Problem 64 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ \frac{\sin \alpha \cos \beta+\cos \alpha \sin \beta}{\cos \alpha \cos \beta-\sin \alpha \sin \beta}=\frac{\tan \alpha+\tan \beta}{1-\tan \alpha \tan \beta} $$

Step-by-Step Solution

Answer

The identity is verified using angle addition formulas for sine and cosine.

Step 1: Recall Trigonometric Identities

We start with the left-hand side (LHS) of the equation. Notice that the numerator \( \sin \alpha \cos \beta + \cos \alpha \sin \beta \) resembles the angle addition formula for sine, \( \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \). The denominator \( \cos \alpha \cos \beta - \sin \alpha \sin \beta \) mirrors the angle addition formula for cosine, \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).

Step 2: Express LHS as a Trigonometric Identity

Substitute the known angle addition formulas into the LHS of the given identity. Rewrite the equation as: \( \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \).

Step 3: Simplify LHS using Tangent Identity

The expression \( \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)} \) is equivalent to \( \tan(\alpha + \beta) \). Therefore, the LHS simplifies to \( \tan(\alpha + \beta) \).

Step 4: Verify with Right-Hand Side (RHS)

We use the tangent addition formula for \( \tan(\alpha + \beta) \), which is given by: \( \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \). Note that the RHS of our original equation is \( \frac{\tan \alpha + \tan \beta}{1-\tan \alpha \tan \beta} \).

Step 5: Confirm Both Sides are Equal

Both \( \tan(\alpha + \beta) \) from the LHS and the expression \( \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta} \) from the RHS are equal according to the tangent addition identity. Therefore, the given identity is verified.

Key Concepts

Angle Addition Formulas

The angle addition formulas are essential tools in trigonometry. They help us express the sine, cosine, and tangent of sums of angles. Let’s break them down:

For sine, the angle addition formula is:
\[\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\]
This means that the sine of the sum of two angles is the sum of the product of their sine and cosine pairs. It’s a handy identity that shows how we can deconstruct complex trigonometric expressions into simpler components.

For cosine, we have a similar formula:
\[\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]
Notice the subtraction here, which is a characteristic feature of the cosine addition formula. It allows us to express the cosine of an angle sum by using the cosines and sines of the individual angles. Both these formulas are pivotal in simplifying and solving trigonometric identities and proofs.

Sine and Cosine Functions

Sine and cosine are fundamental functions in trigonometry, reflecting waves and oscillations found in physics, engineering, and even finance. Each function has unique properties.

The sine function, \( \sin \theta \), starts at zero, reaches a maximum of 1, and a minimum of -1. It's a periodic function with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians.

• Sine represents the y-coordinate of a unit circle.
• It provides the "rise" or vertical component of an angle.

Cosine, denoted as \( \cos \theta \), behaves similarly but starts at 1. It also oscillates between -1 and 1 and has a period of \(2\pi\).

• Cosine represents the x-coordinate of a unit circle.
• It delivers the "run" or horizontal component of an angle.

These functions are not just theoretical. They have practical applications in modeling periodic phenomena like tides, sound waves, and light.

Tangent Function

The tangent function, \( \tan \theta \), is the ratio of the sine to the cosine of an angle:
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
This identity shows how tangent connects to the basic sine and cosine functions. Unlike sine and cosine, the tangent function does not have a maximum or minimum value, as it can approach positive and negative infinity. This occurs when \( \cos \theta = 0 \), causing the function to become undefined.

• Tangent represents the slope of the line formed by the angle with the positive x-axis when viewed on a coordinate plane.
• Its period is \( \pi \), meaning it repeats every \( \pi \) radians.

The tangent function is especially useful in calculus and real-world applications involving slopes and angles. It also plays a key role in the tangent addition formula. For angles \( \alpha \) and \( \beta \), the formula is:
\[\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}\]
This formula helps in simplifying expressions involving sums of angles, as seen in the exercise. It transforms a seemingly complex identity into a more manageable equation, highlighting the power of trigonometric identities.