Stitz-Zeager_College_Algebra_e-book

# Since we encountered no further 2 2 restrictions on t

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Unformatted text preview: se a half angle formula to ﬁnd the exact value of cos (15◦ ). 2. Suppose −π ≤ θ ≤ 0 with cos(θ) = − 3 . Find sin 5 θ 2 . 3. Use the identity given in number 3 of Example 10.4.3 to derive the identity θ 2 tan = sin(θ) 1 + cos(θ) Solution. 1. To use the half angle formula, we note that 15◦ = its cosine is positive. Thus we have 30◦ 2 and since 15◦ is a Quadrant I angle, √ cos (15◦ ) = + √ = 1 + 23 2 √ √ 2+ 3 2+ 3 = 4 2 1 + cos (30◦ ) = 2 1+ 2 3 2 · 2 = 2 Back in Example 10.4.1, we found√ √ ◦ ) by using the diﬀerence formula for cosine. In that cos (15 6+ 2 ◦) = case, we determined cos (15 . The reader is encouraged to prove that these two 4 expressions are equal. 10.4 Trigonometric Identities 2. If −π ≤ θ ≤ 0, then − π ≤ 2 θ 2 665 ≤ 0, which means sin θ 2 < 0. Theorem 10.19 gives =− θ 2 1 − cos (θ) =− 2 =− sin 1+ 2 3 5 · 5 =− 5 3 1 − −5 2 √ 8 25 =− 10 5 3. Instead of our usual approach to verifying identities, namely...
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