Chapter 8: Trigonometric Equations and Applications L8.4 Relationships Among the Functions

Fundamental Trig Identities An identity is true for all values of the variable(s) for which each side of the equation is defined. Five Sets of Fundamental Trig Identities*: Reciprocal Identities (RI) Quotient Identities (QI) Relationship with Negatives (even/odd) (EO) Cofunction Identities (CI) Pythagorean Identities (PI) You should have the fundamental identities in front of you when you work! * Note that the variable is the same throughout each identity. for ex: sin2 x + cos2 x = 1 vs. sin2 x + cos2 y = 1 an identity not an identity The fundamental identities can be used to Find trig values (instead of drawing triangles) Simplify expressions Prove additional trig identities → tomorrow Solve more complex trig equations → (L8.5) → today

1. USING TRIG IDS TO FIND TRIG VALUES Ex: Find all trig function values if sinθ = ¾ and π/2 ≤ θ < π. Up to now, we have drawn a triangle in Q2, solved for the missing side using the Pyth Thm and answered the question. Now, we can use the Pythagorean Identities instead: sin2 θ + cos2 θ = 1, so cos2 θ = 1 – sin2θ = 1 – (¾)2 = Since we are in Q2, cos < 0, so cos = Then tan θ is found from the quotient ids: And the remaining trig functions are reciprocals: cscθ = 4/3, secθ = , and cot = Start w/ cos:

2/3. Simplifying Trig Expressions or Proving Identities IMPORTANT GUIDELINES Proving or Verifying a Trig Identity: 1) Work one side only (generally the more complex side). 2) Be clear which side you have selected. Once you’ve selected a side, you are basically simplifying a trig expression. Simplifying Trig Expressions: 1) Work the entire expression – do not break into pieces, working each independently and joining them together later. 2) At each step, do one thing (occasionally two is OK). Label each step with the initials of the identity being used. Generally, an expression is simplified when it involves only one type of trig function or is a constant. In some ways, proving identities is easier since you know when you’re done! ** If your work can’t be followed, its not very compelling and you won’t receive much credit. **

2. SIMPLIFYING TRIG EXPRESSIONS STRATEGIES: 1. Use the Pythagorean identities for expressions that match trigfcn2θ ± 1 or trigfcn2θ ± trigfcn2θ 2. Factor 3. Fraction Algebra − Combine fractions − Split apart fractions 4. Multiply by conjugates 5. Convert to sine & cosine and simplify (a strategy of last resort) An expression is simplified when it involves only one type of trig function or is a constant. There are many ways to simplify expressions. All are OK, so long as you use the identities properly and use valid algebra.

STRATEGY 1: Use Pythagorean Ids csc2x – sin2x – cos2x

STRATEGY 2: Factor 3. sinθ·cos2θ – sinθ 4. cot2θ·cscθ + cscθ Note: Factoring is used extensively in solving trig equations.

STRATEGY 3 & 4: Fraction Algebra / Conjugates COMBINE FRACTIONS (Find LCD): 5. SPLIT APART FRACTIONS: 6. USE CONJUGATES 7. #6 & 7: Do not work pieces independently! There are many ways to do these problems. See Class Exercise p320 #8

STRATEGY 5: Convert to sine and cosine (a strategy of last resort) 8. sinθ·secθ·cotθ 9. secθ − sinθ·tanθ 10. cscθ(cos3θ·tanθ − sinθ)

OTHER EXAMPLES 11. cotθ·sin(−θ) 12. tanθ·cos(−θ) 13. 1 + tan2(θ – π/2)

SIMPLIFYING TRIG EXPRESSIONS STRATEGIES: 1. Use the Pythagorean identities for expressions that match trigfcn2θ ± 1 or trigfcn2θ ± trigfcn2θ 2. Factor 3. Fraction Algebra − Combine fractions − Split apart fractions 4. Multiply by conjugates 5. Convert to sine & cosine and simplify (a strategy of last resort) An expression is simplified when it involves only one type of trig function or is a constant. There are many ways to simplify expressions. All are OK, so long as you use the identities properly and use valid algebra.