Angles A and B are located in the first quadrant. If sin a =5/13 and cos b= 3/5 , determine the exact value of cos (a+b) .
well, we have
cosa = 12/13
sinb = 4/5
So, just plug them into
cos(a+b) = cosa cosb - sina sinb
I got the answer .22882 is that correct?
To determine the exact value of cos (a+b), we can use the cosine addition formula:
cos (a+b) = cos a * cos b - sin a * sin b
Given that sin a = 5/13 and cos b = 3/5, we need to find the values of cos a and sin b in order to calculate cos (a+b).
To find cos a, we can use the Pythagorean identity: sin^2 a + cos^2 a = 1.
Since sin a = 5/13, we can find cos a as follows:
sin^2 a + cos^2 a = 1
(5/13)^2 + cos^2 a = 1
25/169 + cos^2 a = 1
cos^2 a = 1 - 25/169
cos^2 a = 144/169
cos a = √(144/169)
cos a = 12/13
Next, to find sin b, we can use the Pythagorean identity: sin^2 b + cos^2 b = 1.
Since cos b = 3/5, we can find sin b as follows:
sin^2 b + cos^2 b = 1
sin^2 b + (3/5)^2 = 1
sin^2 b + 9/25 = 1
sin^2 b = 1 - 9/25
sin^2 b = 16/25
sin b = √(16/25)
sin b = 4/5
Now that we have the values of cos a, sin b, sin a, and cos b, we can substitute them into the cosine addition formula to calculate cos (a+b):
cos (a+b) = cos a * cos b - sin a * sin b
cos (a+b) = (12/13) * (3/5) - (5/13) * (4/5)
cos (a+b) = 36/65 - 20/65
cos (a+b) = 16/65
Therefore, the exact value of cos (a+b) is 16/65.
To determine the exact value of cos(a+b), we can use the sum of angles trigonometric identity:
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
Given that sin(a) = 5/13 and cos(b) = 3/5, we can substitute these values into the formula:
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
= (5/13)(3/5) - (5/13)(sin(b))
Now, we need to find sin(b).
Using the Pythagorean identity sin^2(b) + cos^2(b) = 1, we can substitute the given value of cos(b) and solve for sin(b):
sin^2(b) + (3/5)^2 = 1
sin^2(b) + 9/25 = 1
sin^2(b) = 1 - 9/25
sin^2(b) = 25/25 - 9/25
sin^2(b) = 16/25
Since sin(b) is positive in the first quadrant, we take the positive square root:
sin(b) = sqrt(16/25) = 4/5
Now, we can substitute this value into our original equation:
cos(a+b) = (5/13)(3/5) - (5/13)(4/5)
= 15/65 - 20/65
= -5/65
Therefore, the exact value of cos(a+b) is -5/65.