Sr Examen
Simplificación de expresiones/
Descomposición en fracciones simples/
((sin((3*pi)/2-a)^(2))+(cos((3*pi)/2-a)^(2)))/(1/tan(pi-a)^(2)*tan(pi-a)^(2))
¿Cómo vas a descomponer esta ((sin((3*pi)/2-a)^(2))+(cos((3*pi)/2-a)^(2)))/(1/tan(pi-a)^(2)*tan(pi-a)^(2)) expresión en fracciones?
Solución
Ha introducido
[src]
2/3*pi \ 2/3*pi \
sin |---- - a| + cos |---- - a|
\ 2 / \ 2 /
-------------------------------
/ 2 \
|tan (pi - a)|
|------------|
| 2 |
\tan (pi - a)/
$$\frac{\sin^{2}{\left(- a + \frac{3 \pi}{2} \right)} + \cos^{2}{\left(- a + \frac{3 \pi}{2} \right)}}{\frac{1}{\tan^{2}{\left(\pi - a \right)}} \tan^{2}{\left(\pi - a \right)}}$$
(sin((3*pi)/2 - a)^2 + cos((3*pi)/2 - a)^2)/((tan(pi - a)^2/tan(pi - a)^2))
Respuesta numérica
[src]
cos((3*pi)/2 - a)^2 + sin((3*pi)/2 - a)^2
cos((3*pi)/2 - a)^2 + sin((3*pi)/2 - a)^2
Combinatoria
[src]
2 2 cos (a) + sin (a)
$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
cos(a)^2 + sin(a)^2
Unión de expresiones racionales
[src]
2/-2*a + 3*pi\ 2/-2*a + 3*pi\
cos |-----------| + sin |-----------|
\ 2 / \ 2 /
$$\sin^{2}{\left(\frac{- 2 a + 3 \pi}{2} \right)} + \cos^{2}{\left(\frac{- 2 a + 3 \pi}{2} \right)}$$
cos((-2*a + 3*pi)/2)^2 + sin((-2*a + 3*pi)/2)^2
Denominador común
[src]
2 2 cos (a) + sin (a)
$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
cos(a)^2 + sin(a)^2
Potencias
[src]
2 2 cos (a) + sin (a)
$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
2 2 / / 3*pi\ / 3*pi\\ / / 3*pi\ / 3*pi\\ | I*|a - ----| I*|-a + ----|| | I*|a - ----| I*|-a + ----|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |e e | \- e + e / |------------- + --------------| - ----------------------------------- \ 2 2 / 4
$$\left(\frac{e^{i \left(- a + \frac{3 \pi}{2}\right)}}{2} + \frac{e^{i \left(a - \frac{3 \pi}{2}\right)}}{2}\right)^{2} - \frac{\left(e^{i \left(- a + \frac{3 \pi}{2}\right)} - e^{i \left(a - \frac{3 \pi}{2}\right)}\right)^{2}}{4}$$
(exp(i*(a - 3*pi/2))/2 + exp(i*(-a + 3*pi/2))/2)^2 - (-exp(i*(a - 3*pi/2)) + exp(i*(-a + 3*pi/2)))^2/4
Denominador racional
[src]
2 2 cos (a) + sin (a)
$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}$$
cos(a)^2 + sin(a)^2
Compilar la expresión
[src]
2/3*pi \ 2/3*pi \
cos |---- - a| + sin |---- - a|
\ 2 / \ 2 /
$$\sin^{2}{\left(- a + \frac{3 \pi}{2} \right)} + \cos^{2}{\left(- a + \frac{3 \pi}{2} \right)}$$
cos((3*pi)/2 - a)^2 + sin((3*pi)/2 - a)^2
Parte trigonométrica
[src]
2
(1 + sin(a))
---------------------
4
/ /a\\ 4/a\
|1 + tan|-|| *cos |-|
\ \2// \2/
$$\frac{\left(\sin{\left(a \right)} + 1\right)^{2}}{\left(\tan{\left(\frac{a}{2} \right)} + 1\right)^{4} \cos^{4}{\left(\frac{a}{2} \right)}}$$
2/a pi\
4*tan |- + --|
4/a\ 2/a\ \2 4 /
- 4*cos |-| + 4*cos |-| + -------------------
\2/ \2/ 2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2 2 cos (pi - a) + sin (a)
$$\sin^{2}{\left(a \right)} + \cos^{2}{\left(\pi - a \right)}$$
2/a pi\
4 4*tan |- + --|
/ 2/a\\ 8/a\ \2 4 /
2*(1 + cos(a)) - 4*|-1 + cot |-|| *sin |-| + -------------------
