Unit 3: Trigonometric Identities

Test WEDNESDAY 9/26 Unit 3: Trigonometric Identities MA3A5. Students will establish the identities and use them to simplify trigonometric expressions and verify equivalence statements. LG 3-1 Simplifying & Verifying Identities LG 3-2 Applying Trig Identities

Introduction - Complete the Doublets (word identities) and the Trigonometry Triangles (back) • Answer Key: #1 Change TEN to TWO. #2 Change TEN to SIX. T E N T E N T o N T i N T O o S I x T W o S I X #3 Change FIVE to FOUR. #4 Change PIG to STY F I V E P I G F I l E w I G F i l L w A G f a l l w A Y f a i l s A Y F O I L S T Y F O u L F O U R

Answer Key continued… #5 Change TEN to ONE. T E N t i n a metal t i p tilt l i p edge l i d cover a i d help a n d with (a conjuntion) a n t insect a c t perform a c e a card a r e a form of "to be" o r e mineral O N E

Answer Key continued… • Reciprocal Identities • Product Identities • Pythagorean Identities

What is a trigonometric identity? • A trigonometric identity is a trigonometric equation that is valid for all values of the variables for which the expression is defined. • In this unit, you will be manipulating expressions to make them equal something • When simplifying, you won’t know the answer • When verifying, you have the answer and your job is to manipulate one side of an equation to make it look like the other side

Unit 3: Trig Identities • You will now make a booklet with all the identities you are required to learn for this unit. • PLEASE PAY ATTENTION as we fold our booklets • On the FRONT COVER, write the title of the unit and YOUR NAME! • Do NOT lose your booklet! • Look for this symbol so you know when to write in your booklet Booklet

Booklet pg 1 Reciprocal Identities Also work with powers…

Booklet pg 2 Quotient Identities

Pythagorean Identities Draw a right triangle on a separate sheet of paper. Label one of the angles as . Label the hypotenuse with a length of 1 unit. Label the side opposite  as a and the side adjacent to  as b. Use the right-triangle ratio for sine to write an equation for sin(). Solve this equation for a. Use the right-triangle ratio for cosine to write an equation for cos(). Solve this equation for b. Write the formula for Pythagorean Theorem. Then substitute your expressions for a and b above and substitute the length of your hypotenuse for c into the Pythagorean Formula. What you now have is the Pythagorean Identity for Sine and Cosine.

tan2 = sec2 + 1 cot2 + 1 = csc2 Pythagorean Identities cos2 + sin2 = 1 Each of the three Pythagorean Identities can be rearranged into two additional forms by simply adding or subtracting terms on both sides of the equation. Do this now with all three identities. You should have a total of 9 formulas in your booklet.

+ 1 tan2 = sec2 cot2 + 1 = csc2 Booklet pgs 3-4 Pythagorean Identities cos2 + sin2 = 1

Booklet pg 5 Sum and Difference Identities The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like: The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like:

Booklet pg 6 Double-Angle Identities sin (2x) = 2sin x cos x cos(2x) = cos2x - sin2x = 2cos2x – 1 = 1 - 2sin2x

Booklet (back cover) Problem Solving Strategies • Create a monomial denominator • Add fractions (find an LCD) • Factor if possible • Convert everything to sines and cosines • When verifying, work on ONE side only – always pick the more complicated side to transform • Always try SOMETHING! It may not be the right thing but it’s better than nothing!

Example 1: Simplify sin x cot x Substitute using the quotient property. Simplify. Done!

Example 2: Simplify Use the reciprocal and quotient properties to make fractions. Cancel the sines and cosines. Done!

Example 3: Simplify:

Simplify

Unit 3: Trigonometric Identities