Problem 19 - Precalculus: Functions and Graphs

Question

Verify the identity. $$\frac{\sin u+\sin v}{\cos u+\cos v}=\tan \frac{1}{2}(u+v)$$

Step-by-Step Solution

Answer

The identity is verified using sum-to-product identities and simplification.

Step 1: Understand the Trigonometric Identity

The given problem is to verify that \( \frac{\sin u + \sin v}{\cos u + \cos v} \) equals \( \tan \frac{1}{2}(u+v) \). This identity suggests there might be trigonometric transformations involved in proving it. We will confirm if both sides of the equation are equal by using known trigonometric identities.

Step 2: Apply Sum-to-Product Identities

We start by transforming the numerator and the denominator separately using the sum-to-product identities. The identities are: \( \sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2} \) and \( \cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2} \). Applying them to \( \sin u + \sin v \) and \( \cos u + \cos v \), we have:\[\sin u + \sin v = 2\sin\frac{u+v}{2}\cos\frac{u-v}{2}\]\[\cos u + \cos v = 2\cos\frac{u+v}{2}\cos\frac{u-v}{2}\]

Step 3: Simplify the Fraction

Using the identities from Step 2, we simplify the fraction:\[\frac{\sin u + \sin v}{\cos u + \cos v} = \frac{2\sin\frac{u+v}{2}\cos\frac{u-v}{2}}{2\cos\frac{u+v}{2}\cos\frac{u-v}{2}}\]The \(2\cos\frac{u-v}{2}\) terms in the numerator and the denominator cancel out, simplifying to:\[\frac{\sin\frac{u+v}{2}}{\cos\frac{u+v}{2}}\]

Step 4: Conclude with Trigonometric Functions

The fraction \( \frac{\sin\frac{u+v}{2}}{\cos\frac{u+v}{2}} \) is the definition of \( \tan\frac{u+v}{2} \). Therefore, we verified the initial identity:\[\frac{\sin u + \sin v}{\cos u + \cos v} = \tan \frac{1}{2}(u+v)\]

Key Concepts

Sum-to-Product Identities

Trigonometry offers a variety of identities to transform one expression into another, making problem-solving more accessible. Sum-to-Product identities help convert sums of trigonometric terms into products, which can simplify complex expressions. These identities include:

• \(\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}\)
• \(\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}\)

Sum-to-Product identities are handy when handling problems that initially seem challenging to simplify. For example, in our original exercise, converting the sum of sines and cosines into products allowed us to clearly observe hidden patterns that facilitated the simplification process. It's like finding a hidden shortcut in a complex maze, turning a winding path into a straight line.
Applying these identities breaks down an equation, thus playing a crucial role in verification or simplification of trigonometric equations. So, when you encounter sums of sines or cosines, remember the sum-to-product identities for a simpler journey!

Simplifying Trigonometric Expressions

Simplification is the process of transforming complex trigonometric expressions into their simplest form. This often involves applying identities skillfully. In the context of our example, simplifying \(\frac{\sin u + \sin v}{\cos u + \cos v}\) required employing our derived product forms. Initially, our expression was a ratio of two sums of trigonometric functions. By applying sum-to-product identities to both the numerator and denominator, we managed to reframe it in a way that the components in the expression shared common factors. This simplification step allowed for a key cancellation to occur:

• The terms \(2\cos\frac{u-v}{2}\) canceled out from both the numerator and denominator.

After this cancellation, we were left with \(\frac{\sin\frac{u+v}{2}}{\cos\frac{u+v}{2}}\). The ability to simplify complex expressions makes trigonometry not only more understandable but also powerful, providing clear paths to solutions that initially appear daunting.

Verifying Trigonometric Identities

Verification involves proving that two sides of a trigonometric identity are equivalent by using known identities and algebraic manipulation. This process builds deeper understanding and confidence in trigonometric concepts. In our task, the goal was to prove that \(\frac{\sin u + \sin v}{\cos u + \cos v} = \tan \frac{1}{2}(u+v)\).The verification involved transforming the left side of the equation using known identities, particularly the sum-to-product identities, to simplify it into a recognizable form. By simplifying to \(\frac{\sin\frac{u+v}{2}}{\cos\frac{u+v}{2}}\), we connected the dots to its equivalent trigonometric function, \(\tan\frac{1}{2}(u+v)\). Verifying identities bridges the gap between algebraic manipulation and trigonometric relationships, ensuring that they maintain consistency across different contexts.
Through verification, students actively engage with trigonometric properties, gaining a robust understanding that reinforces their learning and problem-solving skills.