1.3.2: Trigonometry Functions
- Page ID
- 111529
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Trigonometric Relations Review
This diagram shows a right triangle with an hypotenuse of length 1. For an angle theta:
sine(theta) = x
cosine(theta) = y
https://commons.wikimedia.org/wiki/F...itskreis_1.svg. Author: Stephan Kulla (User:Stephan Kulla)
This file is made available under the Creative Commons CC0 1.0 Universal Public Domain Dedication
For a general triangle with hypotenuse, h (or circle with radius h) the relations are:
sine(theta) = x/h
cosine(theta) = y/h
tangent(theta) = y/x
The following are the corresponding MATLAB/Octave functions:
sin(), sind()
sin(theta) for theta in radians
sind(angle) for angle in degrees
cos(), cosd()
cos(theta) for theta in radians
cosd(angle) for angle in degrees
tan(), tand()
tan(theta) for theta in radians
tand(angle) for angle in degrees
atan(), atand()
atan(a) is the arctangent function, which is also called the inverse tangent function. atan(a) returns the angle in radians, with values in the interval [-pi/2, pi/2].
atand(a) is the arctangent function, which returns the angle in degrees with values in the interval [-90, 90].
atan2(y, x), atan2d(y, x)
atan2(y,x) allows the user to specify both the x & y (the cos and sin parts) of the tangent ratio. When x & y are known, atand is generally preferred. The 2 input version is advantageous because there is an ambiguity when only the tangent value is entered. This is because tangent is periodic with a period of pi radians (180 degrees).
In other words, because tan(theta) = sin(theta) / cos(theta) also = -sin(theta) / -cos(theta)
this occurs when theata2 = theta1 + pi
For example,
tan(pi/3) = 1.7321
and
tan(-2*pi/3) = 1.7321
In MATLAB and Octave, atan(1.7321) = 1.0472 = pi/3
The following pairs of angles illustrate that they have the same value for the tangent:
theta1 = pi/6 (30 degrees) and theta2 = -5*pi/6 (-150 degrees or +210 degrees):
tan(theta1) = sin(theta1) / cos(theta1) = 0.5000 / 0.8660 = 0.5774
tan(theta2) = sin(theta2) / cos(theta2) = -0.5000 / -0.8660 = 0.5774
theta1 = -pi/6 (-30 degrees) and theta2 = 5*pi/6 (150 degrees):
tan(theta1) = sin(theta1) / cos(theta1) = -0.5000 / 0.8660 = -0.5774
tan(theta2) = sin(theta2) / cos(theta2) = 0.5000 / -0.8660 = -0.5774
atan2() and atan2d() solve this problem because both the numerator and denominator of the tangent are inputs. Examples:
sin(4*pi/3) = -0.8660
cos(4*pi/3) = -0.5000
atan2(-0.8660, -0.5000) = -2.0944 = -2*pi/3
Note that the inputs for the sin() cos() and tan() functions are in radians and the outputs of the atan() and atan2() functions are in radians.
When you want to work in degrees, use atand() and atan2d().
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