θ is an angle such that 180∘<θ<270∘, and sinθ=−12/13. What is the value of
180(sin^2 θ/2+cos^2 θ/2 +tan^2 θ/2)?
To find the value of the expression 180(sin^2 θ/2+cos^2 θ/2 +tan^2 θ/2), we first need to determine the value of θ.
Given that sinθ = -12/13 and 180° < θ < 270°, we can use the fact that sinθ is negative in the second and third quadrants to determine the value of θ.
Since sinθ = -12/13, we can define a right-angled triangle where the opposite side is -12 and the hypotenuse is 13. This gives us a triangle in the third quadrant.
Let's apply the Pythagorean identity sin^2 θ + cos^2 θ = 1 to find the value of cosθ:
cos^2 θ = 1 - sin^2 θ
= 1 - (-12/13)^2
= 1 - 144/169
= 25/169
Taking the square root of both sides, we get:
cosθ = ± √(25/169)
= ± 5/13
Since θ is in the third quadrant, cosθ should be negative, so cosθ = -5/13.
Now, let's find the value of tanθ:
tanθ = sinθ/cosθ
= (-12/13) / (-5/13)
= 12/5
To find the value of θ, we need to determine the arctangent (or inverse tangent) of tanθ:
θ = arctan(12/5)
≈ 68.2°
Now that we know the value of θ, we can substitute it into the expression:
180(sin^2 θ/2+cos^2 θ/2 +tan^2 θ/2)
= 180(sin^2 (68.2°/2)+cos^2 (68.2°/2) +tan^2 (68.2°/2))
Using half-angle identities, we can simplify the expression:
sin^2 (θ/2) = (1 - cosθ)/2
cos^2 (θ/2) = (1 + cosθ)/2
tan^2 (θ/2) = (1 - cosθ)/(1 + cosθ)
Substituting the values of cosθ and simplifying the expression, we get:
180((1 - cosθ)/2 + (1 + cosθ)/2 + (1 - cosθ)/(1 + cosθ))
= 180((2 - 2cosθ + 1 + cosθ + 1 - cosθ)/(2 + cosθ))
Simplifying further:
180(4/2)
= 180(2)
= 360
Therefore, the value of 180(sin^2 θ/2+cos^2 θ/2 +tan^2 θ/2) is 360.