Section 3.6 Basic Trigonometric Identities © Copyright all rights reserved to Homework depot: www.BCMath.ca

I) What is a Trigonometric Identity? A trigonometric identity is an equation that is equal for all values of the variable(s) for which the equation is defined Examples of trigonometric identities Trigonometric equations that are not Identities Both sides are equal for all values of “x” Graphically, if both sides overlap each other completely, then the equation is an identity Equation is true only at certain values of “x” Graphically, both sides only intersect at certain points, then the equation is NOT an identity © Copyright all rights reserved to Homework depot: www.BCMath.ca

II) Odd vs Even Identities: Even Identities: An function that looks the same when reflected over the y-axis (Horizontal Reflection) Odd Identities: A function that looks the same when reflected over both the X and Y axis x y -2p -p p 2p -1 1 x y -2p -p p 2p -1 1 © Copyright all rights reserved to Homework depot: www.BCMath.ca

III) Pythagorean Identities: Review: The coordinates of any point on the circumference of an unit circle can be represented by: X-coordinate Y-coordinate Since: Pythagorean Identity Other Pythagorean Identities can be generated by dividing all terms by either “cos2x” or “sin2x” © Copyright all rights reserved to Homework depot: www.BCMath.ca

IV) Basic Identities Odd- Even Identities Reciprocal Identities Quotient Identities Pythagorean Identities Other identities can be created by manipulating the equation © Copyright all rights reserved to Homework depot: www.BCMath.ca

V) Verifying and Proving Identities There are two ways to Verify an identity Plug variety of numbers into the equation If the equation is equal for all the values, then the equation is an identity OR Graph the equations, if they completely overlap, then it’s an identity Proving an Identity Simplify the equation algebraically and then show that both sides are equal When proving algebraically, first convert all functions into sine or cosine Then simplify using basic identities © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Verify the Following identity: Since the equation is equal for both verifications, then it’s likely to be an identity NUMERICALLY: Pick a random number Pick another number Both sides are equal! Both sides are equal again! GRAPHICALLY: x y -4 -3 -2 -1 1 2 3 4 Since the graphs overlap each other completely, then it must be an identity © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Verify the following identities Numerically: Pick a random number for “x” Make “x” = 10 rad. Both sides are equal Both sides are equal Try another value for “x” Make “x” = 2 rad. Since the equation is equal for both verifications, then it is likely to be an identity The equation is likely to be an identity © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Verify each of the following Identities: Numerically: Numerically: Not an Identity!! Graphically: Graphically: x y -2p -p p 2p -2 -1 1 2 x y -2p -p p 2p -2 -1 1 2 © Copyright all rights reserved to Homework depot: www.BCMath.ca

vi) Proving Identities Algebraically: When proving trigonometric identities: Convert all trig. functions to sine or cosine Use basic trig. Identities to simplify complicated ones Odd/Even, Quotient, Pythagorean Identities Start with the side that looks “more complicated” You may need to rationalize the expression, factor out common factors, or multiply all terms by the LCD Trial and Error (Do whatever it takes) Once the left side and right side are equal then the equation is a trigonometric identity © Copyright all rights reserved to Homework depot: www.BCMath.ca

Vii) Proving Trigonometric Identities By Using Basic Identities: Since the left side looks more complicated, we will prove the identity from the left Cancel out common factors Quotient Identity! © Copyright all rights reserved to Homework depot: www.BCMath.ca

viii) Proving by Adding/Subtracting Identities You can also prove the identity from the other side Split into 2 separate fractions Common Denominator Combine the Fractions The proof may be the different but the result is the same © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Prove the following identities: Get the LCD with the numerator Common Denominator Factor out sin θ Combine the Fractions © Copyright all rights reserved to Homework depot: www.BCMath.ca

ix) Proving Identities Using Fractions : Pythagorean Identities Divide the fractions © Copyright all rights reserved to Homework depot: www.BCMath.ca

x) Proving Identities by Factoring: Difference of squares Factor out the sine function Pythagorean identity Common Denominator Pythagorean Identity Expand FOIL © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Prove the following Identities by Factoring Bracket the last 2 terms Pythagorean Identity Factor the Trinomial Pythagorean identity Factor: Difference of squares © Copyright all rights reserved to Homework depot: www.BCMath.ca

xi) Prove by Conjugating The Expression: Quotient Identities Reciprocal Identity Multiply both top & bottom by the conjugate Expand Pythagorean Identity Pythagorean identity Formula Sheet © Copyright all rights reserved to Homework depot: www.BCMath.ca

Practice: Prove the following Identity by conjugating the expression: Formula Sheet Quotient Identities Reciprocal Identity Multiply both top & bottom by the conjugate Expand Pythagorean Identity Pythagorean identity CLOSE © Copyright all rights reserved to Homework depot: www.BCMath.ca

Homework: Assignment 3.6