key term - Sum and Difference Formulas

Definition

The sum and difference formulas are a set of trigonometric identities that allow for the simplification and verification of trigonometric expressions involving the addition or subtraction of angles. These formulas are essential in the contexts of verifying trigonometric identities and solving trigonometric equations.

5 Must Know Facts For Your Next Test

1. The sum and difference formulas allow for the simplification of expressions involving the addition or subtraction of angles, such as $\sin(A + B)$ or $\cos(A - B)$.
2. These formulas are particularly useful in verifying trigonometric identities, as they provide a way to manipulate the expressions on both sides of the identity to ensure they are equivalent.
3. The sum and difference formulas are also essential in solving trigonometric equations, as they can be used to isolate the variable of interest and simplify the equation for easier solving.
4. The specific sum and difference formulas include $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$, $\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$, $\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$, and $\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$.
5. Understanding and applying the sum and difference formulas is crucial for success in topics related to verifying trigonometric identities and solving trigonometric equations.

Review Questions

• Explain how the sum and difference formulas can be used to simplify trigonometric expressions.
  • The sum and difference formulas allow you to rewrite trigonometric expressions involving the addition or subtraction of angles in a simpler form. For example, the formula $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ can be used to expand a expression like $\sin(30^\circ + 45^\circ)$ into a more manageable form. By applying the appropriate sum or difference formula, you can manipulate the trigonometric functions and simplify the overall expression.
• Describe the role of the sum and difference formulas in verifying trigonometric identities.
  • Verifying trigonometric identities often requires transforming the expressions on both sides of the identity to ensure they are equivalent. The sum and difference formulas provide a powerful tool for this process. By using the appropriate formula to rewrite the trigonometric functions in a different form, you can manipulate the expressions and demonstrate that they are indeed equal, thus verifying the identity. This is an essential skill in the context of 9.1 Verifying Trigonometric Identities.
• Analyze how the sum and difference formulas can be applied to solve trigonometric equations.
  • When solving trigonometric equations, the sum and difference formulas can be used to isolate the variable of interest and simplify the equation. For example, in the equation $\sin(x + 30^\circ) = 0.5$, you can use the formula $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ to rewrite the left-hand side and solve for the unknown angle $x$. By strategically applying the appropriate sum or difference formula, you can transform the trigonometric equation into a more manageable form, facilitating the solving process in the context of 9.5 Solving Trigonometric Equations.

Related terms

Trigonometric Identities:

Trigonometric identities are mathematical equations that hold true for all values of the variables involved, providing a way to simplify and manipulate trigonometric expressions.

Trigonometric Functions:

The basic trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant, which describe the relationships between the sides and angles of a right triangle.

Angle Addition and Subtraction: The process of adding or subtracting angles, which can be simplified using the sum and difference formulas.