14-5 Sum and Difference of Angles Formulas

The Formulas

Example 5-1a Find the exact value of sin 75 . Use the formula. Sum of angles Evaluate each expression. Example:

Example 5-1b Multiply. Simplify. Answer:

Example 5-1c Find the exact value of cos (–75  ). Use the formula Difference of angles Evaluate each expression. Another one:

Example 5-1d Multiply. Simplify. Answer:

Example 5-1e Find the exact value of each expression. a. sin 105  b. cos (–120  ) Answer: More:

Example 5-3a Verify that is an identity. Difference of angles formula Evaluate each expression. Simplify. Original equation Answer: Another one:

Example 5-3b Verify that is an identity. Simplify. Answer: Original equation Difference of angles formula Evaluate each expression. Another one:

Example 5-3c Verify that each of the following is an identity. a. Answer: Another one:

14-6 Double and Half Angle Formulas

Example 6-1a Find the value of if and is between Use the identity First find the value of Subtract. Example:

Example 6-1a Find the square root of each side. Since is in the first quadrant, cosine is positive. Thus,

Example 6-1a Now find Double-angle formula Simplify. Answer: The value of Example:

Example 6-1a Find the value of if and is between Double-angle formula Simplify. Answer: The value of cos 2  Example:

Example 6-1b Find the value of each expression if and is between a. b. Answer: Some more:

Example 6-2a Findis in the second quadrant. we must findfirst. Since Example:

Example 6-2a Simplify. Take the square root of each side. Sinceis in the second quadrant, Half-angle formula

Example 6-2a Simplify the radicand. Rationalize. Multiply.

Example 6-2a Thus, is Answer: Sinceis between positive and equals

Example 6-2b Findis in the fourth quadrant. Answer: Yep, there is more:

Example 6-3a Find the exact value ofby using the half-angle formulas. Example:

Example 6-3a Simplify the radicand. Simplify the denominator. Answer:

Example 6-3a Find the exact value ofby using the half-angle formulas. Example:

Example 6-3a Simplify the radicand. Simplify the denominator.

Example 6-3a Answer: Sinceis in the third quadrant, is negative. Thus,

Example 6-3b Find the exact value of each expression by using the half-angle formulas. a. b. Answer: A few more:

Example 6-4a Verify that is an identity. Answer: Original equation Distributive Property Simplify. Multiply. Example:

Example 6-4b Verify that is an identity. Answer: This could be the last one: