6 1 B Reciprocal Quotient and Pythagorean Identities

6. 1 B Reciprocal, Quotient, and Pythagorean Identities Trigonometric Equations Math 30 -1 1

Proving an Identity Algebraically a) sec x(1 + cos x) = 1 + sec x cos x sec x + 1 1 + sec x Verify Graphically L. S. = R. S. b) sec x = tan x csc x L. S. = R. S. Math 30 -1 2

Verify Trigonometric Equations Numerically, Graphically, and Prove Algebraically Consider the equation a) Use a graph to verify that the equation is an identity. b) Verify numerically that this statement is true for c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. a) Graphically Math 30 -1 3

Verify Numerically for θ = 30°. Prove Algebraically

Using Exact Values to Prove an Identity Consider a) Use a graph to verify that the equation is an identity. b) Verify that this statement is true for x = c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. a) Math 30 -1 5

b) Verify that this statement is true for x = Rationalize the denominator: L. S. = R. S. Therefore, the identity is true for the particular case of 6

c) Use an algebraic approach to prove that the identity is true in general. State any restrictions. Restrictions: Note the left side of the equation has the restriction 1 - cos x ≠ 0 or cos x ≠ 1. Therefore, x ≠ 0 + 2 p n, where n is any integer. The right side of the equation has the restriction sin x ≠ 0. x = 0, p. Therefore, x ≠ 0 + 2 p n and x ≠ p + 2 p n, where n is any integer. L. S. = R. S. Math 30 -1 7

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