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Difference between revisions of "Trigonometry"

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Trigonometry seeks to find the lengths of a [[triangle]]'s sides, given 2 [[angle]]s and a side. Trigonometry is closely related to [[analytic geometry]].
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In [[geometry]], '''Trigonometry''' is the study of the '''trigonometric functions,''' functions that seek to relate the lengths and angles of [[triangles]]. Trigonometry is integral to geometry, as many famous results were proven using trigonometry.
  
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==Basic definitions==
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In contest math, trigonometry's use is not just limited to geometry; problems involving equations with trigonometric functions are very common. These are often solved via clever usage of the trigonometric functions' many [[Trigonometric identities | identities]], which drastically simplify expressions.
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Usually we call an angle <math>\theta</math>, read "theta", but <math>\theta</math> is just a variable. We could just as well call it <math> a</math>.
 
  
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For the following definitions, the "opposite side" is the side opposite of angle <math>\theta</math>, and the "adjacent side" is the side that is part of angle <math>\theta</math>, but is not the hypotenuse.  
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Outside of competition math, trigonometry is the backbone of much of analysis, especially Fourier analysis.
  
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i.e. If ABC is a right triangle with right angle C, and angle A = <math>\theta</math>, then BC is the "opposite side", AC is the "adjacent side", and AB is the hypotenuse.  
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== Right triangle definition ==
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The right triangle definition of trigonometry involves the ratios between edges of a right triangle, with respect to a given angle. The definitions below will be referring to angle <math>A</math>, with side lengths specified in the diagram.
  
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[[Image:306090triangle.gif]]
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* '''Sine''': The sine of angle <math>A</math>, denoted <math>\sin (A)</math>, is defined as the ratio of the side opposite <math>A</math> to [[Image:Trig_triangle.png|thumb|right|350px]] the hypotenuse; <cmath>\sin (A) = \frac{\textrm{opposite}}{\textrm{hypotenuse}} = \frac{a}{c}.</cmath>
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* '''Cosine''': The sine of angle <math>A</math>, denoted <math>\sin (A)</math>, is defined as the ratio of the side adjacent <math>A</math> to the hypotenuse; <cmath>\cos (A) = \frac{\textrm{adjacent}}{\textrm{hypotenuse}} = \frac{b}{c}.</cmath>
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* '''Tangent''': The tangent of angle <math>A</math>, denoted <math>\tan (A)</math>, is defined as the ratio of the side opposite <math>A</math> to the side adjacent to <math>A</math>; <cmath>\tan (A) = \frac{\textrm{opposite}}{\textrm{adjacent}} = \frac{a}{b}.</cmath>
  
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===Sine===
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A common mneumonic to remember this is '''SOH-CAH-TOA''', where '''S'''ine = '''O'''pposite / '''H'''ypotenuse, '''C'''osine = '''A'''djacent / '''H'''ypotenuse  '''T'''angent = '''O'''pposite / '''A'''djacent
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The sine of an angle <math>\theta</math>, abbreviated <math>\sin \theta</math>, is the ratio between the opposite side and the [[hypotenuse]] of a triangle. For instance, in the 30-60-90 triangle above, <math>\sin 30^{\circ}=\frac 12</math>.
 
  
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===Cosine===
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More uncommon are the reciprocals of the trigonometric functions, listed below.
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The cosine of an angle <math>\theta</math>, abbreviated <math>\cos \theta</math>, is the ratio between the adjacent side and the [[hypotenuse]] of a triangle. For instance, in the 30-60-90 triangle above, <math>\cos 30^{\circ} =\frac{\sqrt{3}}{2}</math>.
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* '''Cosecant''': The cosecant of angle <math>A</math>, denoted <math>\csc (A)</math>, is defined as the reciprocal of the sine of <math>A</math>; <cmath>\csc (A) = \frac{1}{\sin (x)} = \frac{\textrm{hypotenuse}}{\textrm{opposite}} = \frac{c}{a}.</cmath>
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* '''Secant''': The secant of angle <math>A</math>, denoted <math>\sec (A)</math>, is defined as the reciprocal of the cosine of <math>A</math>; <cmath>\sec (A) = \frac{1}{\cos (x)} = \frac{\textrm{hypotenuse}}{\textrm{adjacent}} = \frac{c}{b}.</cmath>
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* '''Cotangent''': The cotangent of angle <math>A</math>, denoted <math>\cot (A)</math>, is defined as the reciprocal of the tangent of <math>A</math>; <cmath>\cot (A) = \frac{1}{\tan (x)} = \frac{\textrm{adjacent}}{\textrm{opposite}} = \frac{b}{a}.</cmath>
  
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===Tangent===
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== Unit Circle Definition ==
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The tangent of an angle <math>\theta</math>, abbreviated <math>\tan \theta</math>, is the ratio between the opposite side and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, <math>\tan 30^{\circ}=\frac{\sqrt{3}}{3}</math>. (Note that <math> \tan \theta=\frac{\sin\theta}{\cos\theta}</math>.)
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Consider the [[unit circle]], the circle with radius one 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:
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===Cosecant===
 
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The cosecant of an angle <math>\theta</math>, abbreviated <math>\csc \theta</math>, is the ratio between the [[hypotenuse]] and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, <math>\csc 30^{\circ}=2</math>. (Note that <math> \csc \theta=\frac{1}{\sin \theta}</math>.)
 
