Problem 111 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos 3 \beta=\cos ^{3} \beta-3 \sin ^{2} \beta \cos \beta$$

Step-by-Step Solution

Answer

The given identity \(\cos 3 \beta = \cos ^3 \beta - 3 \sin ^2 \beta \cos \beta\) is indeed correct when we substitute \(\sin^2 \beta\) with \(1 - \cos^2 \beta\) and simplify.

Step 1: Recall Trig Identities

Recall the triple-angle identity for cosine, which says: \( \cos 3x = 4 \cos^3x - 3 \cos x\).

Step 2: Expand the given Identity

In the given identity, \(3 \beta\) is being taken as \(x\) in the triple-angle identity. Therefore, the right side of the equality can be rewritten as: \(4 \cos^3 \beta - 3 \cos \beta\). However, it actually is given as: \(\cos^3 \beta - 3 \sin^2 \beta \cos \beta\) but we know that \(\sin^2 \beta = 1 - \cos^2 \beta\), so we substitute this into our equation.

Step 3: Substitute and Simplify

Substitute \(\sin^2 \beta\) with \(1 - \cos^2 \beta\) in the given identity to get: \( \cos^3 \beta - 3 \cos \beta + 3 \cos^3 \beta\). Simplify this to obtain: \(4 \cos^3 \beta - 3 \cos \beta\). Comparing this with the triple-angle identity for cosine, \( \cos 3x = 4 \cos^3x - 3 \cos x\), it can be noted that both are identical, hence the given identity is verified.

Key Concepts

Triple-angle formulas

Triple-angle formulas are essential trigonometric identities that relate an angle multiplied by three to the trigonometric functions of the original angle. These are especially useful in simplifying expressions and solving equations that involve trig functions of multiple angles. The triple-angle formula for cosine is given by:

• \( \cos 3x = 4\cos^3x - 3\cos x \)

What this identity tells us is how to express \(\cos 3x\) in terms of \(\cos x\). Understanding these identities allows us to break down more complex trigonometric expressions into simpler components. By mastering this, students can navigate complex problems more easily. This is because the identity neatly eliminates any need for sine terms when expanding \(\cos 3x\).
Using this identity correctly requires practice, particularly in recognizing which elements in a math problem relate to it. Be sure to practice on various examples to become adept at identifying where and how to apply this formula.

Cosine function

The cosine function is one of the fundamental periodic functions in trigonometry. It relates the angle of a right triangle to the ratio of the length of the adjacent side over the hypotenuse. Cosine is written mathematically as \(\cos x\) and has a range of values from -1 to 1.
Applications of the cosine function are diverse, ranging from simple triangle calculations to complex waveforms in physics. In trigonometric identities, like the triple-angle formula, cosine plays a central role.
Here's a quick recap of its properties:

• Periodicity: The cosine function is periodic with a period of \(2\pi\).
• Symmetry: It is an even function, meaning \(\cos(-x) = \cos x\).

The understanding of the cosine function is essential, especially when verifying identities as seen in trigonometry exercises. Coupling cosine with other trig identities can simplify problems significantly.

Algebraic verification

Algebraic verification involves proving a mathematical identity by simplifying and manipulating algebraic expressions while respecting mathematical rules and known identities. In trigonometry, this often requires substituting known values or identities and simplifying to confirm both sides of an equation are equal.
For the exercise given, we start with the identity \(\cos 3\beta = \cos^3 \beta - 3\sin^2 \beta \cos \beta\), and through careful algebraic manipulation, simplify different parts while using the Pythagorean identity \(\sin^2\beta = 1 - \cos^2\beta\).
This substitution turns the expression into a form resembling the known triple-angle formula:

• Substitute \(\sin^2 \beta\) with \(1 - \cos^2 \beta\) in the expression.
• Combine and rearrange to achieve \(4\cos^3 \beta - 3\cos \beta\).

Once simplified, both expressions match, confirming the identity is valid. Successfully proving this requires a good grasp of algebra and trigonometric identities and demonstrates the interconnected nature of these mathematical principles.