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\begin_body

\begin_layout Title
Math Trigonometry Review
\end_layout

\begin_layout Author
CHSN Review Project
\end_layout

\begin_layout Publishers
\begin_inset Graphics
	filename ccbysa.png
	scale 50

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\begin_layout Standard
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\begin_layout Standard
This review guide was written by Dara Adib.
 It was designed for a test on trigonometry administered by Jeanine Lennon
 to her Math 12H (4H/Precalculus) class on February 12, 2008, but also applies
 to trigonometry material in Math 11H (3H).
\end_layout

\begin_layout Standard
This is a development version of the text that should be considered a work-in-pr
ogress.
\end_layout

\begin_layout Standard
This review guide and other review material are developed by the CHSN Review
 Project.
\end_layout

\begin_layout Standard
Copyright © 2008-2009 Dara Adib.
 This is a freely licensed work, as explained in the Definition of Free
 Cultural Works (
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

freedomdefined.org
\end_layout

\end_inset

).
 It is licensed under the Creative Commons Attribution-Share Alike 3.0 United
 States License.
 To view a copy of this license, visit 
\begin_inset Flex URL
status collapsed

\begin_layout Plain Layout

http://creativecommons.org/licenses/by-sa/3.0/us/
\end_layout

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 or send a letter to Creative Commons, 171 Second Street, Suite 300, San
 Francisco, California, 94105, USA.
\end_layout

\begin_layout Standard
This review guide is provided 
\begin_inset Quotes eld
\end_inset

as is
\begin_inset Quotes erd
\end_inset

 without warranty of any kind, either expressed or implied.
 You should not assume that this review guide is error-free or that it will
 be suitable for the particular purpose which you have in mind when using
 it.
 In no event shall the CHSN Review Project be liable for any special, incidental
, indirect or consequential damages of any kind, or any damages whatsoever,
 including, without limitation, those resulting from loss of use, data or
 profits, whether or not advised of the possibility of damage, and on any
 theory of liability, arising out of or in connection with the use or performanc
e of this review guide or other documents which are referenced by or linked
 to in this review guide.
\end_layout

\begin_layout Standard
\begin_inset Newpage newpage
\end_inset


\end_layout

\begin_layout Section
Angles and Sectors of Circles
\end_layout

\begin_layout Subsection
Radians
\end_layout

\begin_layout Itemize
To convert degrees to radians, multiply by 
\begin_inset Formula $\frac{\pi}{180}$
\end_inset

.
\end_layout

\begin_layout Itemize
To convert radians to degrees, multiply by 
\begin_inset Formula $\frac{180}{\pi}$
\end_inset

.
\end_layout

\begin_layout Itemize
One radian equals 
\begin_inset Formula $\frac{180}{\pi}$
\end_inset

 or approximately 
\begin_inset Formula $57.296^{\circ}$
\end_inset

.
\end_layout

\begin_layout Subsection
Coterminal Angles
\end_layout

\begin_layout Description
coterminal
\begin_inset space ~
\end_inset

angles different angles that have the same initial and terminal ray
\end_layout

\begin_layout Standard
The differences between coterminal angles are multiples of 
\begin_inset Formula $360^{\circ}$
\end_inset

.
\end_layout

\begin_layout Quote
Examples: 
\begin_inset Formula $60^{\circ}$
\end_inset

, 
\begin_inset Formula $-300^{\circ}$
\end_inset

, 
\begin_inset Formula $420^{\circ}$
\end_inset


\end_layout

\begin_layout Subsection
Sectors
\end_layout

\begin_layout Description
sector part of a circle formed by two radii and an arc
\end_layout

\begin_layout Standard
\begin_inset Formula $s=r\theta$
\end_inset

 (
\begin_inset Formula $\textrm{arc\, length}=\textrm{radius}\times\textrm{angle}$
\end_inset

)
\end_layout

\begin_layout Standard
The angle must be represented in radians.
\end_layout

\begin_layout Description
apparent
\begin_inset space ~
\end_inset

size the angle that an object subtends at one's eyes; this explains why
 an object appears to be smaller when farther away
\end_layout

