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Trigonometric functions are based on relationships present between
the sides and angles of triangles. Many functions are derived from
a right triangle—a triangle containing a right (90˚)
angle. Consider the right triangle below with sides *A, B,* and *C,* and
angles α*,* β*,* and γ.

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Side C, which is the longest side and the side opposite the right
angle, is known as the *hypotenuse of the
triangle.*

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A commonly used trigonometric relationship for right triangles
is the *Pythagorean theorem.* The Pythagorean
theorem is an expression of the relationship between the hypotenuse
and the other two sides of a right triangle:

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The sum of the squares of the lengths of the two sides of a right
triangle is equal to the square of the length of the hypotenuse.

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Using the sides of the labeled triangle yields the following:

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Suppose that sides *A* and *B* are 3 and 4 units long, respectively.
The Pythagorean theorem can be used to solve for the length of side *C:*

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Three trigonometric relationships are based on the ratios of
the lengths of the sides of a right triangle. The sine (abbreviated *sin*) of an angle is defined as the
ratio of the length of the side of the triangle opposite the angle
to the length of the hypotenuse. Using the labeled triangle yields
the following:

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With *A* = 3, *B* = 4, and *C* = 5:

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The cosine (abbreviated *cos*) of
an angle is defined as the ratio of the length of the side of the
triangle adjacent to the angle to the length of the hypotenuse.
Using the labeled triangle yields the following:

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With *A* = 3, *B* = 4, and *C* = 5:

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The third function, the tangent (abbreviated *tan*)
of an angle, is defined as the ratio of the length of the side of
the triangle opposite the angle to that of the side adjacent to
the angle. Using the labeled triangle yields the following:

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With *A* = 3, *B* = 4, and *C* = 5:

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Two useful trigonometric relationships are applicable to *all* triangles. The first is known as
the law of sines:

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The ratio between the length of any side of a triangle and the
angle opposite that side is equal to the ratio between the length
of any other side of the triangle and the angle opposite that side.

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With respect to the labeled triangle, this may be stated as the
following:

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A second trigonometric relationship applicable to *all* triangles is the law of ...