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π
=
3.141592653589793238462643383279502884197169399375105820974944592307
{\displaystyle \pi =3.141592653589793238462643383279502884197169399375105820974944592307\,\!}
Trigonometric functions [ edit ]
sin
θ
=
opposite
hypotenuse
=
o
h
{\displaystyle \sin \theta ={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {o}{h}}}
cos
θ
=
adjacent
hypotenuse
=
a
h
{\displaystyle \cos \theta ={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {a}{h}}}
tan
θ
=
opposite
adjacent
=
o
a
{\displaystyle \tan \theta ={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {o}{a}}}
Remember: SOHCAHTOA
{\displaystyle {\mbox{Remember: SOHCAHTOA}}\,\!}
sin
θ
=
o
/
h
or SOH
{\displaystyle \sin \theta =o/h{\mbox{ or SOH}}\,\!}
cos
θ
=
a
/
h
or CAH
{\displaystyle \cos \theta =a/h{\mbox{ or CAH}}\,\!}
tan
θ
=
o
/
a
or TOA
{\displaystyle \tan \theta =o/a{\mbox{ or TOA}}\,\!}
Trigonometric equations [ edit ] Example [ edit ]
If
sin
x
=
3
2
{\displaystyle {\mbox{If }}\sin x={\frac {\sqrt {3}}{2}}}
then
x
=
{
60
∘
+
360
∘
k
,
k
∈
I
120
∘
+
360
∘
k
,
k
∈
I
{\displaystyle {\mbox{then }}x={\begin{cases}60^{\circ }+360^{\circ }k,k\in \mathbb {I} \\120^{\circ }+360^{\circ }k,k\in \mathbb {I} \end{cases}}}
Trigonometric identities [ edit ] The unit circle can be very helpful Reciprocal [ edit ] 1.
csc
θ
=
1
sin
θ
{\displaystyle \csc \theta ={\frac {1}{\sin \theta }}}
sec
θ
=
1
cos
θ
{\displaystyle \sec \theta ={\frac {1}{\cos \theta }}}
cot
θ
=
1
tan
θ
{\displaystyle \cot \theta ={\frac {1}{\tan \theta }}}
Quotient [ edit ] 4.
tan
θ
=
sin
θ
cos
θ
{\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}
cot
θ
=
cos
θ
sin
θ
{\displaystyle \cot \theta ={\frac {\cos \theta }{\sin \theta }}}
Pythagorean [ edit ] 6.
sin
2
θ
+
cos
2
θ
=
1
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,\!}
tan
2
θ
+
1
=
sec
2
θ
{\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta \,\!}
1
+
cot
2
θ
=
csc
2
θ
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta \,\!}
Example [ edit ]
Prove.
sec
x
+
1
cot
x
=
1
+
sin
x
cos
x
{\displaystyle {\mbox{Prove. }}\sec x+{\frac {1}{\cot x}}={\frac {1+\sin x}{\cos x}}}
(
1
cos
x
)
+
(
sin
x
cos
x
)
=
1
+
sin
x
cos
x
{\displaystyle \left({\frac {1}{\cos x}}\right)+\left({\frac {\sin x}{\cos x}}\right)={\frac {1+\sin x}{\cos x}}}
1
+
sin
x
cos
x
=
1
+
sin
x
cos
x
{\displaystyle {\frac {1+\sin x}{\cos x}}={\frac {1+\sin x}{\cos x}}}
Radians [ edit ] An angle of 1 radian subtends an arc equal in length to the radius of the circle . The radian is a unit of plane angle , equal to 180/π degrees , or about 57.296 degrees. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level.
Important equations [ edit ]
Arc Length
=
2
π
r
(
θ
360
∘
)
{\displaystyle {\mbox{Arc Length}}=2\pi r\left({\frac {\theta }{360^{\circ }}}\right)}
1
complete circle
=
360
∘
(
1
rad
57.296
∘
)
=
2
π
{\displaystyle 1{\mbox{ complete circle}}=360^{\circ }\left({\frac {1{\mbox{ rad}}}{57.296^{\circ }}}\right)=2\pi }
Degrees → Radians [ edit ]
30
∘
(
π
180
∘
)
=
30
π
180
=
π
6
rad
{\displaystyle 30^{\circ }\left({\frac {\pi }{180^{\circ }}}\right)={\frac {30\pi }{180}}={\frac {\pi }{6}}{\mbox{ rad}}}
Radians → Degrees [ edit ]
3
π
4
(
180
∘
π
)
=
135
∘
{\displaystyle {\frac {3\pi }{4}}\left({\frac {180^{\circ }}{\pi }}\right)=135^{\circ }}
