Using Fundamental Identities Objectives: 1.Recognize and write the fundamental trigonometric identities 2.Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions

WHY??? Fundamental trigonometric identities can be used to simplify trigonometric expressions.

Fundamental Trigonometric Identities Reciprocal Identities Quotient Identities

Fundamental Trigonometric Identities Pythagorean Identities Even/Odd Identities

Fundamental Trigonometric Identities Cofunction Identities

Example: If and Ө is in quadrant II, find each function value. a) sec Ө To find the value of this function, look for an identity that relates tangent and secant. Tip: Use Pythagorean Identities.

b) sin Ө 7 c) cot (  Ө ) Example: If and Ө is in quadrant II, find each function value. (Cont.) Tip: Use Quotient Identities. Tip: Use Reciprocal and Negative-Angle Identities.

Use the values cos x > 0 and identities to find the values of all six trigonometric functions. What quadrant will you use? 1st quadrant Example:

Your Turn: Using Identities to Evaluate a Function  Use the given values to evaluate the remaining trigonometric functions  (You can also draw a right triangle)

Solution: #1

Solution: #2

Solution: #3

Simplify an Expression  Simplify cot x cos x + sin x to a single trigonometric function.

Example: Simplify 1.Factor csc x out of the expression.

2.Use Pythagorean identities to simplify the expression in the parentheses.

3.Use Reciprocal identities to simplify the expression.

Your Turn: Simplifying a Trigonometric Expression

Solutions:

Factoring Trigonometric Expressions -Factor the same way you would factor any quadratic. - If it helps replace the “trig” word with x -Factor the same way you would factor

Make it an easier problem. Let a = csc x 2a 2 – 7a + 6 (2a – 3)(a – 2) Now substitute csc x for a.

1.Use Pythagorean identities to get one trigonometric function in the expression.

2.Now factor.

Your Turn: Factoring Trigonometric Expressions

Solutions:

Your Turn: Factor and simplify

Solutions:

Adding Trigonometric Expressions (Common Denominator)

Your Turn: Adding Trigonometric Expressions

Solutions:

Rewriting a Trigonometric Expression so it is not in Fractional Form

Your Turn: Rewriting a Trigonometric Expression

Solution:

Trigonometric Substitution

Your Turn:

Solutions:

Assignment:  Sec 5.1 pg. 357 – 359: #1 – 13 odd, 15 – 26 all, odd, odd