Sr Examen
¿Cómo vas a descomponer esta sin40/(2*cos20) expresión en fracciones?
Solución
Ha introducido
[src]
sin(40) --------- 2*cos(20)
$$\frac{\sin{\left(40 \right)}}{2 \cos{\left(20 \right)}}$$
sin(40)/((2*cos(20)))
Potencias
[src]
/ -40*I 40*I\
-I*\- e + e /
----------------------
/ -20*I 20*I\
2*\e + e /
$$- \frac{i \left(e^{40 i} - e^{- 40 i}\right)}{2 \left(e^{- 20 i} + e^{20 i}\right)}$$
-i*(-exp(-40*i) + exp(40*i))/(2*(exp(-20*i) + exp(20*i)))
Parte trigonométrica
[src]
sin(20)
$$\sin{\left(20 \right)}$$
/ pi\ cos|20 - --| \ 2 /
$$\cos{\left(20 - \frac{\pi}{2} \right)}$$
/ pi\ cos|40 - --| \ 2 / ------------ 2*cos(20)
$$\frac{\cos{\left(40 - \frac{\pi}{2} \right)}}{2 \cos{\left(20 \right)}}$$
2*cot(10)
------------
2
1 + cot (10)
$$\frac{2 \cot{\left(10 \right)}}{1 + \cot^{2}{\left(10 \right)}}$$
sec(20)
--------------
/ pi\
2*sec|40 - --|
\ 2 /
$$\frac{\sec{\left(20 \right)}}{2 \sec{\left(40 - \frac{\pi}{2} \right)}}$$
1
1 + --------
2
cot (10)
-------------------------------------
/ 1 \ / 1 \
|1 + --------|*|1 - --------|*cot(20)
| 2 | | 2 |
\ cot (20)/ \ cot (10)/
$$\frac{\frac{1}{\cot^{2}{\left(10 \right)}} + 1}{\left(1 - \frac{1}{\cot^{2}{\left(10 \right)}}\right) \left(1 + \frac{1}{\cot^{2}{\left(20 \right)}}\right) \cot{\left(20 \right)}}$$
/ 2 \ \1 + tan (10)/*sec(20)*sin(20) ------------------------------ / 2 \ / 2 \ \1 + tan (20)/*\1 - tan (10)/
$$\frac{\left(\tan^{2}{\left(10 \right)} + 1\right) \sin{\left(20 \right)} \sec{\left(20 \right)}}{\left(1 - \tan^{2}{\left(10 \right)}\right) \left(1 + \tan^{2}{\left(20 \right)}\right)}$$
1 ------------ / pi\ sec|20 - --| \ 2 /
$$\frac{1}{\sec{\left(20 - \frac{\pi}{2} \right)}}$$
/ 2 \
| sec (10) |
|1 + -------------|*sec(20)
| 2/ pi\|
| sec |10 - --||
\ \ 2 //
----------------------------------------------------
/ 2 \ / 2 \
| sec (20) | | sec (10) | / pi\
|1 + -------------|*|1 - -------------|*sec|20 - --|
| 2/ pi\| | 2/ pi\| \ 2 /
| sec |20 - --|| | sec |10 - --||
\ \ 2 // \ \ 2 //
$$\frac{\left(\frac{\sec^{2}{\left(10 \right)}}{\sec^{2}{\left(10 - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(20 \right)}}{\left(1 + \frac{\sec^{2}{\left(20 \right)}}{\sec^{2}{\left(20 - \frac{\pi}{2} \right)}}\right) \left(- \frac{\sec^{2}{\left(10 \right)}}{\sec^{2}{\left(10 - \frac{\pi}{2} \right)}} + 1\right) \sec{\left(20 - \frac{\pi}{2} \right)}}$$
2*tan(10)
------------
2
1 + tan (10)
$$\frac{2 \tan{\left(10 \right)}}{\tan^{2}{\left(10 \right)} + 1}$$
/ 2 \
\1 + tan (10)/*sin(20)
-------------------------------------
/ 2 \ / 2 \
\1 + tan (20)/*\1 - tan (10)/*cos(20)
$$\frac{\left(\tan^{2}{\left(10 \right)} + 1\right) \sin{\left(20 \right)}}{\left(1 - \tan^{2}{\left(10 \right)}\right) \left(1 + \tan^{2}{\left(20 \right)}\right) \cos{\left(20 \right)}}$$
/ pi\ csc|-20 + --| \ 2 / ------------- 2*csc(40)
