Download MAT 106 – 040 TRIGONOMETRY LECTURE 33 9.3 — Further

MAT 106 – 040 TRIGONOMETRY
LECTURE 33
WING HONG TONY WONG
9.3 — Further trigonometric identities
Double-angle identities
sin(2x) = 2 sin x cos x
cos(2x) = cos2 x − sin2 x
tan(2x) =
2 tan x
1−tan2 x
Since sin2 x + cos2 x = 1, we also have cos(2x) = 2 cos2 x − 1 = 1 − 2 sin2 x. From these
two identities, we can deduce the half-angle identities.
Half-angle identities
q
x
sin x2 = ± 1−cos
2
cos x2 = ±
Sum-to-product identities
sin x + sin y = 2 sin x+y
cos
2 x+y
cos x + cos y = 2 cos 2 cos
q
tan x2 = ±
1+cos x
2
x−y
2 x−y
2
q
1−cos x
1+cos x
=
1−cos x
sin x
=
sin x
1+cos x
x−y
sin x − sin y = 2 cos x+y
sin
2
2
cos x − cos y = −2 sin x+y
sin x−y
2
2
Product-to-sum identities
sin x cos y = 21 [sin(x + y) + sin(x − y)]
cos x cos y = 21 [cos(x + y) + cos(x − y)]
cos x sin y = 21 [sin(x + y) − sin(x − y)]
sin x sin y = − 12 [cos(x + y) − cos(x − y)]
Example 1. Verify cot x sin 2x = 1 + cos 2x.
Example 2. Write sin 3x in terms of sin x.
Example 3. (9.3.4) If tan θ =
5
3
and sin θ < 0, find sin 2θ and cos 2θ.
Example 4. (9.3.12) Write cos2
number.
Example 5. (9.3.20) Write
number.
π
8
−
1
2
as a single trigonometric function or as a single
4 tan x cos2 x−2 tan x
1−tan2 x
as a single trigonometric function or as a single
Example 6. (9.3.34) Use a half-angle identity to find an exact value for cos x2 , given
sin x = − 45 and 3π
< x < 2π.
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Date: Monday, November 11, 2013.
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Example 7. (9.3.56) Verify cos x =
1−tan2
1+tan2
x
2
x
2
.
Example 8. (9.3.74) Write sin 9B − sin 3B as a product of trigonometric functions.
Example 9. Find the value of sin 67.5◦ − sin 22.5◦ .
Example 10. Find the value of tan 22.5◦ .
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