Trig Identities
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Trigonometric identities are mathematical equations that involve trigonometric functions such as sine, cosine, tangent, and cotangent. These identities are useful in simplifying trigonometric expressions and making calculations easier.
Here are some of the most common trigonometric 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)
These identities are used to relate the trigonometric functions of an angle to the sides of a right triangle with that angle.
2. Even-Odd Identities:
sin(-x) = -sin(x)
cos(-x) = cos(x)
tan(-x) = -tan(x)
These identities refer to the symmetry of the trigonometric functions with respect to the y-axis (even functions) or origin (odd functions).
3. Co-function Identities:
sin(90° – x) = cos(x)
cos(90° – x) = sin(x)
tan(90° – x) = cot(x)
These identities relate the trigonometric functions of an angle with that of its complementary angle.
4. Angle Sum and Difference Identities:
sin(x + y) = sin(x)cos(y) + cos(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))
These identities are used to calculate the trigonometric functions of a sum or difference of two angles.
5. Double Angle Identities:
sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) – sin^2(x)
tan(2x) = (2tan(x))/(1 – tan^2(x))
These identities allow us to find the trigonometric functions for an angle that is twice the original angle.
By using these identities, we can simplify complicated trigonometric expressions, solve trigonometric equations, and express trigonometric functions in terms of other functions, which makes it much easier to understand and work with them.
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