Using a Pythagorean Identities Objective Use fundamental trigonometric
Using a Pythagorean Identities Objective Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities. Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Pythagorean Theorem x 2 + y 2 = r 2 Note: This will also be the equation for the unit circle Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities A derivation for a Pythagorean identity is shown below. x 2 + y 2 = r 2 Pythagorean Theorem Divide both sides by r 2. cos 2 θ + sin 2 θ = 1 Substitute cos θ for and sin θ for Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities y r x Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Reciprocal Identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Quotient Identities y =x x =y Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Do you remember the Unit Circle? • What is the equation for the unit circle? x 2 + y 2 = 1 • What does x = ? What does y = ? (in terms of trig functions) sin 2θ + cos 2θ = 1 1 st Pythagorean Identity! Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos 2θ sin 2θ + cos 2θ = 1. cos 2θ tan 2θ + 1 = sec 2θ Quotient Identity 2 nd Pythagorean Identity Reciprocal Identity Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin 2θ + cos 2θ = 1. sin 2θ 1 + cot 2θ = csc 2θ Quotient Identity 3 rd Pythagorean Identity Reciprocal Identity Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Opposite Angle Identities sometimes these are called even/odd identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Other versions of the Pythagorean identities may be used also. Look at cos 2 θ + sin 2 θ = 1 and solve for cos 2 θ This should leave you with: cos 2 θ = 1 - sin 2 θ Do this again this time solving for sin 2 θ. This should leave you with: sin 2 θ = 1 - cos 2 θ Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Use Pythagorean Identities If cot θ = 2 and cos θ < 0, find sin θ and cos θ. Use the Pythagorean Identity that involves cot θ. θ cot 2 θ + 1 = csc 2 θ Pythagorean Identity (2) 2 + 1 = csc 2 θ = Take cotcsc θ =θ 2 the square 2 θroot of each 5 = csc side. Reciprocal Simplify. Identity θ. Solve for sin Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean 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. Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean 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. Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
Using a Pythagorean Identities Students will learn how to use fundamental trigonometric identities to simplify and rewrite expressions
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