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April 03, 2011 Trig Identities The ones you already know: NEW: PYTHAGOREAN IDENTITIES 2 · What would sin θ equal? sin2θ = 1 - cos2θ 2 · What would cos θ equal? cos2θ = 1 - sin2θ 2. Let's take 2the original Pythagorean identity and divide by sin θ. What is the new identity that we get? 1 + cot2θ = csc2θ 3. Let's take the original Pythagorean identity and 2 divide by cos θ. What is the new identity that we get? tan2θ + 1 = sec2θ April 03, 2011 Examples: HINT 1. Simplify: cotθsecθ in terms of sinθ First: Find an identity to substitute a different function for one of the given functions. (You want to try and get them all to be the same trig function.) 2. Simplify the expression trigonometric function. to a single 3. Simplify: 4. Write this expression as a monomial with a single trigonometric function: April 03, 2011 5. Simplify the expression: Practice: Write the expression as a monomial containing a single function 7. sinθsecθ 6. sinθcotθ 9. secθsinθcscθ 8. secθcotθsinθ 2 2 11. sinθ(cot θ + 1) 10. cscθ(1 - cos θ) 2 12. secθcosθ - cos θ Using Identities in Equation Solving: If there is more than one trig function in the equation, identities are needed to reduce the equation to a single function for solving. 2 1. Solve: 2cos x + 3sinx - 3 = 0 Use the Pythagorean Identities to replace the squared term! 2 2. 2cos x - sinx = 1 2 3. sec x - tanx - 1 = 0 4. cosx = secx 5. 2sinx = cscx April 03, 2011