Given:
cos u = 3/5; 0 < u < pi/2
cos v = 5/13; 3pi/2 < v < 2pi
Find:
sin (v + u)
cos (v - u)
tan (v + u)
First compute or list the cosine and sine of both u and v.
Then use the combination rules
sin (v + u) = sin u cos v + cos v sin u.
cos (v - u) = cos u cos v + sin u sin v
and
tan (u + v) = [tan u + tan v]/[1 - tan u tan v]
We got sin u = 4/5 & sin v = SQRT (7/13)
if cos u=3/5
cos v=5/13
then sin u=4/5 and sin v=12/13
then sin(v+u)=sinv cos u+cos v sin u
sin(v+u)=56/65
cos(v-u)=63/65
Tan u=4/3 and tan v=12/5 then
tan(u+v)=-56/33
To find sin (v + u), we will use the formula sin (v + u) = sin u cos v + cos v sin u.
Given:
sin u = 4/5
sin v = sqrt(7/13)
cos u = 3/5 (given)
cos v = 5/13 (given)
Now substitute the values into the formula:
sin (v + u) = sin u cos v + cos v sin u
= (4/5)(5/13) + (3/5)(sqrt(7/13))
= (20/65) + (3/5)(sqrt(7/13))
To simplify the expression further, convert 20/65 to a simpler fraction.
20/65 = 4/13
= (4/13) + (3/5)(sqrt(7/13))
This is the simplified value of sin (v + u).
Now let's move on to the next part.
To find cos (v - u), we will use the formula cos (v - u) = cos u cos v + sin u sin v.
Given:
sin u = 4/5
sin v = sqrt(7/13)
cos u = 3/5 (given)
cos v = 5/13 (given)
Now substitute the values into the formula:
cos (v - u) = cos u cos v + sin u sin v
= (3/5)(5/13) + (4/5)(sqrt(7/13))
= (15/65) + (4/5)(sqrt(7/13))
And finally, to find tan (v + u), we will use the formula tan (v + u) = (tan u + tan v) / (1 - tan u tan v).
Given:
sin u = 4/5
sin v = sqrt(7/13)
cos u = 3/5 (given)
cos v = 5/13 (given)
To find tan u and tan v, we can use the identities tan u = sin u / cos u and tan v = sin v / cos v.
tan u = (4/5) / (3/5) = 4/3
tan v = (sqrt(7/13)) / (5/13) = sqrt(7/5)
Now substitute the values into the formula:
tan (v + u) = (tan u + tan v) / (1 - tan u tan v)
= (4/3 + sqrt(7/5)) / (1 - (4/3)(sqrt(7/5)))
This is the simplified value of tan (v + u).
So, the final answers are:
sin (v + u) = (4/13) + (3/5)(sqrt(7/13))
cos (v - u) = (15/65) + (4/5)(sqrt(7/13))
tan (v + u) = (4/3 + sqrt(7/5)) / (1 - (4/3)(sqrt(7/5)))
To find sin (v + u), we will use the sine addition formula:
sin (v + u) = sin u cos v + cos u sin v
Given that sin u = 4/5 and sin v = √(7/13), we can substitute these values into the formula:
sin (v + u) = (4/5)(5/13) + (3/5)(√(7/13))
= (20/65) + (3/5)√(7/13)
= 4/13 + (3/5)√(7/13)
Therefore, sin (v + u) = 4/13 + (3/5)√(7/13).
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To find cos (v - u), we will use the cosine subtraction formula:
cos (v - u) = cos u cos v + sin u sin v
Given that cos u = 3/5 and cos v = 5/13, we can substitute these values into the formula:
cos (v - u) = (3/5)(5/13) + (4/5)(√(7/13))
= (15/65) + (4/5)√(7/13)
= 3/13 + (4/5)√(7/13)
Therefore, cos (v - u) = 3/13 + (4/5)√(7/13).
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To find tan (v + u), we will use the tangent addition formula:
tan (v + u) = [tan u + tan v] / [1 - tan u tan v]
Given that tan u = sin u / cos u and tan v = sin v / cos v, we can substitute these values into the formula:
tan (v + u) = [ (sin u / cos u) + (sin v / cos v) ] / [ 1 - (sin u / cos u)(sin v / cos v) ]
= [ (4/5) / (3/5) + (√(7/13)) / (5/13) ] / [ 1 - (4/5)(√(7/13))(5/13) ]
= [ (4/5) / (3/5) + (√(7/13)) / (5/13) ] / [ 1 - (4/5)(√(7/13))(5/13) ]
To simplify further, we can find common denominators:
tan (v + u) = [ (4/5)(13/13) / (3/5)(13/13) + (√(7/13))(5/5) / (5/13)(5/5) ] / [ 1 - (4/5)(√(7/13))(5/13) ]
= [ 52/65 + (5√(7/13)) / 25 ] / [ 1 - (4/5)(√(7/13))(5/13) ]
Therefore, tan (v + u) = [ 52/65 + (5√(7/13)) / 25 ] / [ 1 - (4/5)(√(7/13))(5/13) ].