\ \4// \4/ 2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$2 \left(\cos{\left(a \right)} + 1\right) - 4 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4} \sin^{8}{\left(\frac{a}{4} \right)} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
1 1
------- + ------------
2 2/pi \
sec (a) sec |-- - a|
\2 /
$$\frac{1}{\sec^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\sec^{2}{\left(a \right)}}$$
2
/ 2/a pi\\ 2/a pi\
|1 - tan |- + --|| 4*tan |- + --|
\ \2 4 // \2 4 /
------------------- + -------------------
2 2
/ 2/a pi\\ / 2/a pi\\
|1 + tan |- + --|| |1 + tan |- + --||
\ \2 4 // \ \2 4 //
$$\frac{\left(1 - \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
1 1 ------- + ------------ 2 2 csc (a) sec (pi - a)
$$\frac{1}{\sec^{2}{\left(\pi - a \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}$$
2
/ 2/ a pi\\
2 | csc |- - + --||
/ 1 \ | \ 2 4 /| 4/a pi\
|1 + ------| *|1 + --------------| *csc |- + --|
\ csc(a)/ | 2/a pi\ | \2 4 /
| csc |- + --| |
\ \2 4 / /
------------------------------------------------
4/ a pi\
4*csc |- - + --|
\ 2 4 /
$$\frac{\left(1 + \frac{1}{\csc{\left(a \right)}}\right)^{2} \left(\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \csc^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{4 \csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}$$
2 2/ pi\
sin (a) + sin |a + --|
\ 2 /
$$\sin^{2}{\left(a \right)} + \sin^{2}{\left(a + \frac{\pi}{2} \right)}$$
2/a pi\
4*cos |- - --|
4/a\ 2/a\ \2 4 /
- 4*cos |-| + 4*cos |-| + --------------------------------
\2/ \2/ 2
/ 2/a pi\\
| cos |- - --||
| \2 4 /| 2/a pi\
|1 + ------------| *cos |- + --|
| 2/a pi\| \2 4 /
| cos |- + --||
\ \2 4 //
$$- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} + \frac{4 \cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\left(\frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
1 1
------- + ------------
2 2/pi \
csc (a) csc |-- - a|
\2 /
$$\frac{1}{\csc^{2}{\left(- a + \frac{\pi}{2} \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}$$
2/ a pi\
4*csc |- - + --|
4 4 \ 2 4 /
- ------------ + ------------ + ----------------------------------
4/pi a\ 2/pi a\ 2
csc |-- - -| csc |-- - -| / 2/ a pi\\
\2 2/ \2 2/ | csc |- - + --||
| \ 2 4 /| 2/a pi\
|1 + --------------| *csc |- + --|
| 2/a pi\ | \2 4 /
| csc |- + --| |
\ \2 4 / /
$$\frac{4}{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} - \frac{4}{\csc^{4}{\left(- \frac{a}{2} + \frac{\pi}{2} \right)}} + \frac{4 \csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\frac{\csc^{2}{\left(- \frac{a}{2} + \frac{\pi}{4} \right)}}{\csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \csc^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2
/ 2/a pi\\ 2
|1 + tan |- + --|| *(1 + sin(a))
\ \2 4 //
---------------------------------
4/a pi\
4*tan |- + --|
\2 4 /
$$\frac{\left(\sin{\left(a \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{4 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2
/ 4/a pi\\
| 4*sin |- + --||
| \2 4 /| 2 4
|1 + --------------| *(1 + sin(a)) *cos (a)
| 2 |
\ cos (a) /
-------------------------------------------
8/a pi\
64*sin |- + --|
\2 4 /
