قضیه فیثاغورث ویرایش
cos
2
θ
+
sin
2
θ
=
1
{\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1}
1
+
tan
2
θ
=
sec
2
θ
{\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta }
1
+
cot
2
θ
=
csc
2
θ
{\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }
جمع و تفاضل دو زاویه ویرایش
cos
(
θ
+
β
)
=
cos
θ
.
cos
β
−
sin
θ
.
sin
β
{\displaystyle \cos(\theta +\beta )=\cos \theta .\cos \beta -\sin \theta .\sin \beta \,}
cos
(
θ
−
β
)
=
cos
θ
.
cos
β
+
sin
θ
.
sin
β
{\displaystyle \cos(\theta -\beta )=\cos \theta .\cos \beta +\sin \theta .\sin \beta \,}
sin
(
θ
+
β
)
=
sin
θ
.
cos
β
+
cos
θ
.
sin
β
{\displaystyle \sin(\theta +\beta )=\sin \theta .\cos \beta +\cos \theta .\sin \beta \,}
sin
(
θ
−
β
)
=
sin
θ
.
cos
β
−
cos
θ
.
sin
β
{\displaystyle \sin(\theta -\beta )=\sin \theta .\cos \beta -\cos \theta .\sin \beta \,}
tan
(
θ
+
β
)
=
tan
θ
+
tan
β
1
−
tan
θ
.
tan
β
{\displaystyle \tan(\theta +\beta )={\frac {\tan \theta +\tan \beta }{1-\tan \theta .\tan \beta }}\,}
tan
(
θ
−
β
)
=
tan
θ
−
tan
β
1
+
tan
θ
.
tan
β
{\displaystyle \tan(\theta -\beta )={\frac {\tan \theta -\tan \beta }{1+\tan \theta .\tan \beta }}\,}
cot
(
θ
+
β
)
=
cot
θ
.
cot
β
−
1
cot
θ
+
cot
β
{\displaystyle \cot(\theta +\beta )={\frac {\cot \theta .\cot \beta -1}{\cot \theta +\cot \beta }}\,}
cot
(
θ
−
β
)
=
cot
θ
.
cot
β
+
1
cot
β
−
cot
θ
{\displaystyle \cot(\theta -\beta )={\frac {\cot \theta .\cot \beta +1}{\cot \beta -\cot \theta }}\,}
زاویه دو برابر ویرایش <:
sin
(
2
θ
)
=
2
sin
θ
cos
θ
=
2
tan
θ
1
+
tan
2
θ
{\displaystyle \sin(2\theta )=2\sin \theta \cos \theta ={\frac {2\tan \theta }{1+\tan ^{2}\theta }}}
cos
(
2
θ
)
=
cos
2
θ
−
sin
2
θ
=
2
cos
2
θ
−
1
=
1
−
2
sin
2
θ
=
1
−
tan
2
θ
1
+
tan
2
θ
{\displaystyle \cos(2\theta )=\cos ^{2}\theta -\sin ^{2}\theta =2\cos ^{2}\theta -1=1-2\sin ^{2}\theta ={\frac {1-\tan ^{2}\theta }{1+\tan ^{2}\theta }}}
tan
(
2
θ
)
=
2
tan
θ
1
−
tan
2
θ
{\displaystyle \tan(2\theta )={\frac {2\tan \theta }{1-\tan ^{2}\theta }}}
cot
(
2
θ
)
=
cot
2
θ
−
1
2
cot
θ
{\displaystyle \cot(2\theta )={\frac {\cot ^{2}\theta -1}{2\cot \theta }}}
sec
(
2
θ
)
=
sec
2
θ
2
−
sec
2
θ
{\displaystyle \sec(2\theta )={\frac {\sec ^{2}\theta }{2-\sec ^{2}\theta }}}
csc
(
2
θ
)
=
sec
θ
csc
θ
2
{\displaystyle \csc(2\theta )={\frac {\sec \theta \csc \theta }{2}}}
=== زاویه سه برابر ===>
cos
2
a
=
1
2
(
1
+
cos
2
a
)
{\displaystyle \cos ^{2}a={\frac {1}{2}}\ (1+\cos 2a)}
sin
2
a
=
1
2
(
1
−
cos
2
a
)
{\displaystyle \sin ^{2}a={\frac {1}{2}}\ (1-\cos 2a)}
تبدیل ضرب به جمع ویرایش
sin
a
.
cos
b
=
1
2
[
sin
(
a
+
b
)
+
sin
(
a
−
b
)
]
{\displaystyle \sin a.\cos b={\frac {1}{2}}[\sin(a+b)+\sin(a-b)]}
cos
a
.
cos
b
=
1
2
[
cos
(
a
+
b
)
+
cos
(
a
−
b
)
]
{\displaystyle \cos a.\cos b={\frac {1}{2}}[\cos(a+b)+\cos(a-b)]}
sin
a
.
sin
b
=
−
1
2
[
cos
(
a
+
b
)
−
cos
(
a
−
b
)
]
{\displaystyle \sin a.\sin b=-{\frac {1}{2}}[\cos(a+b)-\cos(a-b)]}
تبدیل جمع به ضرب ویرایش
sin
a
±
sin
b
=
2
sin
(
a
±
b
2
)
.
cos
(
a
∓
b
2
)
{\displaystyle \sin a\pm \sin b=2\sin({\frac {a\pm b}{2}}).\cos({\frac {a\mp b}{2}}\,)}
cos
a
+
cos
b
=
2
cos
(
a
+
b
2
)
.
cos
(
a
−
b
2
)
{\displaystyle \cos a+\cos b=2\cos({\frac {a+b}{2}}).\cos({\frac {a-b}{2}}\,)}
cos
a
−
cos
b
=
−
2
sin
(
a
+
b
2
)
.
sin
(
a
−
b
2
)
{\displaystyle \cos a-\cos b=-2\sin({\frac {a+b}{2}}).\sin({\frac {a-b}{2}}\,)}
tan
a
±
t
a
n
b
=
sin
(
a
±
b
)
cos
a
cos
b
{\displaystyle \tan a\pm \ tanb={\frac {\sin(a\pm b)}{\cos a\cos b}}}
جمع سینوس و کسینوس یک زاویه ویرایش
sin
θ
+
cos
θ
=
2
s
i
n
(
π
4
+
θ
)
{\displaystyle \displaystyle \sin \theta +\cos \theta ={\sqrt {2}}sin({\frac {\pi }{4}}+\theta )}
آرتور کاکسفورد، ترجمه عادل ارشقی. اصول و کاربردهای مثلثات . انتشارات رسا. حسابان . شرکت چاپ و نشر کتاب‌های درسی ایران.
![{\displaystyle \sin a.\cos b={\frac {1}{2}}[\sin(a+b)+\sin(a-b)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/946e0b8508eddf36c4d9c5cce47fcdd1976d2f2f)
![{\displaystyle \cos a.\cos b={\frac {1}{2}}[\cos(a+b)+\cos(a-b)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7752d315943731d316adb33658203f97b4ffc41b)
![{\displaystyle \sin a.\sin b=-{\frac {1}{2}}[\cos(a+b)-\cos(a-b)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/957e3dfaf4eceb20b33517bc6323d8491e7db4aa)
