Addition Identity For Sine MHF4U: Advanced Functions Consider the - - PDF document

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MHF4U: Advanced Functions

Addition and Subtraction Identities

• J. Garvin

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Addition Identity For Sine

Consider the following triangle. Since |AC| = 1, |AB| = | cos x| and |BC| = | sin x|.

• J. Garvin — Addition and Subtraction Identities

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Addition Identity For Sine

Now consider the same triangle, rotated by an angle of y. We wish to determine a formula for sin(x + y).

• J. Garvin — Addition and Subtraction Identities

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Addition Identity For Sine

Note that ∠BCG = y, since ∠AGD = ∠BGC (opposite angles) and ∠ADG = ∠CBG (both 90◦).

• J. Garvin — Addition and Subtraction Identities

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Addition Identity For Sine

In ∆CFB, | cos y| = |CF|

|BC| = |CF| | sin x|, so |CF| = | sin x| · | cos y|.

• J. Garvin — Addition and Subtraction Identities

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Addition Identity For Sine

In ∆ABE, | sin y| = |BE|

|AB| = |BE| | cos x|, so |BE| = | sin y| · | cos x|.

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Addition Identity For Sine

In ∆ACD, |AC| = 1, so | sin(x + y)| = |CD|. |CD| = |CF + FD| = |CF + BE|, so | sin(x + y)| = |CF + BE|.

• J. Garvin — Addition and Subtraction Identities

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Addition Identity For Sine

Putting things together, | sin(x + y)| = | sin x| · | cos y| + | sin y| · | cos x|. This gives us the desired formula for sin(x + y).

Addition Identity For Sine

For any angles x and y, sin(x + y) = sin x cos y + sin y cos x.

• J. Garvin — Addition and Subtraction Identities

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Subtraction Identity For Sine

We can use the previous expression to derive an identity for sin(x − y). First, recall that sin(−x) = − sin x, and cos(−x) = cos x.

• J. Garvin — Addition and Subtraction Identities

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Subtraction Identity For Sine

For sin(x − y), substitute −y for y. sin(x + (−y)) = sin x cos(−y) + sin(−y) cos x sin(x − y) = sin x cos y − sin y cos x

Subtraction Identity For Sine

For any angles x and y, sin(x − y) = sin x cos y − sin y cos x.

• J. Garvin — Addition and Subtraction Identities

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Verifying the Sine Identities

Example

Verify that sin 5π

6 − 2π 3

• = sin 5π

6 cos 2π 3 − sin 2π 3 cos 5π 6 .

sin 5π

6 − 2π 3

• = sin

5π

6 − 4π 6

• = sin π

6

= 1

2

sin 5π

6 cos 2π 3 − sin 2π 3 cos 5π 6 = 1 2

• − 1

2

• −

√ 3 2

• −

√ 3 2

• = − 1

4 + 3 4

= 1

2

• J. Garvin — Addition and Subtraction Identities

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Addition Identity For Cosine

To develop an addition formula for cosine, we can make use

• f the cofunction identities developed earlier.

Since cos(a) = sin π

2 − a

• , let a = x + y.

cos(x + y) = sin π

2 − [x + y]

• = sin

π

2 − x

• − y
• Use the subtraction formula for sine next.

cos(x + y) = sin π

2 − x

• cos y − sin y cos

π

2 − x

• = cos x cos y − sin y sin x

Addition Identity For Cosine

For any angles x and y, cos(x + y) = cos x cos y − sin x sin y.

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Subtraction Identity For Cosine

A similar formula can be developed for subtraction using cosine. cos(x − y) = cos(x + (−y)) = cos x cos(−y) − sin x sin(−y) = cos x cos y + sin x sin y

Subtraction Identity For Cosine

For any angles x and y, cos(x − y) = cos x cos y + sin x sin y.

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Verifying the Cosine Identities

Example

Verify that cos π

3 + π

• = cos π

3 cos π − sin π 3 sin π.

cos π

3 + π

• = cos

π

3 + 3π 3

• = cos 4π

3

= − 1

2

cos π

3 cos π − sin π 3 sin π = 1 2(−1) − √ 3 2 (0)

= − 1

2

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Exact Value of a Trigonometric Ratio

Example

Determine an exact value for cos π

12.

Begin by expressing

π 12 as the sum or difference of two angles

for which we know their exact values.

π 12 = 4π 12 − 3π 12

= π

3 − π 4

Now, use the subtraction formula for cosine. cos π

12 = cos

π

3 − π 4

• = cos π

3 cos π 4 + sin π 3 sin π 4

= 1

2 · √ 2 2 + √ 3 2 · √ 2 2

=

√ 2+ √ 6 4

• J. Garvin — Addition and Subtraction Identities

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Questions?

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