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Category
Trigonometric identities
Summary
In trigonometry, trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities potentially involving angles but also involving side lengths or other lengths of a triangle.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
Pythagorean trigonometric identity
The basic relationship between the sine and cosine is given by the Pythagorean identity:
where means and means
This can be viewed as a version of the Pythagorean theorem, and follows from the equation for the unit circle. This equation can be solved for either the sine or the cosine:
where the sign depends on the quadrant of
Dividing this identity by , , or both yields the following identities:
Using these identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):
By examining the unit circle, one can establish the following properties of the trigonometric functions.
When the direction of a Euclidean vector is represented by an angle this is the angle determined by the free vector (starting at the origin) and the positive -unit vector. The same concept may also be applied to lines in a Euclidean space, where the angle is that determined by a parallel to the given line through the origin and the positive -axis. If a line (vector) with direction is reflected about a line with direction then the direction angle of this reflected line (vector) has the value
The values of the trigonometric functions of these angles for specific angles satisfy simple identities: either they are equal, or have opposite signs, or employ the complementary trigonometric function.
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