If sin theeta=3/5 , cos alpha= -12/13 and theeta,alpha both lie in 2nd quadrant,find the value of tan (theeta+alpha).
To find the value of tan (θ + α), we need to find the values of sin(θ) and cos(α) first.
Given:
sin(θ) = 3/5 (θ lies in the 2nd quadrant)
cos(α) = -12/13 (α lies in the 2nd quadrant)
To find cos(θ), we can use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1.
Since sin(θ) = 3/5, we can find cos(θ) as follows:
cos^2(θ) = 1 - sin^2(θ)
cos^2(θ) = 1 - (3/5)^2
cos^2(θ) = 1 - 9/25
cos^2(θ) = (25 - 9)/25
cos^2(θ) = 16/25
Taking the square root of both sides, we get:
cos(θ) = ±√(16/25)
Since θ is in the 2nd quadrant, where cosine is negative, we have:
cos(θ) = -4/5
Now that we have sin(θ) and cos(θ), we can find tan(θ) using the identity: tan(θ) = sin(θ)/cos(θ):
tan(θ) = (3/5) / (-4/5)
tan(θ) = -3/4
Similarly, we can use the Pythagorean identity to find sin(α):
sin^2(α) + cos^2(α) = 1
sin^2(α) + (-12/13)^2 = 1
sin^2(α) = 1 - (144/169)
sin^2(α) = (169 - 144)/169
sin^2(α) = 25/169
Taking the square root of both sides, we get:
sin(α) = ±√(25/169)
Since α is in the 2nd quadrant, where sine is positive, we have:
sin(α) = 5/13
Now that we have sin(α) and cos(α), we can find tan(α) using the identity: tan(α) = sin(α)/cos(α):
tan(α) = (5/13) / (-12/13)
tan(α) = -5/12
Finally, we can find tan(θ + α) using the identity: tan(θ + α) = (tan(θ) + tan(α)) / (1 - tan(θ)tan(α)):
tan(θ + α) = (tan(θ) + tan(α)) / (1 - tan(θ)tan(α))
tan(θ + α) = (-3/4 + -5/12) / (1 - (-3/4)(-5/12))
tan(θ + α) = (-9/12 - 5/12) / (1 - 15/48)
tan(θ + α) = (-14/12) / (1 - 5/16)
tan(θ + α) = (-7/6) / (16/16 - 5/16)
tan(θ + α) = (-7/6) / (11/16)
To simplify, we can multiply the numerator and denominator of the fraction by 16:
tan(θ + α) = (-7/6) * (16/11)
tan(θ + α) = -112/66
The value of tan(θ + α) is -112/66, which can be simplified further if required.