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.
  
==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.
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>.
 
  
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.
  
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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== Definitions ==
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The trigonometric functions have several definitions. The definition usually taught first is the right triangle definition, for its ease of access. An intermediate to olympiad geometry course usually uses the unit circle definition of trigonometry. In complex analysis, the Taylor series definition of trigonometry is preferred to extend trigonometry to a complex domain, although this is beyond the scope of competition math.
  
[[Image:306090triangle.gif]]
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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. Because angle <math>A</math> must be less than <math>90^{\circ}</math> for the triangle to stay right, these definitions only work for acute angles.
  
===Sine===
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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>
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''': 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>
  
===Cosine===
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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, and '''T'''angent = '''O'''pposite / '''A'''djacent
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>.
 
  
===Tangent===
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More uncommon are the reciprocals of the trigonometric functions, listed below.
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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* '''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>
  
===Cosecant===
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This definition is most commonly taught in introductory geometry classes for its simplicity. However, it is limiting. It only works if <math>\triangle ABC</math> is right, which means that the trigonometric functions are only defined when angle <math>A</math> is acute.
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>.)
 
  
===Secant===
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=== Unit circle definition ===
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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[[IMage:Unit_circle_trig.png|thumb|right|300px]] Consider the [[unit circle]], the circle with radius one centered at the origin. Starting at <math>(1, 0)</math>, walk a distance <math>\theta</math> counterclockwise around the unit circle, as shown in the diagram. The coordinates of this point are defined to be <math>(\cos (\theta), \sin (\theta) )</math>.
  
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As for the other trigonometric functions, <math>\tan (\theta)</math> is defined to be the ratio of <math>\sin (\theta)</math> to <math>\cos (\theta)</math>, and cosecant, secant, and cotangent are defined to be the reciprocals of sine, cosine, and tangent, respectively.
  
===Cotangent===
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The benefit of this definition is that it matches the right triangle definition for acute angles, but extends their domain from acute angles to all real-valued angles. As such, this definition is usually preferred in intermediate to olympiad geometry settings.
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>.)
 
  
==Trigonometry Definitions for non-acute angles==
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=== Taylor series definition ===
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:
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The [[Taylor series]] for sine and cosine are used as the definition of sine and cosine in analysis, particularly complex analysis. Defining the trigonometric functions this way gives a concrete way to extend the definition of trigonometry from the real numbers to the full complex plane. The taylor series for sine and cosine are shown below: <cmath>\sin (x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \cdots</cmath> <cmath>\cos (x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \cdots</cmath>
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These definitions are not used in high school math competitions; however, they do appear in the [[Putnam]] and other university competitions.
  
<cmath>\begin{align*} \sin \theta &= y \\
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== Applications to geometry ==
\cos \theta &= x \\
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Trigonometry sees significant use in intermediate geometry. Aside from returning ratios of right triangles, trigonometry can apply to all triangles, via the law of sines and the law of cosines.
\tan \theta &= \frac{y}{x} \\
 
\csc \theta &= \frac{1}{y} \\
 
\sec \theta &= \frac{1}{x} \\
 
\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>.
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=== Law of sines ===
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The [[law of sines]] states that in any <math>\triangle ABC</math>, <cmath>\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R,</cmath> where <math>a</math> is the side opposite to <math>A</math>, <math>b</math> opposite to <math>B</math>, <math>c</math> opposite to <math>C</math>, and and <math>R</math> is the [[circumradius]] of <math>\triangle ABC</math>. The law of sines is particularly handy in problems involving the circumradius, seeing extremely wide usage in intermediate geometry.
  
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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=== Law of cosines ===
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The [[law of cosines]] states that in any <math>\triangle ABC</math>, <cmath>c^2 = a^2 + b^2 - 2ab\cos (C),</cmath> where <math>a</math> is the side opposite to <math>A</math>, <math>b</math> opposite to <math>B</math>, and <math>c</math> opposite to <math>C</math>. It is a generalization of the [[Pythagorean theorem]] and is used to prove several famous results, such as [[Heron's formula]] and [[Stewart's theorem]]. However, it sees limited applicability compared to the law of sines, as usage of the law of cosines can get very algebra-heavy.It is helpful to memorize "nice" values of sine and cosine as it can come handy in contests, especially if you wish to apply either this law or law of sines.
  
