Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 3*x=28-x
-
Equation 3(x-7)-6x=-x
-
Equation cos(x)^(2)=(1/2)
-
Equation (x+3)^2-16=(1-2x)^2
- Express {x} in terms of y where:
- -9*x-19*y=13
- -13*x-5*y=6
- 14*x+3*y=-4
- -11*x+13*y=10
- Derivative of:
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tgx+ctgx
- 3/2
- Integral of d{x}:
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3/2
- 3/2
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Identical expressions
- tgx+ctgx= three / two
- tgx plus ctgx equally 3 divide by 2
- tgx plus ctgx equally three divide by two
- tgx+ctgx=3 divide by 2
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Similar expressions
- -x*((sqrt(3)/2)-1)=1
- cos(x)=2sqrt(3)/2
- tgx-ctgx=3/2
- (2*x+21*sqrt(3)/2)^2+(2*x+21*sqrt(3)/2)^2=((4*x+63)/sqrt(3))^2
- 3/4+3/16-2*sqrt(3)/2*sqrt(3)/4*x=1+1/16+2*1/4*1/2*1/2
tgx+ctgx=3/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
tan(x) + cot(x) = 3/2
$$\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{3}{2}$$
Detail solution
Given the equation
$$\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{3}{2}$$
transform
$$\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{5}{2} = 0$$
$$\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{5}{2} = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Given the equation:
$$w + \tan{\left(x \right)} - \frac{5}{2} = 0$$
transform:
$$w + \tan{\left(x \right)} - \frac{5}{2} = 0$$
Expand brackets in the left part
Move free summands (without w)
from left part to right part, we given:
$$w + \tan{\left(x \right)} = \frac{5}{2}$$
Divide both parts of the equation by (w + tan(x))/w
We get the answer: w = 5/2 - tan(x)
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$\tan{\left(x \right)} + \cot{\left(x \right)} = \frac{3}{2}$$
transform
$$\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{5}{2} = 0$$
$$\tan{\left(x \right)} + \cot{\left(x \right)} - \frac{5}{2} = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Given the equation:
$$w + \tan{\left(x \right)} - \frac{5}{2} = 0$$
transform:
$$w + \tan{\left(x \right)} - \frac{5}{2} = 0$$
Expand brackets in the left part
-5/2 + w + tanx = 0
Move free summands (without w)
from left part to right part, we given:
$$w + \tan{\left(x \right)} = \frac{5}{2}$$
Divide both parts of the equation by (w + tan(x))/w
w = 5/2 / ((w + tan(x))/w)
We get the answer: w = 5/2 - tan(x)
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
Rapid solution
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/ / ___\\ / / ___\\
| |3 I*\/ 7 || | |3 I*\/ 7 ||
x1 = I*im|atan|- - -------|| + re|atan|- - -------||
\ \4 4 // \ \4 4 //
$$x_{1} = \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}$$
/ / ___\\ / / ___\\
| |3 I*\/ 7 || | |3 I*\/ 7 ||
x2 = I*im|atan|- + -------|| + re|atan|- + -------||
\ \4 4 // \ \4 4 //
$$x_{2} = \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}$$
x2 = re(atan(3/4 + sqrt(7)*i/4)) + i*im(atan(3/4 + sqrt(7)*i/4))
Sum and product of roots
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sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |3 I*\/ 7 || | |3 I*\/ 7 || | |3 I*\/ 7 || | |3 I*\/ 7 ||
I*im|atan|- - -------|| + re|atan|- - -------|| + I*im|atan|- + -------|| + re|atan|- + -------||
\ \4 4 // \ \4 4 // \ \4 4 // \ \4 4 //
$$\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) + \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |3 I*\/ 7 || | |3 I*\/ 7 || | |3 I*\/ 7 || | |3 I*\/ 7 ||
I*im|atan|- - -------|| + I*im|atan|- + -------|| + re|atan|- - -------|| + re|atan|- + -------||
\ \4 4 // \ \4 4 // \ \4 4 // \ \4 4 //
$$\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + \operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |3 I*\/ 7 || | |3 I*\/ 7 ||| | | |3 I*\/ 7 || | |3 I*\/ 7 ||| |I*im|atan|- - -------|| + re|atan|- - -------|||*|I*im|atan|- + -------|| + re|atan|- + -------||| \ \ \4 4 // \ \4 4 /// \ \ \4 4 // \ \4 4 ///
$$\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |3 I*\/ 7 || | |3 I*\/ 7 ||| | | |3 I*\/ 7 || | |3 I*\/ 7 ||| |I*im|atan|- - -------|| + re|atan|- - -------|||*|I*im|atan|- + -------|| + re|atan|- + -------||| \ \ \4 4 // \ \4 4 /// \ \ \4 4 // \ \4 4 ///
$$\left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} - \frac{\sqrt{7} i}{4} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{3}{4} + \frac{\sqrt{7} i}{4} \right)}\right)}\right)$$
(i*im(atan(3/4 - i*sqrt(7)/4)) + re(atan(3/4 - i*sqrt(7)/4)))*(i*im(atan(3/4 + i*sqrt(7)/4)) + re(atan(3/4 + i*sqrt(7)/4)))
Numerical answer
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x1 = 0.785398163397448 - 0.397682730611953*i
x2 = 0.785398163397448 + 0.397682730611953*i
x2 = 0.785398163397448 + 0.397682730611953*i