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Stitz-Zeager_College_Algebra_e-book

# 4 the identity cos2 tsin2 t 1 holds for all t in

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Unformatted text preview: ), we need to write 15◦ as a sum or diﬀerence of angles whose cosines and sines we know. One way to do so is to write 15◦ = 45◦ − 30◦ . cos (15◦ ) = cos (45◦ − 30◦ ) = cos (45◦ ) cos (30◦ ) + sin (45◦ ) sin (30◦ ) √ √ √ 2 3 2 1 = + 2 2 2 2 √ √ 6+ 2 = 4 658 Foundations of Trigonometry 2. In a straightforward application of Theorem 10.13, we ﬁnd cos π −θ 2 π π cos (θ) + sin sin (θ) 2 2 = (0) (cos(θ)) + (1) (sin(θ)) = cos = sin(θ) The identity veriﬁed in Example 10.4.1, namely, cos π − θ = sin(θ), is the ﬁrst of the celebrated 2 ‘cofunction’ identities. These identities were ﬁrst hinted at in Exercise 8 in Section 10.2. From sin(θ) = cos π − θ , we get: 2 π π π − θ = cos − − θ = cos(θ), 2 2 2 which says, in words, that the ‘co’sine of an angle is the sine of its ‘co’mplement. Now that these identities have been established for cosine and sine, the remaining circular functions follow suit. The remaining proofs are left as exercises. sin Theorem 10.14...
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