Trig Identities
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Trig identities are equations that involve trigonometric functions and are true for all values of the variables. They are useful for simplifying expressions involving trigonometric functions and can be used to prove other identities or solve equations involving trigonometric functions. Here are some important trig identities:
1. Pythagorean identities:
– sin^2(x) + cos^2(x) = 1
– 1 + tan^2(x) = sec^2(x)
– 1 + cot^2(x) = csc^2(x)
2. Even/odd identities:
– sin(-x) = -sin(x) -> odd
– cos(-x) = cos(x) -> even
– tan(-x) = -tan(x) -> odd
– csc(-x) = -csc(x) -> odd
– sec(-x) = sec(x) -> even
– cot(-x) = -cot(x) -> odd
3. Angle sum/difference identities:
– sin(x+y) = sin(x)cos(y) + cos(x)sin(y)
– sin(x-y) = sin(x)cos(y) – cos(x)sin(y)
– cos(x+y) = cos(x)cos(y) – sin(x)sin(y)
– cos(x-y) = cos(x)cos(y) + sin(x)sin(y)
– tan(x+y) = [tan(x) + tan(y)] / [1 – tan(x)tan(y)]
– tan(x-y) = [tan(x) – tan(y)] / [1 + tan(x)tan(y)]
4. Double angle identities:
– sin(2x) = 2sin(x)cos(x)
– cos(2x) = cos^2(x) – sin^2(x) = 2cos^2(x) – 1 = 1 – 2sin^2(x)
– tan(2x) = [2tan(x)] / [1 – tan^2(x)]
5. Half angle identities:
– sin(x/2) = ±√ [(1 – cos(x)) / 2]
– cos(x/2) = ±√ [(1 + cos(x)) / 2]
– tan(x/2) = [sin(x)] / [1 + cos(x)]
These are some of the most important trig identities in mathematics. By memorizing these identities and practicing problems, one can become proficient at using them to solve complex trigonometric expressions.
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