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Chapter 5 – Analytic Trigonometry 5.1 – Fundamental Identities

Chapter 5 – Analytic Trigonometry 5.1 – Fundamental Identities. HW: Pg. 451 #1-8. Trigonometry Identities. Reciprocal Identities csc θ = 1/sin θ sec θ =1/cos θ cot θ = 1/tan θ sin θ = 1/csc θ cos θ =1/sec θ tan θ =1/cot θ Quotient Identities Tan θ = sin θ /cos θ cot θ =cos θ /sin θ.

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Chapter 5 – Analytic Trigonometry 5.1 – Fundamental Identities

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  1. Chapter 5 – Analytic Trigonometry5.1 – Fundamental Identities HW: Pg. 451 #1-8

  2. Trigonometry Identities • Reciprocal Identities • cscθ= 1/sinθ • secθ=1/cosθ • cotθ= 1/tanθ • sinθ= 1/cscθ • cosθ=1/secθ • tanθ=1/cotθ • Quotient Identities • Tan θ = sin θ/cos θ • cotθ=cosθ/sinθ

  3. Pythagorean Identities • Try to come up with an identity using the pythagorean theorem (a2 + b2 = c2) and the unit circle

  4. Pythagorean Identities • Cos2θ + Sin2θ = 1 • 1 + tan2θ = Sec2θ • Cot2θ + 1 = csc2θ

  5. Using Identities • Find sinθ and cosθ if tanθ = 5 and cosθ > 0

  6. Cofunction Identities

  7. Confunction Identities • Sin(π/2 – θ) = cos θ • Cos(π/2 – θ) = sin θ • Tan(π/2 – θ) = cot θ • Csc(π/2 – θ) = sec θ • Sec(π/2 – θ) = csc θ • Cot(π/2 – θ) = tan θ

  8. Odd-Even Identities • Sin(-x) = -sinx • Cos(-x) = _______ • Tan(-x) = _______ • Csc(-x) = _______ • Sec(-x) = _______ • Sin(-x) = _______

  9. Using Identities • If cos θ = 0.34, find sin(θ - π/2).

  10. Simplifying Trigonometric Expressions • Simplify by factoring and using identities • Sin3x + sinxcos2x

  11. Simplify by expanding and using identities • [(secx + 1)(secx - 1)]/sin2x

  12. Simplify by combining fractions and using identities

  13. Solving Trigonometric Equations • Find all values of x in the interval [0,2π) that solve cos3x / sinx = cotx

  14. Solving a Trig Equation by factoring • Find all solutions to the trigonometric equation 2sin2x + sinx = 1

  15. PRACTICE 5.1 –Pg. 451-452 #9-16, 23-38 (all) Use Identities to simplify

  16. 5.2 – Proving Trigonometric Identities HW: PG. 460 #12-34e

  17. Proving an Algebraic Identity • Prove the algebraic identity : • (x2-1)/(x-1) – (x2-1)/(x+1) = 2

  18. General Proof Strategies I • The proof begins with the expression on one side of the identity. • The proof ends with the expression on the other side • The proof in between consists of showing a sequence of expressions, each one easily seen to be equivalent to its previous expression.

  19. Proving an Identity: • tan x + cot x = sec x csc x

  20. General Strategies II • Begin with the more complicated expression and work toward the less complicated expression. • If no other move suggests itself, convert the entire expression to one involving sines and cosines. • Combine fractions by combining them over a common denominator.

  21. Identifying and Proving an Identity • Match the function f(x)=1/(secx - 1) + 1/(secx + 1) with (i) 2cot x csc x or (ii) 1/secx

  22. Setting up a Difference of Squares • Prove the identity: cost/(1 – sint) = (1 + sint)/cost

  23. General Strategies III • Use the algebraic identity (a + b)(a – b) = a2 – b2 to set up applications of the Pythagorean identities. • Always keep in mind what your goal is (expression), and favor manipulations that bring you closer to your goal.

  24. Working from Both Sides • Prove the identity: • Cot2u/(1 + cscu) = (cotu)(secu – tanu)

  25. Disproving Non-Identities - EXPLORATION Prove or disprove that this is an identity: cos2x = 2cosx • Graph y=cos2x and y=2cosx in the same window. Interpret the graphs to make a conclusion about whether or not the equation is an identity. • With the help of the graph, find a value of x for which cos2x ≠ 2cosx. • Does the existence of the x value in part (2) prove that the equation is not an identity? • Graph y = cos2x and y = cos2x – sin2x in the same window. Interpret the graphs to make a conclusion about whether or not cos2x = cos2x – sin2x is an identity. • Do the graphs in part (4) prove that cos2x = cos2x – sin2x is an identity? Explain your answer.

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