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Mastery Objectives 5.2 Verify trigonometric identities. Determine whether equations are identities.

A.cos x B cos x C.2 sin x D.2 csc x Simplify.

Rewrite as an expression that does not involve a fraction. Pythagorean Identity Reciprocal Identity Quotient Identity Reciprocal Identity

Answer: tan 2 x So, = tan 2 x.

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Verifying Trigonometric Identities To verify an identity means to prove that both sides of the equation are equal for all values of the variable for which both sides are defined. Transform the expression on one side of the identity into the expression on the other side through a sequence of intermediate expressions that are each equivalent to the first. Begin on the side with the MORE complicated expression and work toward the LESS complicated expression

The left-hand side of this identity is more complicated, so transform that expression into the one on the right. Verify that. Pythagorean Identity Reciprocal Identity Simplify. Answer:

Example 1 Verify that 2 – cos 2 x = 1 + sin 2 x. A.2 – cos 2 x = –(sin 2 x + 1) + 2 = 1 + sin 2 x B.2 – cos 2 x = 2 – (sin 2 x + 1) = 1 + sin 2 x C.2 – cos 2 x = 2 – (1 + sin 2 x) + 2 = 1 + sin 2 x D.2 – cos 2 x = 2 – (1 – sin 2 x) = 1 + sin 2 x

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Verify a Trigonometric Identity by Combining Fractions Verify that.

Verify a Trigonometric Identity by Multiplying Verify that.

Answer:

Verify that.

Verify a Trigonometric Identity by Factoring Verify that cos x sec 2 x tan x – cos x tan 3 x = sin x. Answer:cos x sec 2 x tan x – cos x tan 3 x = cos x tan x (sec 2 x – tan 2 x) = cos x tan x (1) = = sin x

Verify that csc x – cos x csc x – cos x cot x + cot x = sin x.

Verify an Identity by Working Each Side Separately Both sides look complicated, but there is a clear first step for the expression on the left. So, start with the expression on the left. Verify that. Simplify and apply a Reciprocal Identity. Write as the sum of two fractions.

Simplify. Factor. Pythagorean Identity From here, it is unclear how to transform 1 + cot x into, so start with the right side and work to transform it into the intermediate form 1 + cot x.

Divide out the common factor of 1 – cot x. To complete the proof, work backward to connect the two parts of the proof. Simplify and apply a Reciprocal Identity. Multiply by Write as the sum of two fractions.

Simplify. Simplify. Pythagorean Identity Answer:

Verify that tan 2 x – sin 2 x = sin 2 x tan 2 x.

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The equation appears to be an identity because the graphs of the related functions over [– 2π, 2π] scl: π by [–1, 3] scl: 1 coincide. Verify this algebraically. Determine Whether an Equation is an Identity Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.

Determine Whether an Equation is an Identity Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal.

The graphs of the related functions do not coincide for all values of x for which the both functions are defined. When, Y 1 ≈ 1.43 but Y 2 ≈ –0.5. The equation is not an identity. Answer: When, Y 1 ≈ 1.43 but Y 2 = –0.5. The equation is not an identity.

A.The equation appears to be an identity because the graphs of the related functions over [–2 π, 2 π ] scl: π by [–3, 3] scl: 1 coincide. B.When, Y 1 ≈ 0.71 but Y 2 ≈ The equation is not an identity. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find a value for which both sides are defined but not equal.

Homework 5.2: P 324: 3 – 18 thirds