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Unformatted text preview: the circular functions for angles, they were also useful in simplifying expressions
involving the circular functions. In this section, we introduce several collections of identities which
have uses in this course and beyond. Our ﬁrst set of identities is the ‘Even / Odd’ identities.1
Theorem 10.12. Even / Odd Identities: For all applicable angles θ,
• cos(−θ) = cos(θ) • sin(−θ) = − sin(θ) • tan(−θ) = − tan(θ) • sec(−θ) = sec(θ) • csc(−θ) = − csc(θ) • cot(θ) = − cot(θ) In light of the Quotient and Reciprocal Identities, Theorem 10.6, it suﬃces to show cos(−θ) = cos(θ)
and sin(−θ) = − sin(θ). The remaining four circular functions can be expressed in terms of cos(θ)
and sin(θ) so the proofs of their Even / Odd Identities are left as exercises. Consider an angle θ
plotted in standard position. Let θ0 be the angle coterminal with θ with 0 ≤ θ0 < 2π . (We can
construct the angle θ0 by rotating counter-clockwise from the positive x-axis to the terminal side
of θ as pictured below.) Since θ and θ0 are co...
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