==Trigonometric Identities==
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== Trigonometric identities ==
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[[Trigonometric identities]] are expressions involving trigonometric functions that are true for all inputs. The trigonometric functions are known for their identities. The most commonly used identities in contest math are:
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* Pythagorean identities
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* Angle addition identities
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* Double angle identities
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* Half angle identities
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* Sum-to-product identities
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* Product-to-sum identities
  
There are many identities that are based on trigonometric functions.
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== See also ==
 
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* [[Trigonometric identities]]
===Pythagorean Identities===
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* [[Law of sines]]
 
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* [[Law of cosines]]
*<math>\sin^2\theta+\cos^2\theta=1</math> 
 
*<math>1+\tan^2\theta=\sec^2\theta</math> 
 
*<math>1+\cot^2\theta=\csc^2\theta</math>
 
 
 
===Double-Angle Identities===
 
 
 
*<math>\sin 2\theta=2\sin\theta\cos\theta</math>
 
*<math>\cos 2\theta=\cos^2\theta-\sin^2\theta</math>
 
*<math>\tan 2\theta=\frac{2\tan\theta}{1-\tan^2\theta}</math>
 
 
 
===Half-Angle Identities===
 
 
 
*<math>\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}</math>
 
*<math>\cos\frac{\theta}{2}=\pm\sqrt{\frac{1+\cos\theta}{2}}</math>
 
*<math>\tan\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{1+\cos\theta}}</math>
 
  
==See also==
 
* [[Trigonometric identities]]
 
* [[Trigonometric substitution]]
 
* [[Geometry]]
 
 
[[Category:Trigonometry]]
 
[[Category:Trigonometry]]
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[[Category:Definition]]

Revision as of 16:55, 31 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.

Definitions

The trigonometric functions have several definitions. The definition usually taught first is the right triangle definition, for its ease of access. An intermediate to olympiad geometry course usually uses the unit circle definition of trigonometry. In complex analysis, the Taylor series definition of trigonometry is preferred to extend trigonometry to a complex domain, although this is beyond the scope of competition math.

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. Because angle $A$ must be less than $90^{\circ}$ for the triangle to stay right, these definitions only work for acute angles.

  • 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, and 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 $A$. \[\cot (A) = \frac{1}{\tan (x)} = \frac{\textrm{adjacent}}{\textrm{opposite}} = \frac{b}{a}.\]

This definition is most commonly taught in introductory geometry classes for its simplicity. However, it is limiting. It only works if $\triangle ABC$ is right, which means that the trigonometric functions are only defined when angle $A$ is acute.

Unit circle definition

Unit circle trig.png

Consider the unit circle, the circle with radius one centered at the origin. Starting at $(1, 0)$, walk a distance $\theta$ counterclockwise around the unit circle, as shown in the diagram. The coordinates of this point are defined to be $(\cos (\theta), \sin (\theta) )$.

As for the other trigonometric functions, $\tan (\theta)$ is defined to be the ratio of $\sin (\theta)$ to $\cos (\theta)$, and cosecant, secant, and cotangent are defined to be the reciprocals of sine, cosine, and tangent, respectively.

The benefit of this definition is that it matches the right triangle definition for acute angles, but extends their domain from acute angles to all real-valued angles. As such, this definition is usually preferred in intermediate to olympiad geometry settings.

Taylor series definition

The Taylor series for sine and cosine are used as the definition of sine and cosine in analysis, particularly complex analysis. Defining the trigonometric functions this way gives a concrete way to extend the definition of trigonometry from the real numbers to the full complex plane. The taylor series for sine and cosine are shown below: \[\sin (x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} \cdots\] \[\cos (x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \cdots\] These definitions are not used in high school math competitions; however, they do appear in the Putnam and other university competitions.

Applications to geometry

Trigonometry sees significant use in intermediate geometry. Aside from returning ratios of right triangles, trigonometry can apply to all triangles, via the law of sines and the law of cosines.

Law of sines

The law of sines states that in any $\triangle ABC$, \[\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R,\] where $a$ is the side opposite to $A$, $b$ opposite to $B$, $c$ opposite to $C$, and and $R$ is the circumradius of $\triangle ABC$. The law of sines is particularly handy in problems involving the circumradius, seeing extremely wide usage in intermediate geometry.

Law of cosines

The law of cosines states that in any $\triangle ABC$, \[c^2 = a^2 + b^2 - 2ab\cos (C),\] where $a$ is the side opposite to $A$, $b$ opposite to $B$, and $c$ opposite to $C$. It is a generalization of the Pythagorean theorem and is used to prove several famous results, such as Heron's formula and Stewart's theorem. However, it sees limited applicability compared to the law of sines, as usage of the law of cosines can get very algebra-heavy.It is helpful to memorize "nice" values of sine and cosine as it can come handy in contests, especially if you wish to apply either this law or law of sines.

Trigonometric identities

Trigonometric identities are expressions involving trigonometric functions that are true for all inputs. The trigonometric functions are known for their identities. The most commonly used identities in contest math are:

  • Pythagorean identities
  • Angle addition identities
  • Double angle identities
  • Half angle identities
  • Sum-to-product identities
  • Product-to-sum identities

See also