$$\frac{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(a \right)}} + 1\right)^{2} \left(\sin{\left(a \right)} + 1\right)^{2} \cos^{4}{\left(a \right)}}{64 \sin^{8}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
1 1 ------- + ------- 2 2 csc (a) sec (a)
$$\frac{1}{\sec^{2}{\left(a \right)}} + \frac{1}{\csc^{2}{\left(a \right)}}$$
2
/ /a\ \
2 | 2*tan|-| |
/ 2/a pi\\ | \2/ |
|1 + tan |- + --|| *|1 + -----------|
\ \2 4 // | 2/a\|
| 1 + tan |-||
\ \2//
--------------------------------------
4/a pi\
4*tan |- + --|
\2 4 /
$$\frac{\left(1 + \frac{2 \tan{\left(\frac{a}{2} \right)}}{\tan^{2}{\left(\frac{a}{2} \right)} + 1}\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}{4 \tan^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2
/ 2/a pi\\
2 | sec |- + --||
/ 1 \ | \2 4 /| 4/a pi\
|1 + -----------| *|1 + ------------| *sec |- - --|
| / pi\| | 2/a pi\| \2 4 /
| sec|a - --|| | sec |- - --||
\ \ 2 // \ \2 4 //
---------------------------------------------------
4/a pi\
4*sec |- + --|
\2 4 /
$$\frac{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}\right)^{2} \left(1 + \frac{1}{\sec{\left(a - \frac{\pi}{2} \right)}}\right)^{2} \sec^{4}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{4 \sec^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2/a pi\
4 4*tan |- + --|
/ 2/a\\ 8/a\ \2 4 /
2*(1 + cos(a)) - 4*|1 - tan |-|| *cos |-| + -------------------
\ \4// \4/ 2
/ 2/a pi\\
|1 + tan |- + --||
\ \2 4 //
$$- 4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4} \cos^{8}{\left(\frac{a}{4} \right)} + 2 \left(\cos{\left(a \right)} + 1\right) + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/a pi\\
| cos |- - --|| 2
| \2 4 /| / / pi\\ 4/a pi\
|1 + ------------| *|1 + cos|a - --|| *cos |- + --|
| 2/a pi\| \ \ 2 // \2 4 /
| cos |- + --||
\ \2 4 //
---------------------------------------------------
4/a pi\
4*cos |- - --|
\2 4 /
$$\frac{\left(\frac{\cos^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}{\cos^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}} + 1\right)^{2} \left(\cos{\left(a - \frac{\pi}{2} \right)} + 1\right)^{2} \cos^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{4 \cos^{4}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
1
$$1$$
2 2 cos (a) + sin (-a + 2*pi)
$$\sin^{2}{\left(- a + 2 \pi \right)} + \cos^{2}{\left(a \right)}$$
4 2
/ 2/a\\ / 2/a\\
4*|-1 + cot |-|| 4*|-1 + cot |-||
\ \4// \ \4// 4
- ----------------- + ----------------- + --------------------------------
4 2 2
/ 2/a\\ / 2/a\\ / 1 \ 2/a pi\
|1 + cot |-|| |1 + cot |-|| |1 + ------------| *cot |- + --|
\ \4// \ \4// | 2/a pi\| \2 4 /
| cot |- + --||
\ \2 4 //
$$- \frac{4 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{4}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}} + \frac{4 \left(\cot^{2}{\left(\frac{a}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{4}{\left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2} \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}$$
2 2/ pi\
cos (a) + cos |a - --|
\ 2 /
$$\cos^{2}{\left(a \right)} + \cos^{2}{\left(a - \frac{\pi}{2} \right)}$$
2
/ 2/a pi\\ 2/a pi\
|-1 + cot |- + --|| 4*cot |- + --|