$$\frac{\csc{\left(-20 + \frac{\pi}{2} \right)}}{2 \csc{\left(40 \right)}}$$
sin(40)
--------------
/ pi\
2*sin|20 + --|
\ 2 /
$$\frac{\sin{\left(40 \right)}}{2 \sin{\left(\frac{\pi}{2} + 20 \right)}}$$
1 ------- csc(20)
$$\frac{1}{\csc{\left(20 \right)}}$$
/ 4 \
2 | 4*sin (10)|
2*sin (20)*|1 + ----------|
| 2 |
\ sin (20) /
-----------------------------------------
/ 4 \ / 4 \
| 4*sin (10)| | 4*sin (20)|
|1 - ----------|*|1 + ----------|*sin(40)
| 2 | | 2 |
\ sin (20) / \ sin (40) /
$$\frac{2 \left(\frac{4 \sin^{4}{\left(10 \right)}}{\sin^{2}{\left(20 \right)}} + 1\right) \sin^{2}{\left(20 \right)}}{\left(1 + \frac{4 \sin^{4}{\left(20 \right)}}{\sin^{2}{\left(40 \right)}}\right) \left(- \frac{4 \sin^{4}{\left(10 \right)}}{\sin^{2}{\left(20 \right)}} + 1\right) \sin{\left(40 \right)}}$$
sec(20) --------- 2*csc(40)
$$\frac{\sec{\left(20 \right)}}{2 \csc{\left(40 \right)}}$$
/ 2/ pi\\
| csc |-10 + --||
| \ 2 /| / pi\
|1 + --------------|*csc|-20 + --|
| 2 | \ 2 /
\ csc (10) /
-------------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| csc |-20 + --|| | csc |-10 + --||
| \ 2 /| | \ 2 /|
|1 + --------------|*|1 - --------------|*csc(20)
| 2 | | 2 |
\ csc (20) / \ csc (10) /
$$\frac{\left(\frac{\csc^{2}{\left(-10 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(10 \right)}} + 1\right) \csc{\left(-20 + \frac{\pi}{2} \right)}}{\left(1 + \frac{\csc^{2}{\left(-20 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(20 \right)}}\right) \left(- \frac{\csc^{2}{\left(-10 + \frac{\pi}{2} \right)}}{\csc^{2}{\left(10 \right)}} + 1\right) \csc{\left(20 \right)}}$$
/ 2/ pi\\
| cos |10 - --||
| \ 2 /| / pi\
|1 + -------------|*cos|20 - --|
| 2 | \ 2 /
\ cos (10) /
-----------------------------------------------
/ 2/ pi\\ / 2/ pi\\
| cos |20 - --|| | cos |10 - --||
| \ 2 /| | \ 2 /|
|1 + -------------|*|1 - -------------|*cos(20)
| 2 | | 2 |
\ cos (20) / \ cos (10) /
$$\frac{\left(\frac{\cos^{2}{\left(10 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(10 \right)}} + 1\right) \cos{\left(20 - \frac{\pi}{2} \right)}}{\left(1 + \frac{\cos^{2}{\left(20 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(20 \right)}}\right) \left(- \frac{\cos^{2}{\left(10 - \frac{\pi}{2} \right)}}{\cos^{2}{\left(10 \right)}} + 1\right) \cos{\left(20 \right)}}$$
/ 2 \
\1 + tan (10)/*tan(20)
-----------------------------
/ 2 \ / 2 \
\1 + tan (20)/*\1 - tan (10)/
$$\frac{\left(\tan^{2}{\left(10 \right)} + 1\right) \tan{\left(20 \right)}}{\left(1 - \tan^{2}{\left(10 \right)}\right) \left(1 + \tan^{2}{\left(20 \right)}\right)}$$
/ 2 \
\1 + cot (10)/*cot(20)
------------------------------
/ 2 \ / 2 \
\1 + cot (20)/*\-1 + cot (10)/
$$\frac{\left(1 + \cot^{2}{\left(10 \right)}\right) \cot{\left(20 \right)}}{\left(-1 + \cot^{2}{\left(10 \right)}\right) \left(\cot^{2}{\left(20 \right)} + 1\right)}$$
(1 + cot(10)^2)*cot(20)/((1 + cot(20)^2)*(-1 + cot(10)^2))