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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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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