = sqrt(1 - (-5/13)^2) = sqrt(1 - 25/169) = sqrt(144/169) = 12/13 (which is positive because it's in Quadrant 3).
Now, let's substitute the values of sin A, cos A, sin B, and cos B into the trigonometric identity:
cos(A + B) = cos A × cos B - sin A × sin B
= (-3/5) × (-5/13) - (-4/5) × (12/13)
= 15/65 + 48/65
= 63/65
Therefore, the exact value of the expression cos(A + B) is 63/65.