Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Verify that each equation is an identity. $$\sin 4 \alpha=4 \sin \alpha \cos \alpha \cos 2 \alpha$$

Step-by-Step Solution

Answer

The equation is verified as an identity.

Step 1: Understand the Given Equation

We are tasked with verifying whether \(\sin 4\alpha = 4 \sin \alpha \cos \alpha \cos 2\alpha\) is an identity. This means proving that both sides of the equation are equal for all permissible values of \(\alpha\).

Step 2: Expand LHS using Double Angle Formulas

To begin, we use the double angle identity for sine: \( \sin 2A = 2 \sin A \cos A \). We can extend this for \(\sin 4\alpha\) by using \(\sin 4\alpha = \sin (2 \times 2\alpha) = 2 \sin 2\alpha \cos 2\alpha\).

Step 3: Substitute Double Angle Identity

Substitute \(\sin 2\alpha\) on the right side of the equation using the identity: \(\sin 2\alpha = 2\sin \alpha \cos \alpha\). Thus, we have \(2(2 \sin \alpha \cos \alpha) \cos 2\alpha\).

Step 4: Simplify LHS Expression

Simplifying the expression, \(2(2 \sin \alpha \cos \alpha) \cos 2\alpha = 4 \sin \alpha \cos \alpha \cos 2\alpha\). Now, this matches the right side of the original equation.

Step 5: Conclusion

Since the simplified form of the left-hand side equals the right-hand side, we have verified that \(\sin 4\alpha = 4 \sin \alpha \cos \alpha \cos 2\alpha\) is indeed an identity.

Key Concepts

Double Angle Formulas

When dealing with trigonometric functions, the double angle formulas can be incredibly useful. These formulas allow us to express trigonometric functions of double angles in terms of single angles.
For sine, the double angle formula is:

• \(\sin 2\theta = 2 \sin \theta \cos \theta\)

For cosine, you have a few options depending on the problem's requirements:

• \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\)
• \(\cos 2\theta = 2 \cos^2 \theta - 1\)
• \(\cos 2\theta = 1 - 2 \sin^2 \theta\)

In the problem we are solving, the double angle formula for sine helps us express \(\sin 4\alpha\) as \(2 \sin 2\alpha \cos 2\alpha\). By continuing to apply these identities, we break complex expressions down into simpler components that are easier to manage.

Sine Function

The sine function, often denoted as \(\sin(x)\), is one of the primary trigonometric functions. It is a periodic function representing the y-coordinate of a point on the unit circle as an angle x is traced.Si

• It's defined as the ratio of the opposite side to the hypotenuse in a right triangle.
• The range of sine is from -1 to 1.

Moreover, it is an odd function, which means that \(\sin(-x) = -\sin(x)\).
In trigonometric identities like the one we're verifying, \(\sin 2\alpha\) formula \(= 2 \sin \alpha \cos \alpha\) comes into play to transform expressions into equivalent forms, which can simplify our calculations.

Cosine Function

The cosine function, or \(\cos(x)\), is another key trigonometric function that pairs with sine to describe the properties of angles in circles and triangles. It represents the x-coordinate of a point on the unit circle.

• In a right triangle, it's the ratio of the adjacent side to the hypotenuse.
• The function's range is also from -1 to 1.

It is even, meaning \(\cos(-x) = \cos(x)\).
In the equation \(\sin 4\alpha = 4 \sin \alpha \cos \alpha \cos 2\alpha\), cosine is essential for expressing double angles. Using \(\cos 2\alpha\) as part of the identity allows us to maintain proportionate relationships within the trigonometric functions, serving to bridge the gap between complex expressions.

Verifying Identities

Verifying trigonometric identities is a vital skill in understanding trigonometry. The task involves proving that two expressions are equivalent for all values of the involved variable (here, \(\alpha\)).

• First, simplify one or both sides of the equation using known identities and algebraic manipulation.
• Utilize trigonometric formulas such as the Pythagorean identity or double angle formulas to substitute specific functions.

This exercise can include trying different approaches, such as factoring and substituting known identities like \(\sin 2\alpha = 2 \sin \alpha \cos \alpha\).
By the end, the goal is to show both sides are the same, indicating the identity is true. It's an excellent way to deepen understanding and conceptualize how different trigonometric functions interrelate.