\ \2 4 // \2 4 /
-------------------- + -------------------
2 2
/ 2/a pi\\ / 2/a pi\\
|1 + cot |- + --|| |1 + cot |- + --||
\ \2 4 // \ \2 4 //
$$\frac{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} - 1\right)^{2}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}} + \frac{4 \cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2/a pi\
4*sec |- + --|
4 4 \2 4 /
- ------- + ------- + --------------------------------
4/a\ 2/a\ 2
sec |-| sec |-| / 2/a pi\\
\2/ \2/ | sec |- + --||
| \2 4 /| 2/a pi\
|1 + ------------| *sec |- - --|
| 2/a pi\| \2 4 /
| sec |- - --||
\ \2 4 //
$$\frac{4}{\sec^{2}{\left(\frac{a}{2} \right)}} - \frac{4}{\sec^{4}{\left(\frac{a}{2} \right)}} + \frac{4 \sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(1 + \frac{\sec^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}\right)^{2} \sec^{2}{\left(\frac{a}{2} - \frac{\pi}{4} \right)}}$$
4/a pi\
16*sin |- + --|
4/pi a\ 2/pi a\ \2 4 /
- 4*sin |-- + -| + 4*sin |-- + -| + -----------------------------
\2 2/ \2 2/ 2
/ 4/a pi\\
| 4*sin |- + --||
| \2 4 /| 2
|1 + --------------| *cos (a)
| 2 |
\ cos (a) /
$$- 4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} + 4 \sin^{2}{\left(\frac{a}{2} + \frac{\pi}{2} \right)} + \frac{16 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\frac{4 \sin^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\cos^{2}{\left(a \right)}} + 1\right)^{2} \cos^{2}{\left(a \right)}}$$
2
/ /a\ \
2 | 2*cot|-| |
/ 1 \ | \2/ | 4/a pi\
|1 + ------------| *|1 + -----------| *cot |- + --|
| 2/a pi\| | 2/a\| \2 4 /
| cot |- + --|| | 1 + cot |-||
\ \2 4 // \ \2//
---------------------------------------------------
4
$$\frac{\left(1 + \frac{2 \cot{\left(\frac{a}{2} \right)}}{\cot^{2}{\left(\frac{a}{2} \right)} + 1}\right)^{2} \left(1 + \frac{1}{\cot^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}\right)^{2} \cot^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{4}$$
4 2
/ 2/a\\ / 2/a\\ 2/a pi\
4*|1 - tan |-|| 4*|1 - tan |-|| 4*tan |- + --|
\ \4// \ \4// \2 4 /
- ---------------- + ---------------- + -------------------
4 2 2
/ 2/a\\ / 2/a\\ / 2/a pi\\
|1 + tan |-|| |1 + tan |-|| |1 + tan |- + --||
\ \4// \ \4// \ \2 4 //
$$- \frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{4}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{4}} + \frac{4 \left(1 - \tan^{2}{\left(\frac{a}{4} \right)}\right)^{2}}{\left(\tan^{2}{\left(\frac{a}{4} \right)} + 1\right)^{2}} + \frac{4 \tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{\left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2}}$$
2
/ 2/a pi\\ 2 4/a pi\
|1 + tan |- + --|| *(1 + sin(a)) *cot |- + --|
\ \2 4 // \2 4 /
----------------------------------------------
4
$$\frac{\left(\sin{\left(a \right)} + 1\right)^{2} \left(\tan^{2}{\left(\frac{a}{2} + \frac{\pi}{4} \right)} + 1\right)^{2} \cot^{4}{\left(\frac{a}{2} + \frac{\pi}{4} \right)}}{4}$$
2 4/a\ 2/a\
cos (a) - 4*cos |-| + 4*cos |-|
\2/ \2/
$$- 4 \cos^{4}{\left(\frac{a}{2} \right)} + 4 \cos^{2}{\left(\frac{a}{2} \right)} + \cos^{2}{\left(a \right)}$$
cos(a)^2 - 4*cos(a/2)^4 + 4*cos(a/2)^2