Answer :
I can assist you with finding the tangent of the difference between angles a and b (tan(a - b)) given that sin(a) = 3/5 and cos(b) = 5/13.
To find tan(a - b), we can use the tangent difference formula:
tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a) tan(b))
However, we're only given sine and cosine values, not tangent values. We can address this by using the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
This identity allows us to find one trigonometric ratio from another. In this case, we can solve for tan(a) and tan(b):
tan(a) = sin(a) / cos(a) = (3/5) / (sqrt(1 - (5/13)^2))
tan(b) = sin(b) / cos(b) = sqrt(1 - (5/13)^2) / (5/13) (Assuming sin(b) is positive in this context)
Now that we have expressions for tan(a) and tan(b), we can plug them into the tangent difference formula and simplify:
tan(a - b) = ((3/5) / (sqrt(1 - (5/13)^2))) - ((sqrt(1 - (5/13)^2)) / (5/13))) / (1 + ((3/5) / (sqrt(1 - (5/13)^2))) * ((sqrt(1 - (5/13)^2)) / (5/13)))
After simplifying, you'll obtain the value of tan(a - b).
To find tan(a - b), we can use the tangent difference formula:
tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a) tan(b))
However, we're only given sine and cosine values, not tangent values. We can address this by using the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
This identity allows us to find one trigonometric ratio from another. In this case, we can solve for tan(a) and tan(b):
tan(a) = sin(a) / cos(a) = (3/5) / (sqrt(1 - (5/13)^2))
tan(b) = sin(b) / cos(b) = sqrt(1 - (5/13)^2) / (5/13) (Assuming sin(b) is positive in this context)
Now that we have expressions for tan(a) and tan(b), we can plug them into the tangent difference formula and simplify:
tan(a - b) = ((3/5) / (sqrt(1 - (5/13)^2))) - ((sqrt(1 - (5/13)^2)) / (5/13))) / (1 + ((3/5) / (sqrt(1 - (5/13)^2))) * ((sqrt(1 - (5/13)^2)) / (5/13)))
After simplifying, you'll obtain the value of tan(a - b).