Can i have a solution on how i can derive on this answer 4cos^3x - 3 cos x.....please..
after doing what ?
Ah. If you just wanted to verify that
cos 3x = 4cos^3 x - 3cos x, then consider
cos 3x = cos(2x+x)
= cos(2x)cos(x) - sin(2x)sin(x)
= (2cos^2(x)-1)(cos x) - (2sin x cos x)(sin x)
= 2cos^3(x)-cos(x) - 2sin^2(x)cos(x)
= 2cos^3(x) - cos(x) - 2(1-cos^2(x))cos(x)
= 2cos^3x - cosx - 2cosx + 2cos^3x
= 4cos^3x - 3cosx
Of course! I can help you derive the given expression, which is 4cos^3(x) - 3cos(x). To do this, we'll need to use the derivative rules for trigonometric functions. Let's break down the steps:
Step 1: Rewrite the expression using a power rule.
- cos^3(x) is equivalent to (cos(x))^3
Step 2: Use the chain rule for the power function.
- Take the derivative of the outer function, which is the power function.
- Multiply it by the derivative of the inner function, which is the cosine function.
Step 3: Differentiate each term separately.
- The derivative of 4(cos(x))^3 is 12(cos(x))^2 * (-sin(x)), using the chain rule.
- The derivative of 3cos(x) is -3sin(x), using the derivative of cosine.
Step 4: Combine the differentiated terms.
- Putting the derived terms together, we have: 12(cos(x))^2 * (-sin(x)) - 3sin(x).
- Simplifying further, we get: -12cos^2(x)sin(x) - 3sin(x).
Therefore, the derivative of the expression 4cos^3(x) - 3cos(x) is -12cos^2(x)sin(x) - 3sin(x).
If you want to know the solutions to
4cos^3x - 3 cos x = 0,
cos x (4cos^2 x - 3) = 0
so,
cos x = 0
or
cos x = ±√3/2
You should recognize those values, as they come up often.