\begin_layout Section
Trigonometric Functions
\end_layout

\begin_layout Subsection
Reciprocal Functions
\end_layout

\begin_layout Standard
The following are reciprocal functions.
\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle \csc x=\frac{1}{\sin x}}$
\end_inset

 
\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle \sec x=\frac{1}{\cos x}}$
\end_inset

 
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\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle \cot x=\frac{1}{\tan x}}$
\end_inset


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\end_inset


\end_layout

\begin_layout Standard
As a result, the following are true as well.
\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle \sin x=\frac{1}{\csc x}}$
\end_inset

 
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\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle \cos x=\frac{1}{\sec x}}$
\end_inset

 
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\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula ${\displaystyle \tan x=\frac{1}{\cot x}}$
\end_inset


\end_layout

\begin_layout Subsection
Cofunctions
\end_layout

\begin_layout Standard
The following are cofunctions.
\end_layout

\begin_layout Itemize
\begin_inset Formula $\sin x=\cos(90-x)$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\tan x=\cot(90-x)$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\sec x=\csc(90-x)$
\end_inset


\end_layout

\begin_layout Standard
As a result, the following are true as well.
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\begin_layout Itemize
\begin_inset Formula $\cos x=\sin(90-x)$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\cot x=\tan(90-x)$
\end_inset


\end_layout

\begin_layout Itemize
\begin_inset Formula $\csc x=\sec(90-x)$
\end_inset


\end_layout

\begin_layout Section
Special Angles
\end_layout

\begin_layout Subsection
Quadrantal Angles
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\begin_layout Description
quadrantal
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angles angles that have a terminal side coinciding with a coordinate axis
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\begin_layout Standard
The value of the trigonometric function (i.e.
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 By definition, the point 
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 on a unit circle corresponds to 
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.
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As a result, quadrantal angles can be determined easily.
 Table 
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 lists quadrantal angles and the values of trigonometric functions.
 Since 
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, the tangent function of an angle can be found by dividing the sine function
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quadrantal angles
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Subsection
Angles greater than 
\begin_inset Formula $90^{\circ}$
\end_inset


\end_layout

\begin_layout Standard
Angles greater than 
\begin_inset Formula $90^{\circ}$
\end_inset

 may be in different quadrants.
 The trigonometric functions vary over whether the trigonometric function
 of a certain angle is positive or negative.
 It is useful to remember the mnemonic device 
\begin_inset Quotes eld
\end_inset

All Students Take Calculus.
\begin_inset Quotes erd
\end_inset


\end_layout

\begin_layout Itemize
All trigonometric functions are positive in the first quadrant
\end_layout

\begin_layout Itemize
Sine and cosecant are positive in the second quadrant
\end_layout

\begin_layout Itemize
Tangent and cotangent are positive in the third quadrant
\end_layout

\begin_layout Itemize
Cosine and secant are positive in the fourth quadrant
\end_layout

\begin_layout Itemize
The remaining trigonometric functions in each quadrant are negative.
\end_layout

\begin_layout Subsection
Reference Angles
\end_layout

\begin_layout Standard
The use of reference angles is a way to simplify the calculation of the
 values of trigonometric functions in different quadrants.
 Refer to Table 
\begin_inset CommandInset ref
LatexCommand ref
reference "tab:reference-angles"

\end_inset

, where 
\begin_inset Formula $\theta$
\end_inset

 is the angle (in degrees) and 
\begin_inset Formula $\beta$
\end_inset

 is the reference angle.
\end_layout

\begin_layout Standard
\noindent
\align center
\begin_inset Float table
wide false
sideways false
status open

\begin_layout Plain Layout
\noindent
\align center
\begin_inset Tabular
<lyxtabular version="3" rows="5" columns="2">
<features>
<column alignment="center" valignment="top" width="0">
<column alignment="center" valignment="top" width="0">
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Quadrant
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
Reference Angle (
\begin_inset Formula $\beta$
\end_inset

)
\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
I
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=\theta$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
II
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=180^{\circ}-\theta$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
III
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=\theta-180^{\circ}$
\end_inset


\end_layout

\end_inset
</cell>
</row>
<row>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
IV
\end_layout

\end_inset
</cell>
<cell alignment="center" valignment="top" topline="true" bottomline="true" leftline="true" rightline="true" usebox="none">
\begin_inset Text

