Cosine-Sine Product Formula

Definition

The cosine-sine product formula is a trigonometric identity that expresses the product of the cosine and sine functions in terms of the sum and difference of two angles. This formula is particularly useful in the context of the topics covered in Section 9.4: Sum-to-Product and Product-to-Sum Formulas.

5 Must Know Facts For Your Next Test

1. The cosine-sine product formula states that $\cos(x)\sin(y) = \frac{1}{2}\sin(x+y) - \frac{1}{2}\sin(x-y)$.
2. This formula is derived from the angle sum and difference formulas for sine and cosine.
3. The cosine-sine product formula is useful for simplifying expressions involving the product of cosine and sine, and for transforming between product and sum/difference forms.
4. The cosine-sine product formula is a special case of the more general product-to-sum formulas, which can be used to express the product of any two trigonometric functions.
5. Understanding the cosine-sine product formula is essential for mastering the techniques covered in Section 9.4, which involve converting between product and sum/difference forms of trigonometric expressions.

Review Questions

• Explain how the cosine-sine product formula is derived from the angle sum and difference formulas for sine and cosine.
  • The cosine-sine product formula can be derived by applying the angle sum and difference formulas for sine and cosine. Specifically, the formula $\cos(x)\sin(y) = \frac{1}{2}\sin(x+y) - \frac{1}{2}\sin(x-y)$ is obtained by using the formula $\sin(A+B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ and rearranging the terms to isolate the product $\cos(x)\sin(y)$.
• Describe how the cosine-sine product formula can be used to transform between product and sum/difference forms of trigonometric expressions.
  • The cosine-sine product formula provides a way to convert a product of cosine and sine functions into a sum or difference of sine functions. This can be useful when simplifying trigonometric expressions or when working with identities and formulas that involve the product of cosine and sine. Conversely, the formula can also be used to express a sum or difference of sine functions as a product of cosine and sine, which is important for the techniques covered in Section 9.4 on converting between product and sum/difference forms.
• Analyze the significance of the cosine-sine product formula within the context of the topics covered in Section 9.4: Sum-to-Product and Product-to-Sum Formulas.
  • The cosine-sine product formula is a crucial tool for understanding and applying the techniques discussed in Section 9.4, which focus on converting between product and sum/difference forms of trigonometric expressions. By providing a way to express the product of cosine and sine as a sum or difference of sine functions, the cosine-sine product formula allows students to manipulate and simplify trigonometric expressions using the sum-to-product and product-to-sum formulas. This formula is essential for mastering the skills and concepts covered in this section, as it enables students to fluently move between different representations of trigonometric functions.

Related terms

Trigonometric Identity:

An equation involving trigonometric functions that is true for all values of the variables.

Angle Sum and Difference Formulas: Formulas that express the sine, cosine, and tangent of the sum or difference of two angles in terms of the sine, cosine, and tangent of the individual angles.

Product-to-Sum Formulas:

Formulas that express the product of two trigonometric functions as the sum or difference of two other trigonometric functions.