Problem 70 - Algebra and Trigonometry

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 $$ \sin x=\cos \left(\frac{\pi}{2}-x\right) $$ $$\cos x=\sin \left(\frac{\pi}{2}-x\right)$$ to prove the Addition Formula for sine. \([\text { Hint: To get started, }\) 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 is proven as \( \sin(s+t) = \sin(s)\cos(t) + \cos(s)\sin(t) \).

Step 1: Apply Cofunction Identity

We start by using the given cofunction identity for sine. To express \( \sin(s+t) \) as something involving cosine, we use the identity \( \sin x = \cos \left(\frac{\pi}{2} - x\right) \). So, we write: \[\sin(s+t) = \cos\left(\frac{\pi}{2} - (s + t)\right).\] This step redefines \( \sin(s+t) \) in terms of cosine.

Step 2: Simplify Angle in Cosine Function

Next, simplify the angle in the cosine function using the expression obtained in Step 1:\[\cos\left(\frac{\pi}{2} - (s + t)\right) = \cos\left(\left(\frac{\pi}{2} - s\right) - t\right).\] This sets up the expression so that we can apply the Subtraction Formula for cosine.

Step 3: Apply Subtraction Formula for Cosine

Use the Subtraction Formula for cosine, \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), on the expression from Step 2:\[\cos\left(\left(\frac{\pi}{2} - s\right) - t\right) = \cos\left(\frac{\pi}{2} - s\right)\cos(t) + \sin\left(\frac{\pi}{2} - s\right)\sin(t).\]

Step 4: Use Cofunction Identities

Substitute in the cofunction identities \( \cos\left(\frac{\pi}{2} - s\right) = \sin(s) \) and \( \sin\left(\frac{\pi}{2} - s\right) = \cos(s) \): \[\sin(s+t) = \sin(s)\cos(t) + \cos(s)\sin(t).\] This step completes the transformation using the cofunction identities.

Key Concepts

Cofunction Identities

Cofunction identities are quite useful in trigonometry because they allow us to relate sine and cosine functions to each other. This relationship hinges on the concept of complementary angles. Complementary angles are two angles that add up to 90 degrees or \( \frac{\pi}{2} \) radians.

• The basic cofunction identity for sine is: \( \sin x = \cos \left( \frac{\pi}{2} - x \right) \). This means that the sine of an angle is equivalent to the cosine of its complement.
• Similarly, the cofunction identity for cosine is: \( \cos x = \sin \left( \frac{\pi}{2} - x \right) \).

These identities are incredibly useful when we need to transform one trigonometric function into another, especially in proofs and simplifying expressions. In the exercise, understanding and using the cofunction identities allowed for expressing \( \sin(s+t) \) in terms of cosine, paving the way for applying the subtraction formula for cosine.

Subtraction Formula for Cosine

The subtraction formula for cosine is an identity that helps us express the cosine of the difference between two angles, \( a - b \). This formula is written as:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]This identity is derived from the unit circle and is a core part of solving trigonometric equations. It allows us to break down more complex trigonometric expressions into simpler parts based on known angle measures.

• For example, if you know the cosine and sine values of individual angles \( a \) and \( b \), you can find the cosine of their difference by plugging those values into the formula above.
• In our exercise, we used the subtraction formula for cosine to expand \( \cos \left( \left(\frac{\pi}{2} - s\right) - t \right) \), which was crucial in transforming the expression into the standard addition formula for sine.

By understanding and applying this formula, solving trigonometric problems becomes more manageable and comprehensible.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the included variables. They are the backbone of any trigonometry course and are used extensively to simplify complex expressions.
Some of the key identities include the Pythagorean identity, reciprocal identities, and angle sum and difference identities, among others.

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \)
• Angle Sum Identities: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \) and, of course, the subtraction formula for cosine which we have already discussed.
• Reciprocal Identities: These relate the basic functions with their reciprocals, like \( \csc x = \frac{1}{\sin x} \).

In the exercise, we used these identities to derive the addition formula for sine from the subtraction formula for cosine. The relationships between these identities were leveraged to transform and simplify the expression, showcasing the power and utility of trigonometric identities in mathematical proofs and problem-solving.