Difference between revisions of "Trigonometry"

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Consider a [[unit circle]] that is centered at the origin. By picking a point on the circle, and dropping a perpendicular line to the x-axis, a right triangle is formed with a [[hypotenuse]] 1 unit long. Letting the angle at the origin be <math>\theta </math> and the coordinates of the point we picked to be <math>(x,y) </math>, we have:
 
Consider a [[unit circle]] that is centered at the origin. By picking a point on the circle, and dropping a perpendicular line to the x-axis, a right triangle is formed with a [[hypotenuse]] 1 unit long. Letting the angle at the origin be <math>\theta </math> and the coordinates of the point we picked to be <math>(x,y) </math>, we have:
  
<math>\sin \theta = y </math>
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<cmath>\begin{align*} \sin \theta &= y \\
  
<math>\cos \theta = x </math>
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\cos \theta &= x \\
  
<math>\tan \theta = \frac{y}{x} </math>
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\tan \theta &= \frac{y}{x} \\
  
<math>\csc \theta = \frac{1}{y} </math>
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\csc \theta &= \frac{1}{y} \\
  
<math>\sec \theta = \frac{1}{x} </math>
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\sec \theta &= \frac{1}{x} \\
  
<math>\cot \theta = \frac{x}{y} </math>
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\cot \theta &= \frac{x}{y} \end{align*}</cmath>
  
 
Note that <math>(x,y) </math> is the rectangular coordinates for the point <math> (1,\theta) </math>.
 
Note that <math>(x,y) </math> is the rectangular coordinates for the point <math> (1,\theta) </math>.
  
This is true for all angles, even negative angles and angles greater than 360 degrees. Due to the way trig ratios are defined for non-acute angles, the value of a trig ratio could be positive or negative, or even 0.  
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This is true for all angles, even negative angles and angles greater than 360 degrees. Due to the way trig ratios are defined for non-acute angles, the value of a trig ratio could be positive or negative, or even 0.
  
 
==Trigonometric Identities==
 
==Trigonometric Identities==

Revision as of 13:17, 19 October 2013

Trigonometry seeks to find the lengths of a triangle's sides, given 2 angles and a side. Trigonometry is closely related to analytic geometry.

Basic definitions

Usually we call an angle $\theta$, read "theta", but $\theta$ is just a variable. We could just as well call it $a$.

For the following definitions, the "opposite side" is the side opposite of angle $\theta$, and the "adjacent side" is the side that is part of angle $\theta$, but is not the hypotenuse.

i.e. If ABC is a right triangle with right angle C, and angle A = $\theta$, then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.

306090triangle.gif

Sine

The sine of an angle $\theta$, abbreviated $\sin \theta$, is the ratio between the opposite side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\sin 30^{\circ}=\frac 12$.

Cosine

The cosine of an angle $\theta$, abbreviated $\cos \theta$, is the ratio between the adjacent side and the hypotenuse of a triangle. For instance, in the 30-60-90 triangle above, $\cos 30^{\circ} =\frac{\sqrt{3}}{2}$.

Tangent

The tangent of an angle $\theta$, abbreviated $\tan \theta$, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\tan 30^{\circ}=\frac{\sqrt{3}}{3}$. (Note that $\tan \theta=\frac{\sin\theta}{\cos\theta}$.)

Cosecant

The cosecant of an angle $\theta$, abbreviated $\csc \theta$, is the ratio between the hypotenuse and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\csc 30^{\circ}=2$. (Note that $\csc \theta=\frac{1}{\sin \theta}$.)

Secant

The secant of an angle $\theta$, abbreviated $\sec \theta$, is the ratio between the hypotenuse and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, $\sec 30^{\circ}=\frac{2\sqrt{3}}{3}$. (Note that $\sec \theta=\frac{1}{\cos \theta}$.)


Cotangent

The cotangent of an angle $\theta$, abbreviated $\cot \theta$, is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, $\cot 30^{\circ}=\sqrt{3}$. (Note that $\cot \theta=\frac{\cos\theta}{\sin\theta}$ or $\cot \theta = \frac{1}{\tan \theta}$.)

Trigonometery Definitions for non-acute angles

Consider a unit circle that is centered at the origin. By picking a point on the circle, and dropping a perpendicular line to the x-axis, a right triangle is formed with a hypotenuse 1 unit long. Letting the angle at the origin be $\theta$ and the coordinates of the point we picked to be $(x,y)$, we have:

\begin{align*} \sin \theta &= y \\

\cos \theta &= x \\

\tan \theta &= \frac{y}{x} \\

\csc \theta &= \frac{1}{y} \\

\sec \theta &= \frac{1}{x} \\

\cot \theta &= \frac{x}{y} \end{align*} (Error compiling LaTeX. ! Paragraph ended before \align* was complete.)

Note that $(x,y)$ is the rectangular coordinates for the point $(1,\theta)$.

This is true for all angles, even negative angles and angles greater than 360 degrees. Due to the way trig ratios are defined for non-acute angles, the value of a trig ratio could be positive or negative, or even 0.

Trigonometric Identities

There are many identities that are based on trigonometric functions.

Pythagorean Identities

  • $\sin^2\theta+\cos^2\theta=1$
  • $1+\tan^2\theta=\sec^2\theta$
  • $1+\cot^2\theta=\csc^2\theta$

Double-Angle Identities

  • $\sin 2\theta=2\sin\theta\cos\theta$
  • $\cos 2\theta=\cos^2\theta-\sin^2\theta$
  • $\tan 2\theta=\frac{2\tan\theta}{1-\tan^2\theta}$

Half-Angle Identites

  • $\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}$
  • $\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}$
  • $\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}$

See also

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