If sinA=1/2 then find cos^3A-3cosA.
To find the value of cos^3A - 3cosA when sinA = 1/2, we need to use a trigonometric identity.
First, we should find the value of cosA. Since sinA = 1/2, we know that A corresponds to an angle in the first or second quadrant, where sin is positive. In the first or second quadrant, there are two possible angles that have a sine value of 1/2: A = 30 degrees (π/6 radians) or A = 150 degrees (5π/6 radians).
Let's consider the angle A = 30 degrees (or π/6 radians). To find cosA, we can use the trigonometric identity: sin^2A + cos^2A = 1.
Since sinA = 1/2, we have:
(1/2)^2 + cos^2A = 1
1/4 + cos^2A = 1
cos^2A = 3/4
Taking the square root of both sides, we get:
cosA = ±√(3/4)
We know that A is in the first or second quadrant, so cosA must be positive, which gives:
cosA = √(3/4) = √3/2
Now, let's substitute the value of cosA into the expression cos^3A - 3cosA:
(√3/2)^3 - 3(√3/2)
Simplifying, we have:
(√3)^3 / (2)^3 - 3(√3/2)
(3√3)/8 - (3√3)/2
To simplify further, we need a common denominator:
(3√3)/(8) - (24√3)/(8)
Combining the terms, we get:
(-21√3) / 8
Therefore, when sinA = 1/2, the value of cos^3A - 3cosA is (-21√3) / 8.
To find the value of cos^3A - 3cosA, we can use the Pythagorean identity:
sin^2A + cos^2A = 1.
Since sinA = 1/2, we can substitute this value into the equation to find the value of cosA:
(1/2)^2 + cos^2A = 1.
1/4 + cos^2A = 1.
Now, we can solve this equation to find the value of cosA:
cos^2A = 1 - 1/4.
cos^2A = 3/4.
Taking the square root of both sides:
cosA = ± √(3/4).
cosA = ± √3/2.
Now, we can substitute this value of cosA into the expression cos^3A - 3cosA:
(√3/2)^3 - 3(√3/2).
(√27/8) - (√3/2).
Simplifying further, (√27/8) can be expressed as (√(9*3)/8) = (√9 * √3)/8 = (3√3/8).
So, the final answer is (3√3/8) - (√3/2).
if sin A = 1/2 then cos A = (sqrt 3)/2
because that is a 30, 60, 90 triangle
(1/8) 3 * 3^1/2- (3/2) * 3^1/2
= -(9/8) * 3^1/2