Problem 36 - Contemporary Precalculus

Question

Prove the subtraction identity for sine: $$ \sin (x-y)=\sin x \cos y-\cos x \sin y $$ I Hint: Use the first cofunction identity* $$ \sin (x-y)=\cos \left[\frac{\pi}{2}-(x-y)\right]=\cos \left[\left(\frac{\pi}{2}-x\right)+y\right] $$ and the addition identity for cosine. \(]\)

Step-by-Step Solution

Answer

Question: Prove the subtraction identity for sine using cofunction identity for sine and addition identity for cosine. Answer: The subtraction identity for sine is given by: $$ \sin(x-y) = \sin x \cos y - \cos x \sin y $$

Step 1: Write down the given subtraction identity and cofunction identity

We are given two identities: 1. Subtraction identity for sine (to be proved): \(\sin (x-y) = \sin x \cos y - \cos x \sin y\) 2. Cofunction identity for sine: \(\sin (x-y) = \cos\left(\frac{\pi}{2} - (x-y)\right) = \cos\left(\left(\frac{\pi}{2} - x\right) + y\right)\) Now, we will use the cofunction identity to find the expression for \(\sin(x-y)\) and then apply the addition identity for cosine.

Step 2: Write down the addition identity for cosine

The addition identity for cosine is given by: \(\cos (a+b) = \cos a \cos b - \sin a \sin b\) Here, \(a = \left(\frac{\pi}{2} - x\right)\) and \(b = y\). We will substitute these values in the above equation.

Step 3: Apply the addition identity for cosine

Plugging in the values of \(a\) and \(b\), we get: \(\cos\left(\left(\frac{\pi}{2} - x\right) + y\right) = \cos\left(\frac{\pi}{2} - x\right) \cos y - \sin\left(\frac{\pi}{2} - x\right) \sin y\) Now, let's rewrite \(\cos\left(\frac{\pi}{2} - x\right)\) and \(\sin\left(\frac{\pi}{2} - x\right)\) using cofunction identities. \(\cos\left(\frac{\pi}{2} - x\right) = \sin x\) \(\sin\left(\frac{\pi}{2} - x\right) = \cos x\) Substituting these back into the equation, we get: \(\sin(x-y) = \sin x \cos y - \cos x \sin y\)

Step 4: Conclusion

We have shown that by using the cofunction identity for sine and the addition identity for cosine, we obtain the subtraction identity for sine, as desired. Thus, the identity holds true: $$ \sin(x-y) = \sin x \cos y - \cos x \sin y $$

Key Concepts

Subtraction Identity for Sine

Understanding the subtraction identity for sine can be immensely helpful in solving trigonometric equations and simplifying complex expressions. This identity states that \[ \sin(x-y) = \sin x \cos y - \cos x \sin y \]The beauty of this formula lies in its ability to break down the sine of a difference of angles into more manageable products of sines and cosines.
- The subtraction identity is derived by transforming the difference of two angles into an expression involving basic sine and cosine functions.- It is especially valuable when dealing with angles that are not readily interpretable.- This identity simplifies calculations and aids in solving trigonometric equations.To grasp this concept deeply, it is crucial to understand how it is derived. The formula comes from the transformations involving cofunction identities and the addition identity for cosine. This derivation provides a foundation for understanding not only this identity but also the broader landscape of trigonometric manipulations.

Cofunction Identities

Cofunction identities provide a link between sine and cosine, showing how each function can be expressed in terms of the other. These identities are pivotal, especially when simplifying expressions or proving other identities. For sine, the cofunction identity is: \[ \sin(x) = \cos\left(\frac{\pi}{2} - x\right) \]
This identity tells us that the sine of an angle is equal to the cosine of its complement. Similarly, \[ \cos(x) = \sin\left(\frac{\pi}{2} - x\right) \]These relationships highlight the periodic nature and symmetry in trigonometric functions.- Cofunction identities are critical for transforming sine expressions to cosine and vice versa.- They give insight into the complementary nature of trigonometric functions.- Using cofunction identities can help solve equations where angles have complementary relationships.The elegance of cofunction identities lies in their simplicity and versatility, making them core tools in trigonometry.

Addition Identity for Cosine

The addition identity for cosine is an invaluable tool in trigonometry. It allows expressions of the cosine of a sum of angles in terms of simpler functions. The formula is:\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
This identity is instrumental when working with multiple angles, giving a method to express these compound angles in more fundamental trigonometric functions.- It is similar in form to the subtraction identity for sine, emphasizing the deep connections between trigonometric identities.- This identity allows for the breaking down of complex angles into simpler calculations, aiding in graphing and equation-solving tasks.- When used together with subtraction identities and cofunction identities, it broadens the scope of problems that can be tackled by trigonometry.The addition identity for cosine is more than just a formula; it is a gateway to understanding the interactions between trigonometric functions and solving real-world problems tied to angles and measurements.