If tan A = 2 and A belongs to [pie,3pie/2] then the expression cos A divided by sin cubed A +cos cubed A is equal to what?
tanA = 2 in QIII means
sinA = -2/√5
cosA = -1/√5
cosA/(sin^3A + cos^3A)
= -1/√5 / (-8/5√5 - 1/5√5)
= -1/√5 / (-9/5√5)
= -1/√5 * -5√5/9
= 5/9
that's PI, not PIE!
To evaluate the expression cos A divided by sin cubed A + cos cubed A, we need to find the values of cos A and sin A.
Given: tan A = 2 and A belongs to [pi, 3pi/2]
Since tan A = 2, we can use the identity tan A = sin A / cos A to express sin A and cos A in terms of tan A.
Let's solve for sin A first:
tan A = sin A / cos A
2 = sin A / cos A
sin A = 2 cos A
Now, we can substitute this relationship into the Pythagorean Identity for sin A and cos A:
sin^2 A + cos^2 A = 1
(2 cos A)^2 + cos^2 A = 1
4 cos^2 A + cos^2 A = 1
5 cos^2 A = 1
cos^2 A = 1/5
Taking the square root of both sides, we find:
cos A = ± √(1/5)
Since A belongs to [pi, 3pi/2], we know that cos A < 0 for this range. Therefore, we take the negative value:
cos A = - √(1/5)
Now, substitute this value of cos A into the expression cos A divided by sin cubed A + cos cubed A:
cos A / (sin^3 A + cos^3 A)
(-√(1/5)) / (sin^3 A + cos^3 A)
To simplify further, we need to express sin^3 A and cos^3 A in terms of cos A:
sin^3 A = (2 cos A)^3
= 8 cos^3 A
cos^3 A = (-√(1/5))^3
= -√(1/125)
= -1/√125
Substituting back into the expression:
(-√(1/5)) / (8 cos^3 A + cos^3 A)
(-√(1/5)) / (9/√125)
(-√5 / 5) / (9√125)
-√5 / (45√5)
Finally, simplifying by canceling out the square roots:
-1 / 45
Therefore, the expression cos A divided by sin cubed A + cos cubed A is equal to -1/45.