Trigonometric Functions: Sum and Difference Formulas

Sum and Difference Formulas for the Sine Function

The sum and difference formulas for sine allow us to compute the sine of the sum or difference of two angles using the sines and cosines of the individual angles. These formulas are fundamental in trigonometry and are frequently used in precalculus to simplify expressions and solve equations.

• Sum Formula:

• Difference Formula:

Example: Find the exact value of using sum and difference formulas.

• Let

• Apply the difference formula:

Substitute known values:

Calculate:

Sum and Difference Formulas for the Cosine Function

The sum and difference formulas for cosine are used to find the cosine of the sum or difference of two angles. These formulas are essential for simplifying trigonometric expressions and solving equations involving cosine.

• Sum Formula:

• Difference Formula:

Example: If with , and with , find the exact values of , , , and .

• Find and using the Pythagorean identity .

• For : (since is in the second quadrant, cosine is negative).

• For : (since is in the first quadrant, cosine is positive).

• Apply the sum formula:

Sum and Difference Formulas for the Tangent Function

The sum and difference formulas for tangent are used to find the tangent of the sum or difference of two angles. These formulas are particularly useful for simplifying expressions and solving trigonometric equations involving tangent.

• Sum Formula:

• Difference Formula:

Example: Establish the identity .

• Start with the definition:

• Apply sum formulas for sine and cosine:

• Divide numerator and denominator by :

Summary Table: Sum and Difference Formulas

The following table summarizes the sum and difference formulas for sine, cosine, and tangent:

Applications: These formulas are used to simplify trigonometric expressions, solve equations, and find exact values for trigonometric functions at non-standard angles.