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