\begin_layout Plain Layout
\begin_inset Formula $\beta=360^{\circ}-\theta$
\end_inset


\end_layout

\end_inset
</cell>
</row>
</lyxtabular>

\end_inset


\end_layout

\begin_layout Plain Layout
\begin_inset Caption

\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "tab:reference-angles"

\end_inset

reference angles
\end_layout

\end_inset


\end_layout

\end_inset


\end_layout

\begin_layout Section
Sum and Difference Formulas
\end_layout

\begin_layout Subsection
Sum Formulas
\end_layout

\begin_layout Subsubsection
Sine
\end_layout

\begin_layout Standard
\begin_inset Formula $\sin(A+B)=\sin A\cos B+\cos A\sin B$
\end_inset


\end_layout

\begin_layout Subsubsection
Cosine
\end_layout

\begin_layout Standard
\begin_inset Formula $\cos(A+B)=\cos A\cos B-\sin A\sin B$
\end_inset


\end_layout

\begin_layout Subsubsection
Tangent
\end_layout

\begin_layout Standard
The derivation for the tangent of the sum of two angles follows.
\end_layout

\begin_layout Standard
\begin_inset Formula $\tan(A+B)$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula ${\displaystyle \frac{\sin(A+B)}{\cos(A+B)}}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit

\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula ${\displaystyle \frac{\sin A\cos B+\cos A\sin B}{\cos A\cos B-\sin A\sin B}}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit

\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \frac{{\displaystyle \frac{\sin A\cos B}{\cos A\cos B}+\frac{\cos A\sin B}{\cos A\cos B}}}{{\displaystyle \frac{\cos A\cos B}{\cos A\cos B}-\frac{\sin A\sin B}{\cos A\cos B}}}}$
\end_inset


\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula ${\displaystyle \frac{\tan A+\tan B}{1-\tan A\tan B}}$
\end_inset


\end_layout

\begin_layout Subsection
Difference Formulas
\end_layout

\begin_layout Subsubsection
Sine
\end_layout

\begin_layout Standard
\begin_inset Formula $\sin(A-B)=\sin A\cos B-\cos A\sin B$
\end_inset


\end_layout

\begin_layout Subsubsection
Cosine
\end_layout

\begin_layout Standard
\begin_inset Formula $\cos(A-B)=\cos A\cos B+\sin A\sin B$
\end_inset


\end_layout

\begin_layout Subsubsection
Tangent
\end_layout

\begin_layout Standard
The derivation for the tangent of the difference of two angles follows.
\end_layout

\begin_layout Standard
\begin_inset Formula $\tan(A-B)$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula ${\displaystyle \frac{{\displaystyle \sin(A-B)}}{\cos(A-B)}}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit

\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula ${\displaystyle \frac{\sin A\cos B-\cos A\sin B}{\cos A\cos B+\sin A\sin B}}$
\end_inset


\family default
\series default
\shape default
\size default
\emph default
\bar default
\noun default
\color inherit

\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \frac{{\displaystyle \frac{\sin A\cos B}{\cos A\cos B}-\frac{\cos A\sin B}{\cos A\cos B}}}{{\displaystyle \frac{\cos A\cos B}{\cos A\cos B}+\frac{\sin A\sin B}{\cos A\cos B}}}}$
\end_inset


\begin_inset VSpace bigskip
\end_inset


\end_layout

\begin_layout Standard

\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\noun off
\color none
\begin_inset Formula ${\displaystyle \frac{\tan A-\tan B}{1+\tan A\tan B}}$
\end_inset


\end_layout

\begin_layout Section
Identities
\end_layout

\begin_layout Subsection
Pythagorean Identities
\end_layout

\begin_layout Enumerate
\begin_inset Formula $\sin^{2}\theta+\cos^{2}\theta=1$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula $1+\tan^{2}\theta=\sec^{2}\theta$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula $1+\cot^{2}\theta=\csc^{2}\theta$
\end_inset


\end_layout

\begin_layout Subsection
Quotient Identities
\end_layout

\begin_layout Enumerate
\begin_inset Formula ${\displaystyle \tan\theta=\frac{\sin\theta}{\cos\theta}}$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Enumerate
\begin_inset Formula ${\displaystyle \cot\theta=\frac{\cos\theta}{\sin\theta}}$
\end_inset


