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Trigonometric Identities

Trigonometric Identities. We begin by listing some of the basic trigonometric identities. Simplifying Trigonometric Expressions. Identities enable us to write the same expression in different ways.

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Trigonometric Identities

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  1. Trigonometric Identities • We begin by listing some of the basic trigonometric identities.

  2. Simplifying Trigonometric Expressions • Identities enable us to write the same expression in different ways. • To simplify algebraic expressions use factoring, common denominators, and the Special Product Formulas. • To simplify trigonometric expressions, use these same techniques together with the fundamental trigonometric identities.

  3. Example 1 – Simplifying a Trigonometric Expression • Simplify the expression cos t + tan t sin t. • Solution:We start by rewriting the expression in terms of sine andcosine: • cos t + tan t sin t = cos t + sin t • = • = • = sec t Reciprocal identity Common denominator Pythagorean identity Reciprocal identity

  4. Trigonometric Identities • Many identities follow from the fundamental identities. • First, it’s easy to decide when a given equation is not an identity. • All we need to do is show that the equation does not hold for some value of the variable (or variables).

  5. Proving Trigonometric Identities • Thus the equation • sin x + cos x = 1 • is not an identity, because when x =  /4, we have • To verify that a trigonometric equation is an identity, we transform one side of the equation into the other side by a series of steps, each of which is itself an identity.

  6. Proving Trigonometric Identities

  7. Example 3 – Proving an Identity by Rewriting in Terms of Sine and Cosine • Verify the equation cos (sec – cos) = sin2. • and confirm graphically that the equation is an identity. • (a) We start with the LHS and try to transform it into the right-hand side: • LHS = cos (sec – cos) • = cos • = 1 – cos2Expand • = sin2 = RHS Reciprocal identity Pythagorean identity

  8. Example 3 – Solution cont’d • (b) Graph each side of the equation to see whether the graphs coincide. From Figure 1 we see that the graphs of y = cos (sec – cos) and y =sin2 are identical. Figure 1

  9. Example 5 – Proving an Identity by Introducing Something Extra • Verify the identity = sec u + tan u. • Start with the LHS and introduce “something extra” • And multiply the numerator and denominator by 1 + sin u: • LHS = Multiply numerator and denominator by 1 + sin u

  10. Example 5 – Solution cont’d • = • = • = • = • = sec u + tan u Expand denominator Pythagorean identity Cancel common factor Separate into two fractions Reciprocal identities

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