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Fundamental FundamentalTrigonometric Identities Trigonometric Identities HoltMcDougal Algebra 2Algebra 2 Holt Fundamental Trigonometric Identities Objective Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities. Holt McDougal Algebra 2 Fundamental Trigonometric Identities A derivation for a Pythagorean identity is shown below. x2 + y2 = r2 Pythagorean Theorem Divide both sides by r2. cos2 θ + sin2 θ = 1 Holt McDougal Algebra 2 Substitute cos θ for sin θ for and Fundamental Trigonometric Identities To prove that an equation is an identity, alter one side of the equation until it is the same as the other side. Justify your steps by using the fundamental identities. Holt McDougal Algebra 2 Fundamental Trigonometric Identities Other versions of the Pythagorean identities may be used also. Look at cos2 θ + sin2 θ = 1 and solve for cos2 θ This should leave you with: cos2 θ = 1 - sin2 θ Do this again this time solving for sin2 θ. This should leave you with: sin2 θ = 1 - cos2 θ Holt McDougal Algebra 2 Fundamental Trigonometric Identities Prove each trigonometric identity. Choose the right-hand side to modify. Reciprocal identities. Simplify. Ratio identity. Holt McDougal Algebra 2 Fundamental Trigonometric Identities Helpful Hint You may start with either side of the given equation. It is often easier to begin with the more complicated side and simplify it to match the simpler side. If you get stuck, try converting all of the trigonometric functions to sine and cosine functions. Holt McDougal Algebra 2 Fundamental Trigonometric Identities Prove each trigonometric identity. sin θ cot θ = cos θ cos θ cos θ = cos θ Holt McDougal Algebra 2 Choose the left-hand side to modify. Ratio identity. Simplify. Fundamental Trigonometric Identities Prove each trigonometric identity. sec θ (1 – sin2θ)=cos θ Substitute. Multiply. cos θ Holt McDougal Algebra 2 Simplify. Fundamental Trigonometric Identities Verify. sinθ cosθ(tanθ + cotθ) =1 Substitute. Multiply. sin2θ + cos2θ 1 Holt McDougal Algebra 2 Simplify. Pythagorean identity. Fundamental Trigonometric Identities Verify = Pythagorean identity. Factor the difference of two squares. Simplify. Holt McDougal Algebra 2 Fundamental Trigonometric Identities Other strategies you may need to use are: 1. Get a common denominator. 2. Split up a fraction that has a monomial in the denominator. 3. Distribute. 4. Simplify complex fractions. Holt McDougal Algebra 2 Fundamental Trigonometric Identities Simplify sin2θ + cos2 θ + tan2 θ Holt McDougal Algebra 2 Fundamental Trigonometric Identities Verify sin cos 1 cot sin Holt McDougal Algebra 2 Fundamental Trigonometric Identities Cwk Pg. 359 17-25 odd Hwk Pg. 359 27-31 odd, 32-37 Holt McDougal Algebra 2