Sum And Difference Identities

What are the Sum and Difference Identities?

Sum and difference identities are a set of trigonometric formulas that allow you to express trigonometric functions of the sum or difference of two angles in terms of trigonometric functions of those individual angles. These identities are useful for simplifying expressions, solving equations, and finding exact values of trigonometric functions.

The following diagram shows the Sum and Difference Identities for sin, cos and tan. Scroll down the page for more examples and solutions on how to use the identities.

 

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How to Remember the Sum and Difference Identities? (Mnemonics & Patterns)
It is often necessary to memorize these identities for quick application. Here are some common ways to remember them:

Sine (Same Sign): “Sin-Cos, Cos-Sin, and the sign’s the same."
\(sin(A \pm B) = sin(A)cos(B) \pm cos(B)sin(A)\)

Cosine (Opposite Sign): “Cos-Cos, Sin-Sin, and the sign is opposite."
\(cos(A \pm B) = cos(A)cos(B) \mp sin(A)sin(B)\)

Tangent: Think of the numerator as having the “sum/difference” behavior of the original angles, and the denominator as “1 opposite product."
\(tan(A + B) = \frac{tan(A) + tan(B)}{1 - tan(A)tan(B)}\)
\(tan(A - B) = \frac{tan(A) - tan(B)}{1 + tan(A)tan(B)}\)

Example:

Solution:
Given that cos(α + β) = cos α cos β – sin α sin β, then

Example:

Solution:

How to use the sum and difference identities for sin, cos, and tan?
Example:

1. Find sin(105°) exactly
2. Find cos(105°) exactly
3. Find tan(105°) exactly

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How to use Sum and Difference Identities to find exact trig values?
Example:

1. Find \(\cos \left( {\frac{{3\pi }}{4},\frac{\pi }{3}} \right)\) exactly
2. Find cos(42°)cos(18°) - sin(42°)sin(18°) exactly
3. Find \(\frac{{\tan 80^\circ - \tan 35^\circ }}{{1 + \tan 80^\circ \tan 35^\circ }}\) exactly
4. Find cos(u + v) exactly if sin(u) = 3/5 and sin(v) = 12/13 where u and v are acute angles (quadrant I)

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How to use the Sum and Difference Identities to Prove Other Identities
Example:
Prove sin(α + β) - sin(α - β) = 2cosαsinβ

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Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 1:
If sin x = 12/13 and x is in the first quadrant, find tan(2x)

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Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 2:
If tan x = 5/3 and x is in the first quadrant, find sin(2x)

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Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 3:
Simplify 1 - 16sin²x cos²x

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Key Applications

1. Finding Exact Values:
   These identities can be used to find the exact values of trigonometric functions for angles that are not directly on the unit circle. For example, you can find sin(75°) by expressing it as sin(45° + 30°).
2. Simplifying Expressions:
   They can help simplify complex trigonometric expressions, making them easier to work with.
3. Solving Trigonometric Equations:
   These identities are essential tools for solving various trigonometric equations.

Important Notes:

• It’s crucial to remember the signs in these formulas, as they can easily lead to errors.
• These identities are valid for any angles A and B.

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