Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity algebraically. Use a graphing utility to check your result graphically. $$\sqrt{\frac{1-\cos \beta}{1+\cos \beta}}=\frac{1-\cos \beta}{|\sin \beta|}$$

Step-by-Step Solution

Answer

After simplifying and reordering, it's apparent that the left and right sides of the equation are equal, thus verifying the identity algebraically and this is also confirmed graphically by plotting the two sides of the equation.

Step 1: Rearrange The Identity

Start by squaring both sides, which gets rid of the square root on the left-hand side and gives: \[ \frac{1-\cos \beta}{1+\cos \beta} = \left(\frac{1-\cos \beta}{|\sin \beta|}\right)^2 \]

Step 2: Simplify the Right-Side

Next, rewrite the absolute value sign and simplify the right side to get: \[ \frac{1-\cos \beta}{1+\cos \beta} = \frac{(1-\cos \beta)^2}{\sin^2 \beta} \]

Step 3: Substitute with Trigonometric Identity

Replace \(1 - \cos^2 \beta\) with \(\sin^2 \beta\) to obtain: \[ \frac{1-\cos \beta}{1+\cos \beta} = \frac{\sin^2 \beta - 2 \cos \beta \sin^2 \beta + \cos^2 \beta \sin^2 \beta}{\sin^2 \beta} \] The right side simplifies to: \[ \frac{\sin^2 \beta - 2 \cos \beta \sin^2 \beta + \cos^2 \beta \sin^2 \beta}{\sin^2 \beta} = 1 - 2\cos \beta + \cos^2 \beta \] which is equal to the left side and confirms that the identity is true.

Step 4: Graphically Verification

In order to verify the identity graphically, plot the functions \(y = \sqrt{\frac{1-\cos \beta}{1+\cos \beta}}\) and \(y = \frac{1-\cos \beta}{|\sin \beta|}\). If the graphs overlap for all values of \(\beta\), then the original identity has been proven true.

Key Concepts

Trigonometric Simplification

Trigonometric simplification is all about using trigonometric identities to make an expression easier to work with. In the original problem, this involves reducing the given expression through a series of steps that utilize essential identities.
The first step in simplification was squaring both sides of the equation, which helps eliminate the square root, making the equation easier to manipulate. By squaring, we turn the left-hand side into \[ \frac{1-\cos \beta}{1+\cos \beta} \] and leave the right-hand side as a squared fraction.
This process used some crucial algebraic skills:

• Transforming an expression to reveal simpler components
• Using identities such as \( \sin^2 \beta + \cos^2 \beta = 1 \) to alter terms

Simplifying trigonometric expressions can save time and reduce complexity when solving identities or evaluating expressions.

Graphical Verification

Graphical verification involves using a graphing utility to confirm mathematical identities visually. It's a powerful tool that offers insight into how equations behave for various values, catching mistakes that might be missed through algebraic manipulation alone.
For the given problem, two expressions are graphed:

• \( y = \sqrt{\frac{1-\cos \beta}{1+\cos \beta}} \)
• \( y = \frac{1-\cos \beta}{|\sin \beta|} \)

You look for overlap between the graphs. If the plots cover the same points, it hints at the correctness of the original identity across all feasible values of \( \beta \).
This method is particularly useful when dealing with complex expressions or when the algebra does not yield an obvious simplification.

Square Root Properties

Understanding square roots is essential, especially when they appear in equations like the ones in this trigonometric problem. Square roots can often complicate algebraic expressions, but their properties can be leveraged to simplify problems.
In trigonometry, square roots are frequently seen in identities, as they help handle fractions and radical terms. Squaring the sides of an equation, as done in this problem, gets rid of the square root initially present, streamlining the expression significantly.
Essential properties of square roots include:

• The square root of a product, \( \sqrt{ab} = \sqrt{a} \sqrt{b} \)
• The square root of a quotient, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \)

Utilizing these properties correctly facilitates tackling more challenging algebraic identities.

Absolute Value in Trigonometry

The use of absolute values in trigonometry is critical when it involves expressions with potential positive or negative outcomes. Absolute values ensure the positivity of a term, which is essential in many trigonometric identities.
In the given exercise, the absolute value appears on the denominator of the right-hand side as \(|\sin \beta|\). This is necessary because sine's value can change from positive to negative depending on the angle \(\beta\). Applying the absolute value guards against negative results, providing a consistent positive denominator.
The takeaway from using absolute values in trigonometry includes:

• Stabilizing expressions regardless of input angle
• Guaranteeing mathematical accuracy and reducing unexpected behavior in identities

Grasping this concept is vital for students to effectively work with identities and avoid common pitfalls related to trigonometric sign changes.