\end_layout

\begin_layout Section
Graphing Trigonometric Functions
\end_layout

\begin_layout Subsection
General Form
\end_layout

\begin_layout Standard
The general form of a trigonometric function is one of the following (sine
 or cosine).
\end_layout

\begin_layout Standard
\begin_inset Formula $y=a\sin b(x\pm c)\pm d$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $y=a\cos b(x\pm c)\pm d$
\end_inset


\end_layout

\begin_layout Standard
In other words, it can be understood to mean the following.
\end_layout

\begin_layout Standard
\begin_inset Formula $y=\textrm{amplitude}\times\sin\,[\textrm{frequency}\,(x-\textrm{horizontal\, translation})]+\textrm{vertical\, translation}$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $y=\textrm{amplitude}\times\cos\,[\textrm{frequency}\,(x-\textrm{horizontal\, translation})]+\textrm{vertical\, translation}$
\end_inset


\end_layout

\begin_layout Standard
A negative amplitude means a reflection over the 
\begin_inset Formula $x$
\end_inset

-axis.
 The vertical translation may be kept in front only to prevent the ambiguity
 that the number may be part of the trigonometric function.
\end_layout

\begin_layout Subsection
Frequency and Period
\end_layout

\begin_layout Standard
The following statements are true regarding the relationship between frequency
 and period.
\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \textrm{frequency}=\frac{2\pi}{\textrm{period}}}$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \textrm{period}=\frac{2\pi}{\textrm{frequency}}}$
\end_inset


\begin_inset VSpace medskip
\end_inset


\end_layout

\begin_layout Subsection
Sine and Cosine
\end_layout

\begin_layout Standard
The sine function follows the form zero, maximum, zero, minimum, before
 any translation.
\end_layout

\begin_layout Standard
The cosine function follows the form maximum, zero, minimum, zero, before
 any translation.
\end_layout

\begin_layout Standard
As a result, sine and cosine are horizontal translations of each other.
\end_layout

\begin_layout Section
Double and Half Angle Formulas
\end_layout

\begin_layout Subsection
Double Angle Formulas
\end_layout

\begin_layout Standard
The following are determined by plugging in an angle twice into the sum
 formulas.
\end_layout

\begin_layout Subsubsection
Sine
\end_layout

\begin_layout Standard
\begin_inset Formula $\sin2A=2\sin A\cos A$
\end_inset


\end_layout

\begin_layout Subsubsection
Cosine
\end_layout

\begin_layout Standard
\begin_inset Formula $\cos2A=\cos^{2}A-\sin^{2}A$
\end_inset


\end_layout

\begin_layout Paragraph
More Cosine Formulas
\end_layout

\begin_layout Standard
However, more formulas for the double of an angle with cosine can be determined
 since 
\begin_inset Formula $\sin^{2}A+\cos^{2}A=1$
\end_inset

 (first Pythagorean identity).
\end_layout

\begin_layout Standard
\begin_inset Formula $\cos2A=1-2\sin^{2}A$
\end_inset


\end_layout

\begin_layout Standard
\begin_inset Formula $\cos2A=2\cos^{2}A-1$
\end_inset


\end_layout

\begin_layout Subsubsection
Tangent
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \tan2A=\frac{2\tan A}{1-\tan^{2}A}}$
\end_inset


\end_layout

\begin_layout Subsection
Half Angle Formulas
\end_layout

\begin_layout Subsubsection
Sine
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \sin\frac{1}{2}A=\pm\sqrt{\frac{1-\cos A}{2}}}$
\end_inset


\end_layout

\begin_layout Subsubsection
Cosine
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \cos\frac{1}{2}A=\pm\sqrt{\frac{1+\cos A}{2}}}$
\end_inset


\end_layout

\begin_layout Subsubsection
Tangent
\end_layout

\begin_layout Standard
\begin_inset Formula ${\displaystyle \tan\frac{1}{2}A=\pm\sqrt{\frac{1-\cos A}{1+\cos A}}}$
\end_inset


\end_layout

\end_body
\end_document