SchoolTube Community
Cosine Sum and Difference Identities: A Comprehensive Guide
In the realm of trigonometry, understanding the relationships between trigonometric functions is crucial. One such relationship is embodied in the cosine sum and difference identities. These identities provide a way to express the cosine of the sum or difference of two angles in terms of the cosines and sines of the individual angles.
Cosine of a Sum Identity
The cosine of a sum identity states that the cosine of the sum of two angles is equal to the product of the cosines of the individual angles minus the product of their sines:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
Cosine of a Difference Identity
Similarly, the cosine of a difference identity states that the cosine of the difference of two angles is equal to the product of the cosines of the individual angles plus the product of their sines:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Derivation of the Identities
These identities can be derived using the unit circle and the angle addition formula. The angle addition formula states that:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
By substituting B with -B, we get the cosine of a difference identity:
cos(A - B) = cos(A)cos(-B) - sin(A)sin(-B)
Since cosine is an even function and sine is an odd function, we have:
cos(-B) = cos(B) and sin(-B) = -sin(B)
Substituting these values, we obtain the cosine of a difference identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Applications of the Identities
These identities have numerous applications in trigonometry and other fields. They are used to:
- Simplify trigonometric expressions
- Solve trigonometric equations
- Derive other trigonometric identities
- Calculate the exact values of trigonometric functions for certain angles
Examples
Let's illustrate the use of these identities with a couple of examples:
Example 1: Finding cos(75°)
We can use the cosine of a sum identity to find cos(75°) by expressing 75° as the sum of 45° and 30°:
cos(75°) = cos(45° + 30°)
Applying the identity, we get:
cos(75°) = cos(45°)cos(30°) - sin(45°)sin(30°)
Substituting the known values, we have:
cos(75°) = (√2/2)(√3/2) - (√2/2)(1/2)
Simplifying the expression, we obtain:
cos(75°) = (√6 - √2)/4
Example 2: Simplifying the expression cos(x + π/4)
Using the cosine of a sum identity, we can simplify this expression:
cos(x + π/4) = cos(x)cos(π/4) - sin(x)sin(π/4)
Substituting the known values, we get:
cos(x + π/4) = cos(x)(√2/2) - sin(x)(√2/2)
Factoring out √2/2, we have:
cos(x + π/4) = (√2/2)(cos(x) - sin(x))
Conclusion
The cosine sum and difference identities are fundamental tools in trigonometry. By understanding and applying these identities, you can simplify trigonometric expressions, solve equations, and gain a deeper understanding of the relationships between trigonometric functions. These identities are essential for various applications in mathematics, physics, engineering, and other fields.