Introduction:
The cosine triple angle formula is a fundamental trigonometric identity that relates the cosine of three times an angle to a combination of cosines of the angle itself and its double angle. In this article, we will explore the proof of this formula, providing a step-by-step explanation of the derivation.
Proof:
Let's consider an angle 'x'. We aim to express cos(3x) in terms of cos(x) and cos(2x).
Recall the double angle formula:
cos(2x) = 2cos^2(x) - 1 ...(1)
Now, let's expand cos(3x) using the angle addition formula:
cos(3x) = cos(2x + x)
Applying the angle addition formula for cosine, we have:
cos(3x) = cos(2x)cos(x) - sin(2x)sin(x)
To proceed further, we need to express sin(2x) in terms of cos(x) and sin(x). Using the double angle formula for sine:
sin(2x) = 2sin(x)cos(x) ...(2)
Substitute equation (2) into equation (3):
cos(3x) = cos(2x)cos(x) - 2sin(x)cos(x)sin(x)
Now, substitute equation (1) into equation (4):
cos(3x) = (2cos^2(x) - 1)cos(x) - 2sin(x)cos(x)sin(x)
Distribute and simplify:
cos(3x) = 2cos^3(x) - cos(x) - 2sin(x)cos(x)sin(x)
Apply the double angle formula for cosine:
cos(3x) = 2cos^3(x) - cos(x) - sin^2(x)cos(x)
Recall the Pythagorean identity: sin^2(x) = 1 - cos^2(x)
cos(3x) = 2cos^3(x) - cos(x) - (1 - cos^2(x))cos(x)
Simplify further:
cos(3x) = 2cos^3(x) - cos(x) - cos(x) + cos^3(x)
Combine like terms:
cos(3x) = 3cos^3(x) - 2cos(x)
Conclusion:
Through a series of algebraic manipulations and the use of double angle formulas, we have successfully derived the cosine triple angle formula: cos(3x) = 3cos^3(x) - 2cos(x). This identity provides a convenient way to express the cosine of three times an angle in terms of the cosine of the angle itself. Understanding and applying trigonometric formulas like this one can greatly simplify trigonometric calculations and problem-solving in various mathematical and scientific contexts.
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