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===Secant===
 
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The secant of an angle <math>\theta</math>, abbreviated <math>\sec \theta</math>, is the ratio between the [[hypotenuse]] and the adjacent side of a triangle. For instance, in the 30-60-90 triangle above, <math>\sec 30^{\circ}=\frac{2\sqrt{3}}{3}</math>. (Note that <math> \sec \theta=\frac{1}{\cos \theta}</math>.)
 
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===Cotangent===
 
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The cotangent of an angle <math>\theta</math>, abbreviated <math>\cot \theta</math>, is the ratio between the adjacent side and the opposite side of a triangle. For instance, in the 30-60-90 triangle above, <math>\cot 30^{\circ}=\sqrt{3}</math>. (Note that <math> \cot \theta=\frac{\cos\theta}{\sin\theta}</math> or <math> \cot \theta = \frac{1}{\tan \theta}</math>.)
 
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==Trigonometry Definitions for non-acute angles==
 
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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:
 
  
 
<cmath>\begin{align*} \sin \theta &= y \\
 
<cmath>\begin{align*} \sin \theta &= y \\
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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.
 
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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==Trigonometric Identities==
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== Trigonometric Identities ==
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There are many identities that are based on trigonometric functions.
 
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===Pythagorean Identities===
 
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*<math>\sin^2\theta+\cos^2\theta=1</math> 
 
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*<math>1+\tan^2\theta=\sec^2\theta</math> 
 
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*<math>1+\cot^2\theta=\csc^2\theta</math>
 
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===Double-Angle Identities===
 
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*<math>\sin 2\theta=2\sin\theta\cos\theta</math>
 
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*<math>\cos 2\theta=\cos^2\theta-\sin^2\theta</math>
 
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*<math>\tan 2\theta=\frac{2\tan\theta}{1-\tan^2\theta}</math>
 
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===Half-Angle Identities===
 
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*<math>\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}</math>
 
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*<math>\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}</math>
 
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*<math>\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}</math>
 
  
 
==See also==
 
==See also==
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* [[Trigonometric substitution]]
 
* [[Trigonometric substitution]]
 
* [[Geometry]]
 
* [[Geometry]]
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[[Category:Geometry]]
 
[[Category:Trigonometry]]
 
[[Category:Trigonometry]]
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[[Category:Definition]]

Revision as of 12:42, 30 May 2021

In geometry, Trigonometry is the study of the trigonometric functions, functions that seek to relate the lengths and angles of triangles. Trigonometry is integral to geometry, as many famous results were proven using trigonometry.

In contest math, trigonometry's use is not just limited to geometry; problems involving equations with trigonometric functions are very common. These are often solved via clever usage of the trigonometric functions' many identities, which drastically simplify expressions.

Outside of competition math, trigonometry is the backbone of much of analysis, especially Fourier analysis.

Right triangle definition

The right triangle definition of trigonometry involves the ratios between edges of a right triangle, with respect to a given angle. The definitions below will be referring to angle $A$, with side lengths specified in the diagram.

  • Sine: The sine of angle $A$, denoted $\sin (A)$, is defined as the ratio of the side opposite $A$ to
    Trig triangle.png
    the hypotenuse; \[\sin (A) = \frac{\textrm{opposite}}{\textrm{hypotenuse}} = \frac{a}{c}.\]
  • Cosine: The sine of angle $A$, denoted $\sin (A)$, is defined as the ratio of the side adjacent $A$ to the hypotenuse; \[\cos (A) = \frac{\textrm{adjacent}}{\textrm{hypotenuse}} = \frac{b}{c}.\]
  • Tangent: The tangent of angle $A$, denoted $\tan (A)$, is defined as the ratio of the side opposite $A$ to the side adjacent to $A$; \[\tan (A) = \frac{\textrm{opposite}}{\textrm{adjacent}} = \frac{a}{b}.\]

A common mneumonic to remember this is SOH-CAH-TOA, where Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse Tangent = Opposite / Adjacent

More uncommon are the reciprocals of the trigonometric functions, listed below.

  • Cosecant: The cosecant of angle $A$, denoted $\csc (A)$, is defined as the reciprocal of the sine of $A$; \[\csc (A) = \frac{1}{\sin (x)} = \frac{\textrm{hypotenuse}}{\textrm{opposite}} = \frac{c}{a}.\]
  • Secant: The secant of angle $A$, denoted $\sec (A)$, is defined as the reciprocal of the cosine of $A$; \[\sec (A) = \frac{1}{\cos (x)} = \frac{\textrm{hypotenuse}}{\textrm{adjacent}} = \frac{c}{b}.\]
  • Cotangent: The cotangent of angle $A$, denoted $\cot (A)$, is defined as the reciprocal of the tangent of $A$; \[\cot (A) = \frac{1}{\tan (x)} = \frac{\textrm{adjacent}}{\textrm{opposite}} = \frac{b}{a}.\]

Unit Circle Definition

Consider the unit circle, the circle with radius one 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*}

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

See also

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