Problem 103 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\cos ^{2} 2 \alpha-\sin ^{2} 2 \alpha=\cos 4 \alpha$$

Step-by-Step Solution

Answer

Using the identity \( \sin^{2} \theta + \cos^{2} \theta = 1 \), it was shown that \( \cos^{2} (2 \alpha) - \sin^{2} (2\alpha) \) equals to \( 2\cos^{2}(2\alpha) - 1 \), which aligns with the expression for \( \cos(4\alpha) \). Thus, the given equation is verified algebraically. A graphical check would provide a visual confirmation of this result.

Step 1: Rewrite the expression

Start with the given equation, \( \cos^{2} (2 \alpha) - \sin^{2} (2\alpha) = \cos(4\alpha) \). The given expression can be rewritten using the identity \( \cos(4\alpha) = 2\cos^{2}(2\alpha) - 1\). So, our task is to show that \( \cos^{2} (2 \alpha) - \sin^{2} (2\alpha) \) equals to \( 2\cos^{2}(2\alpha) - 1 \).

Step 2: Substitute the identity

Knowing that \( \sin^{2} \theta + \cos^{2} \theta = 1 \) for any angle \( \theta \), we can substitute \( \sin^{2} (2\alpha) \) in our expression with \( 1 - \cos^{2} (2\alpha) \). Our equation becomes \( \cos^{2} (2 \alpha) - (1 - \cos^{2} (2\alpha)) \).

Step 3: Simplify the equation

Simplify the right side of the equation, and we get \( 2\cos^{2} (2\alpha) - 1 \), which is what the right side of the original equation was rewritten to in Step 1.

Step 4: Check graphically

To confirm the identity, it's always a good practice to graph the functions on either side of the equation to see if they're identical. If they cover the same graph, then it verifies the identity graphically.

Key Concepts

Cosine Function

Cosine, often abbreviated as "cos," is one of the basic trigonometric functions. Cosine is particularly important in the study of triangles, waves, and oscillations. It provides a relationship between the angle of a triangle and the adjacent side to the hypotenuse. The cosine function is periodic and oscillates between -1 and 1. This means that for any angle \( \theta \), \( \cos(\theta) \) will produce a value within that range.

• The value of \( \cos(0)\) is 1, while \( \cos(90^\circ)\) or \( \cos\left(\frac{\pi}{2}\right) \) is 0. This highlights a key feature of the cosine wave, where it reaches its maximum or minimum at these intervals.
• Cosine has various properties and identities, such as the Pythagorean Identity \( \cos^2 \theta + \sin^2 \theta = 1 \) where \(\sin\) is the sine function.

Managing these identities and using them efficiently is crucial when exploring other functions or solving complex trigonometric equations, especially when simplifying or verifying identities.

Double Angle Formulas

Double angle formulas are essential tools in trigonometry used to simplify expressions where angles are doubled. This notably helps in solving equations and verifying identities. For instance, the double angle formula for cosine is \( \cos(2\theta) = \cos^2 \theta - \sin^2 \theta \), which can be alternatively expressed using other well-known identities, such as \( \cos(2\theta) = 2\cos^2 \theta - 1 \) or \( \cos(2\theta) = 1 - 2\sin^2 \theta \).

• These formulas are derived from addition formulas such as \( \cos(a+b)\), which simplifies to \( \cos a \cos b - \sin a \sin b \).
• Knowing multiple forms of the double angle formulas is immensely helpful when manipulating and solving equations involving trigonometric functions.

Using the double angle formula in the cosine identity allows us to transform expressions to make them easier to handle, analyze, or verify.

Identity Verification

Identity verification in mathematics involves demonstrating that two expressions are equivalent under all circumstances, using known properties and logical steps. In trigonometry, verifying identities often means proving that a complex expression involving trigonometric functions simplifies to match another known form.

• This process typically starts with one side of the equation and manipulates it using known identities, like the Pythagorean identities or double angle formulas, to transform it into the other side.
• For example, in the given problem, to verify \( \cos^{2}(2\alpha) - \sin^{2}(2\alpha) = \cos(4\alpha) \), we use an identity for cosine of a double angle, showing the identity via algebraic substitution and simplification.

Graphical verification can also provide insight, where plotting both sides of the equation should yield the same graph, confirming the truth of the identity if it holds consistently across various angles. This approach not only fosters a deeper understanding of mathematical relationships but also helps build confidence in algebraic manipulation skills.