Trigonometric function

In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers. All of these approaches will be presented below.

Right triangle definitions

In order to define the trigonometric functions for the angle A, start with an arbitrary right triangle that contains the angle A:

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We use the following names for the sides of the triangle:

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• The hypotenuse is the side opposite the right angle, or defined as the longest side of a right-angled triangle, in this case h.
• The opposite side is the side opposite to the angle we are interested in, in this case a.
• The adjacent side is the side that is in contact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.

All triangles are taken to exist in the Euclidean plane so that the inside angles of each triangle sum to π radians (or 180°). Then,

Related Topics:
Euclidean plane - Radian - °

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1) The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

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:sin A = rac { extrm{opposite}} { extrm{hypotenuse}} = rac {a} {h}.

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Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.

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2) The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

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:cos A = rac { extrm{adjacent}} { extrm{hypotenuse}} = rac {b} {h}.

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3) The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

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: an A = rac { extrm{opposite}} { extrm{adjacent}} = rac {a} {b} .

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The remaining three functions are best defined using the above three functions.

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4) The cosecant csc(A) is the multiplicative inverse of sin(A), i.e. the ratio of the length of the hypotenuse to the length of the opposite side:

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:csc A = rac { extrm{hypotenuse}} { extrm{opposite}} = rac {h} {a} .

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5) The secant sec(A) is the multiplicative inverse of cos(A), i.e. the ratio of the length of the hypotenuse to the length of the adjacent side:

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:sec A = rac { extrm{hypotenuse}} { extrm{adjacent}} = rac {h} {b} .

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6) The cotangent cot(A) is the multiplicative inverse of tan(A), i.e. the ratio of the length of the adjacent side to the length of the opposite side:

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:cot A = rac { extrm{adjacent}} { extrm{opposite}} = rac {b} {a} .

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Mnemonics

There are a number of mnemonics for the above definitions, for example

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SOHCAHTOA (sounds like "soak a toe-a", can be read as "soccer tour"). It means:

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• SOH ... sin = opposite/hypotenuse
• CAH ... cos = adjacent/hypotenuse
• TOA ... tan = opposite/adjacent.

Many other such words and phrases have been contrived. For more see: trigonometry mnemonics.

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Slope definitions

Equivalent to the right-triangle definitions, the trigonometric functions can be defined in terms of the rise, run, and slope of a line segment relative to some horizontal line. The slope is commonly taught as "rise over run" or rise/run. The three main trigonometric functions are commonly taught in the order sine, cosine, tangent. With a unit circle, this gives rise to the following matchings:

Related Topics:
Slope - Unit circle

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• Sine is first, rise is first. Sine takes an angle and tells the rise.
• Cosine is second, run is second. Cosine takes an angle and tells the run.
• Tangent is the slope formula that combines the rise and run. Tangent takes an angle and tells the slope.

This shows the main use of tangent and arctangent, which is converting between the two ways of telling how slanted a line is: angles and slopes.

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While the radius of the circle makes no difference for the slope (the slope doesn't depend on the length of the slanted line), it does affect rise and run. To adjust and find the actual rise and run, just multiply the sine and cosine by the radius. For instance, if the circle has radius 5, the run at an angle of 1 is 5 cos(1).

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