Problem 36
Question
Use the power-reducing formulas to rewrite each expression as an equivalent expression that does not contain powers of trigonometric functions greater than 1. $$ 10 \cos ^{4} x $$
Step-by-Step Solution
Verified Answer
The simplified equivalent expression for \(10 \cos ^{4} x\) using power-reducing formulas is \(\frac{10 + 10 \cos 2x + 1 + 2 \cos 4x }{8}\)
1Step 1: Apply the Power-Reducing Formula
Recognize that the power-reducing formula for \(\cos ^{2} x\) is \(\frac{1 + \cos 2x}{2}\). This will be applied to the original expression. But first, rewrite \(10 \cos ^{4} x\) as \((10 \cos ^{2} x) * (\cos ^{2} x)\) to make the formula applicable.
2Step 2: Simplifying the Resulting Expression
Substituting \(\cos ^{2} x\) using power-reducing formula in \((10 \cos ^{2} x) * (\cos ^{2} x)\), we have \((10 * \frac{1 + \cos 2x}{2}) * (\frac{1 + \cos 2x}{2})\). Simplifying by multiplying the terms, we have \(\frac{10 + 10 \cos 2x + \cos ^{2} 2x}{4}\).
3Step 3: Apply the Power-Reducing Formula Again
Again applying power-reducing formula for \(\cos ^{2} 2x\) in \(\frac{10 + 10 \cos 2x + \cos ^{2} 2x}{4}\), we get \(\frac{10 + 10 \cos 2x + \frac{1 + \cos 4x}{2}}{4}\). Subtracting the fraction and simplifying gives us the final expression \(\frac{10 + 10 \cos 2x + 1 + 2 \cos 4x }{8}\).
Key Concepts
Trigonometric IdentitiesSimplifying Trigonometric ExpressionsCosine Function
Trigonometric Identities
Trigonometric identities are mathematical statements that express one trigonometric function in terms of another. These identities are essential tools for simplifying trigonometric expressions. They include fundamental relations like the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), reciprocal identities like \( \sec x = \frac{1}{\cos x} \) and \( \csc x = \frac{1}{\sin x} \), and ratio identities like \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \).
Power-reducing formulas are a special set of trigonometric identities used to simplify expressions involving high powers of trigonometric functions to powers of one or less. One of these formulas expresses \( \cos^{2} x \) as \( \frac{1 + \cos 2x}{2} \) and is particularly useful when dealing with even powers of the cosine function, as seen in the provided exercise. By repeatedly applying these identities, we can break down complex expressions into much simpler terms, facilitating further mathematical operations such as integration, differentiation, or solving equations.
Power-reducing formulas are a special set of trigonometric identities used to simplify expressions involving high powers of trigonometric functions to powers of one or less. One of these formulas expresses \( \cos^{2} x \) as \( \frac{1 + \cos 2x}{2} \) and is particularly useful when dealing with even powers of the cosine function, as seen in the provided exercise. By repeatedly applying these identities, we can break down complex expressions into much simpler terms, facilitating further mathematical operations such as integration, differentiation, or solving equations.
Simplifying Trigonometric Expressions
The process of simplifying trigonometric expressions often involves the strategic use of trigonometric identities to transform the original problem into a more manageable form. When dealing with powers of trigonometric functions greater than one, power-reducing formulas come into play. By replacing each squared term with its power-reducing counterpart, the original expression can be rewritten in terms of first-degree trigonometric functions and constants.
In the step-by-step solution of the exercise, the power-reducing formula is applied sequentially to break down the fourth power of the cosine function into expressions with no trigonometric functions raised to a power greater than one. This process isn't just about applying formulas mechanically; it's about understanding when and how to use these identities to simplify the expression at hand effectively. Key to simplification is careful algebraic manipulation post-identity application, involving distribution, combining like terms, and simplifying fractions, resulting in an equivalent, yet simplified, expression.
In the step-by-step solution of the exercise, the power-reducing formula is applied sequentially to break down the fourth power of the cosine function into expressions with no trigonometric functions raised to a power greater than one. This process isn't just about applying formulas mechanically; it's about understanding when and how to use these identities to simplify the expression at hand effectively. Key to simplification is careful algebraic manipulation post-identity application, involving distribution, combining like terms, and simplifying fractions, resulting in an equivalent, yet simplified, expression.
Cosine Function
The cosine function is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the adjacent side to the length of the hypotenuse. The mechanical definition of the cosine function on a unit circle is the x-coordinate of a point determined by the angle x, measured from the positive x-axis.
The power-reducing formulas transform expressions with high powers of the cosine function by employing angles that are multiples of the original angle, such as 2x or 4x. For instance, the provided exercise used the power-reducing formula for \(\cos^{2} x\) to rewrite \(\cos^{4} x\). This demonstrates the cyclic nature of the cosine function, which repeats every \(2\pi\) radians, and how leveraging properties like evenness and periodicity of the function can be essential in simplifying complex trigonometric expressions. It is important to note the input angle may change as a byproduct of using these formulas, hence a student must not just manipulate the function, but also manage the transformations of the angle itself to secure the expression's equivalence.
The power-reducing formulas transform expressions with high powers of the cosine function by employing angles that are multiples of the original angle, such as 2x or 4x. For instance, the provided exercise used the power-reducing formula for \(\cos^{2} x\) to rewrite \(\cos^{4} x\). This demonstrates the cyclic nature of the cosine function, which repeats every \(2\pi\) radians, and how leveraging properties like evenness and periodicity of the function can be essential in simplifying complex trigonometric expressions. It is important to note the input angle may change as a byproduct of using these formulas, hence a student must not just manipulate the function, but also manage the transformations of the angle itself to secure the expression's equivalence.
Other exercises in this chapter
Problem 36
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