Sum-Difference Rule
d/dx [f(x) + g(x)] = f'(x) + g'(x)
The Sum-Difference Rule is a trigonometric identity used to express the trigonometric functions of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. It states:
sin(x ± y) = sin(x)cos(y) ± cos(x)sin(y)
cos(x ± y) = cos(x)cos(y) ∓ sin(x)sin(y)
The sign in the second term depends on whether it is a sum or difference.
The Sum-Difference Rule is helpful in simplifying trigonometric expressions and in solving trigonometric equations. It allows us to transform expressions of the form sin(x ± y) or cos(x ± y) into products of trigonometric functions.
For example, to simplify sin(π/4 + π/6), we can use the Sum-Difference Rule:
sin(π/4 + π/6) = sin(π/4)cos(π/6) + cos(π/4)sin(π/6)
= (1/√2)(√3/2) + (1/√2)(1/2)
= (√6 + √2)/4
Similarly, we can use the Sum-Difference Rule to simplify expressions like cos(π/3 – π/6) or sin(2π/3 + π/6).
Overall, the Sum-Difference Rule is an important identity in trigonometry that allows us to manipulate and simplify trigonometric expressions involving sums and differences of angles.
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