3 4 Fundamental Trigonometric Identities Mrs Lisa Allen

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(3 -4) Fundamental Trigonometric Identities Mrs. Lisa Allen Honors Pre-calculus Saucon Valley High School

(3 -4) Fundamental Trigonometric Identities Mrs. Lisa Allen Honors Pre-calculus Saucon Valley High School

Old Familiar Identities n 2+3=3+2 n 6(x) = x(6) n n Commutative Property Addition

Old Familiar Identities n 2+3=3+2 n 6(x) = x(6) n n Commutative Property Addition Commutative Property Multiplication (½ + 3/2) + 3 = ½ + (3/2 + 3) Associative Property of Addition 4 – 3 = 4 + -3 Definition of Subtraction

Definition: Trigonometric Identities n n A trigonometric identity is a trigonometric equation that is

Definition: Trigonometric Identities n n A trigonometric identity is a trigonometric equation that is true for all values. Reciprocal Identities from Chapter 1 include:

Chapter 1 Basic Trigonometry Review

Chapter 1 Basic Trigonometry Review

Example 1 Prove: sec cos = 1 (1) Given (2) Reciprocal Identities (3) Simplification

Example 1 Prove: sec cos = 1 (1) Given (2) Reciprocal Identities (3) Simplification (4) Simplification

Prove Ratio (or Quotient) Identities n Example 2: Prove (1) Prove: (2) Definition of

Prove Ratio (or Quotient) Identities n Example 2: Prove (1) Prove: (2) Definition of sine and cosine (3) Simplification of a complex fraction (4) Definition of cotangent OR And

Example 3 Prove sin = tan cos sin tan cos (1) Prove this is

Example 3 Prove sin = tan cos sin tan cos (1) Prove this is so! (2) Definition of tangent and cosine (3) Simplification (4) Simplification (5) Definition of sine

!Notation!

!Notation!

Example 4 Prove: 1 (1) PROVE (2) DEFN: SINE & COSINE (3) POWER OF

Example 4 Prove: 1 (1) PROVE (2) DEFN: SINE & COSINE (3) POWER OF A QUOTIENT RULE (4) ADDITION OF TWO FRACTIONS (5) UNIT CIRCLE PYTHAGOREAN TRIPLE x, y, AND r (6) SIMPLIFICATION 1=

Quotient Identities Pythagorean Identities

Quotient Identities Pythagorean Identities

Unit Circle P(x, y) r x y

Unit Circle P(x, y) r x y

Pythagorean Identities n n n There are three of these, but memorization of the

Pythagorean Identities n n n There are three of these, but memorization of the first one will help with the other two. We have already proved this one. So, in the next part we’ll try to derive the other two Pythagorean identities

Pythagorean Identity #2 BEGIN WITH PYTHAGOREAN #1 We HAVE WORKED WITH THE RATIO IDENTITIES…so….

Pythagorean Identity #2 BEGIN WITH PYTHAGOREAN #1 We HAVE WORKED WITH THE RATIO IDENTITIES…so…. DIVIDE EACH TERM BY sine squared SIMPLIFICATION

Pythagorean Identity #2 BEGIN WITH PYTHAGOREAN #1 Again using the RATIO IDENTITIES DIVIDE EACH

Pythagorean Identity #2 BEGIN WITH PYTHAGOREAN #1 Again using the RATIO IDENTITIES DIVIDE EACH TERM BY cosine squared SIMPLIFICATION

Even and Odd Identities Odd Function Even Function Odd Function

Even and Odd Identities Odd Function Even Function Odd Function

Prove: y P(x, y) tan(- ) -tan r x - r -tan P(x, -y)

Prove: y P(x, y) tan(- ) -tan r x - r -tan P(x, -y)

Example 6: Verify the following identities for = 45 a) b) Verifying uses specific

Example 6: Verify the following identities for = 45 a) b) Verifying uses specific values

Homework assignment n P. 133 #s 3 -30 (multiples of 3) #37 & #38

Homework assignment n P. 133 #s 3 -30 (multiples of 3) #37 & #38