Problem 70 - Precalculus Mathematics for Calculus

Question

Addition Formula for Sine In the text we proved only the Addition and Subtraction Formulas for cosine. Use these formulas and the cofunction identities $$ \begin{aligned} \sin x &=\cos \left(\frac{\pi}{2}-x\right) \\ \cos x &=\sin \left(\frac{\pi}{2}-x\right) \end{aligned} $$ to prove the Addition Formula for Sine. [Hint: To get started, use the first cofunction identity to write $$ \begin{aligned} \sin (s+t) &=\cos \left(\frac{\pi}{2}-(s+t)\right) \\ &=\cos \left(\left(\frac{\pi}{2}-s\right)-t\right) \end{aligned} $$ and use the Subtraction Formula for cosine. \(]\)

Step-by-Step Solution

Answer

The Addition Formula for Sine: \( \sin(s+t) = \sin(s)\cos(t) + \cos(s)\sin(t) \).

Step 1: Use Cofunction Identity

Start by expressing the sine addition using the cofunction identity. For \( \sin(s+t) \), write it in terms of cosine: \[ \sin(s+t) = \cos\left(\frac{\pi}{2} - (s+t)\right) \] Use the given identity.

Step 2: Apply Cofunction Identity Rearrangement

Rearrange the expression using the identity: \[ \cos\left(\frac{\pi}{2} - (s+t)\right) = \cos\left((\frac{\pi}{2} - s) - t\right) \] This allows the use of the cosine subtraction formula.

Step 3: Use the Cosine Subtraction Formula

Apply the subtraction formula for cosine to \( \cos((\frac{\pi}{2} - s) - t) \): \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] where \( A = \frac{\pi}{2} - s \) and \( B = t \). Substitute these into the equation.

Step 4: Substitute Values

Substitute \( A = \frac{\pi}{2} - s \) and \( B = t \) into the cosine subtraction formula:\[ \cos((\frac{\pi}{2} - s) - t) = \cos(\frac{\pi}{2} - s)\cos(t) + \sin(\frac{\pi}{2} - s)\sin(t) \]

Step 5: Simplify Using Cofunction Identities

Simplify using the cofunction identities: \[ \cos(\frac{\pi}{2} - s) = \sin(s) \quad \text{and} \quad \sin(\frac{\pi}{2} - s) = \cos(s) \] Substitute these back into the equation.

Step 6: Simplify Further

Substituting the identities, we get: \[ \sin(s+t) = \sin(s)\cos(t) + \cos(s)\sin(t) \] This is the sine addition formula.

Key Concepts

Cofunction Identities

Cofunction identities are a fascinating part of trigonometry that relate sine and cosine functions. These identities utilize the fact that sine and cosine are cofunctions of each other, meaning they complement one another. A cofunction identity can be expressed as:

• \( \sin x = \cos\left(\frac{\pi}{2} - x\right) \)
• \( \cos x = \sin\left(\frac{\pi}{2} - x\right) \)

These equations demonstrate how the sine of an angle is equal to the cosine of its complement, and vice versa. In many trigonometry problems, the cofunction identities help simplify expressions and provide alternate perspectives on angles.
For instance, if you need to find \( \sin(s+t) \), it can be rewritten using the cofunction identity as \( \cos\left(\frac{\pi}{2}-(s+t)\right) \). This clever strategy allows us to connect sine and cosine, enriching our toolkit for solving trigonometric equations.

Cosine Subtraction Formula

The cosine subtraction formula is a powerful and versatile tool in trigonometry. It allows us to express the cosine of a difference of two angles \( A \) and \( B \) in terms of individual sines and cosines of these angles. The formula is as follows:

• \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)

This formula is instrumental in verifying identities and simplifying expressions in trigonometry. For example, when proving the sine addition formula, this cosine subtraction formula comes in handy.
In the given problem, we start with \( \cos\left(\left(\frac{\pi}{2} - s\right) - t\right) \). By applying the cosine subtraction formula, we explore the individual components \( \cos\left(\frac{\pi}{2} - s\right) \) and \( \cos(t) \), making the original expression more manageable.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variable. They are essential tools in simplifying and calculating trigonometric expressions, solving equations, and proving properties like the addition formulas.

• They include basic identities such as \( \sin^2 x + \cos^2 x = 1 \).
• They help transform complex expressions into simpler forms.

By using these identities strategically, you can break down even the toughest trigonometric equations. In the exercise, the use of trigonometric identities is critical to demonstrate how one form can translate into another. Specifically, cofunction identities transform sine to cosine, and subtraction formulas for cosine help express differences between angles.
Understanding and mastering these identities is vital for anyone studying math, as they lay the groundwork for solving a broad range of problems effectively.