7.3 – Day 1
Double-Angle & Half-Angle
Formulas
Objectives
► Double-Angle Formulas
► Half-Angle Formulas
► Simplifying Expressions Involving Inverse
Trigonometric Functions
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Double-Angle & Half-Angle Formulas
The identities we consider in this section are consequences
of the addition formulas.
The Double-Angle Formulas allow us to find the values of
the trigonometric functions at 2x from their values at x.
The Half-Angle Formulas relate the values of the
trigonometric functions at x to their values at x.
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Double-Angle Formulas
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Double-Angle Formulas
Proof of Double-Angle formulas:
Begin with the addition formula for sine:
Begin with the addition formula for tangent:
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Double-Angle Formulas
Proof of Double-Angle formulas:
Begin with the addition formula for cosine:
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Double-Angle Formulas
The formulas in the following box are immediate
consequences of the addition formulas:
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Example 2 – A Triple-Angle Formula
Write cos 3x in terms of cos x.
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Extra Example A
3
Given sin x  , and x is in Quadrant III, find:
5
a)
sin 2x
b)
cos 4 x


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Extra Example A
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Double-Angle Formulas
Example 2 shows that cos 3x can be written as a
polynomial of degree 3 in cos x.
The identity cos 2x = 2 cos2 x – 1 shows that cos 2x is a
polynomial of degree 2 in cos x.
In fact, for any natural number n, we can write cos nx as a
polynomial in cos x of degree n.
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Half-Angle Formulas
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Half-Angle Formulas
Proof of the Lowering Powers Formulas:
1) sin2x – begin with the Double-Angle formula for cosine:
2) cos2x – begin with the Double-Angle formula for cosine:
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Half-Angle Formulas
Proof of the Lowering Powers Formulas:
3) tan2x – begin with sin2x / cos2x:
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Half-Angle Formulas
The following formulas allow us to write any trigonometric
expression involving even powers of sine and cosine in
terms of the first power of cosine only.
This technique is important in calculus. The Half-Angle
Formulas are immediate consequences of these formulas.
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Example 4 – Lowering Powers in a Trigonometric Expression
Express sin2x cos2x in terms of the first power of cosine.
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Example 4 – Lowering Powers in a Trigonometric Expression
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Double-Angle & Half-Angle Formulas
Practice:
p. 514-515
#1-15o, 29, 31
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7.3 – Day 2
Double-Angle & Half-Angle
Formulas
19
Objectives
► Double-Angle Formulas
► Half-Angle Formulas
► Simplifying Expressions Involving Inverse
Trigonometric Functions
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Half-Angle Formulas
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Half-Angle Formulas
Substitute x = u/2 into the Formulas for Lowering
Powers, take the square root of each side, and you will
get:
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Example 5 – Using a Half-Angle Formula
Find the exact value of sin 22.5
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Evaluating Expressions Involving
Inverse Trigonometric Functions
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Evaluating Expressions Involving Inverse Trigonometric Functions
Expressions involving trigonometric functions and their
inverses arise in calculus. In the next example, we illustrate
how to evaluate such expressions.
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Example 8 – Evaluating an Expression Involving Inverse Trigonometric Functions
Evaluate sin 2, where cos  =
with  in Quadrant II.
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Extra Example B
Evaluate:
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Double-Angle & Half-Angle Formulas
Practice:
p. 514-515
#19, 27, 33, 37, 41, 45, 47, 53,
73-83o
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