Trigonometric Identities 11-3 Fundamental Trigonometric Identities 11-3 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2
Warm Up Simplify. 1. 2. cos A 1
Objective Use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities.
You can use trigonometric identities to simplify trigonometric expressions. Recall that an identity is a mathematical statement that is true for all values of the variables for which the statement is defined.
A derivation for a Pythagorean identity is shown below. x2 + y2 = r2 Pythagorean Theorem Divide both sides by r2. Substitute cos θ for and sin θ for cos2 θ + sin2 θ = 1
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.
Example 1A: Proving Trigonometric Identities Prove each trigonometric identity. Choose the right-hand side to modify. Reciprocal identities. Simplify. Ratio identity.
Example 1B: Proving Trigonometric Identities Prove each trigonometric identity. Choose the right-hand side to modify. 1 – cot θ = 1 + cot(–θ) Reciprocal identity. Negative-angle identity. = 1 + (–cotθ) Reciprocal identity. = 1 – cotθ Simplify.
You may start with either side of the given equation 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. Helpful Hint
Check It Out! Example 1a Prove each trigonometric identity. sin θ cot θ = cos θ Choose the left-hand side to modify. cos θ Ratio identity. cos θ = cos θ Simplify.
Check It Out! Example 1b Prove each trigonometric identity. Choose the left-hand side to modify. 1 – sec(–θ) = 1 – secθ Reciprocal identity. Negative-angle identity. Reciprocal Identity.
You can use the fundamental trigonometric identities to simplify expressions. If you get stuck, try converting all of the trigonometric functions to sine and cosine functions. Helpful Hint
Example 2A: Using Trigonometric Identities to Rewrite Trigonometric Expressions Rewrite each expression in terms of cos θ, and simplify. sec θ (1 – sin2θ) Substitute. Multiply. Simplify. cos θ
Example 2B: Using Trigonometric Identities to Rewrite Trigonometric Expressions Rewrite each expression in terms of sin θ, cos θ, and simplify. sinθ cosθ(tanθ + cotθ) Substitute. Multiply. sin2θ + cos2θ Simplify. 1 Pythagorean identity.
Check It Out! Example 2a Rewrite each expression in terms of sin θ, and simplify. Pythagorean identity. Factor the difference of two squares. Simplify.
Check It Out! Example 2b Rewrite each expression in terms of sin θ, and simplify. cot2θ csc2θ – 1 Pythagorean identity. Substitute. Simplify.
Example 3: Physics Application At what angle will a wooden block on a concrete incline start to move if the coefficient of friction is 0.62? Set the expression for the weight component equal to the expression for the force of friction. mg sinθ = μmg cosθ sinθ = μcosθ Divide both sides by mg. sinθ = 0.62 cosθ Substitute 0.62 for μ.
Example 3 Continued Divide both sides by cos θ. Ratio identity. tanθ = 0.62 θ = 32° Evaluate inverse tangent. The wooden block will start to move when the concrete incline is raised to an angle of about 32°.
Check It Out! Example 3 Use the equation mg sinθ = μmg cosθ to determine the angle at which a waxed wood block on a wood incline with μ = 0.4 begins to slide. Set the expression for the weight component equal to the expression for the force of friction. mg sinθ = μmg cosθ sinθ = μcosθ Divide both sides by mg. sinθ = 0.4 cosθ Substitute 0.4 for μ.
Check It Out! Example 3 Continued Divide both sides by cos θ. Ratio identity. tanθ = 0.4 θ = 22° Evaluate inverse tangent. The wooden block will start to move when the concrete incline is raised to an angle of about 22°.
Lesson Quiz: Part I Prove each trigonometric identity. 1. sinθ secθ = 2. sec2θ = 1 + sin2θ sec2θ = 1 + tan2θ = sec2θ
Lesson Quiz: Part II Rewrite each expression in terms of cos θ, and simplify. 3. sin2θ cot2θ secθ cosθ 4. 2 cosθ