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Trigonometry

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Trigonometry (literally "measurement of figures with three angles") is the area of mathematics that deals with relationships between the side lengths and angles of triangles.

Although one can consider triangles on curved surfaces (using spherical and hyperbolic trigonometry), the term trigonometry usually refers to planar trigonometry — that is, the study of triangles in a plane (two-dimensional Euclidean space).

After considering the measurement of angles in degrees and radians, one may define the following six basic trigonometric functions using any right triangle containing a specified acute angle \theta:

  • \sin\theta=\frac{\mbox{opp}}{\mbox{hyp}}
  • \cos\theta=\frac{\mbox{adj}}{\mbox{hyp}}
  • \tan\theta=\frac{\mbox{opp}}{\mbox{adj}}
  • \csc\theta=\frac{\mbox{hyp}}{\mbox{opp}}
  • \sec\theta=\frac{\mbox{hyp}}{\mbox{adj}}
  • \cot\theta=\frac{\mbox{adj}}{\mbox{opp}}

Here "opp" and "adj" represent, respectively, the lengths of the sides opposite and adjacent to the angle \theta, and "hyp" represents the length of the hypotenuse (the side opposite the right angle).

The names of the functions above are abbreviated forms of the words sine, cosine, tangent, cotangent, secant and cosecant.

The values of these six functions for non-acute angles (less than or equal to zero, or equalgreater than a right angle) can be found by reference to the unit circle.

There are many properties of these functions that are true for any given angle. The most fundamental of these are:

  • \tan\theta=\frac{\sin\theta}{\cos\theta}
  • \cot\theta=\frac{\cos\theta}{\sin\theta}= \frac{1}{\tan\theta}
  • \sec\theta=\frac{1}{\cos\theta}
  • \csc\theta=\frac{1}{\sin\theta}
  • \sin^2\theta+\cos^2\theta=1

For other similar properties, see our list of trigonometric identities.

Resource: Online sine, cosine, tangent chart

TablesEdit

Can the function be used? \theta=0^\circ \theta=90^\circ \theta=180^\circ \theta=270^\circ Everything else
\sin\theta Yes Yes Yes Yes Yes
\cos\theta Yes Yes Yes Yes Yes
\tan\theta Yes No Yes No Yes
\csc\theta No Yes No Yes Yes
\sec\theta Yes No Yes No Yes
\cot\theta No Yes No Yes Yes
\mathrm{c}i\mathrm{s}\; \theta Yes Yes Yes Yes Yes
Multiply... \sin\theta \cos\theta \tan\theta
\sin\theta \sin\theta^2 \frac{\tan\theta}{hyp^2} \frac{opp^2}{\cos\theta}
\cos\theta \frac{\tan\theta}{hyp^2} \cos\theta^2
\tan\theta \tan\theta^2

External LinksEdit

Trigonometry Index at Math is Fun

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