Mister Exam
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Equation (3x-2)*(3x+2)+(4x-5)^2=10x+21
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1/2
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1/2
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1/2
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Identical expressions
- Cos(x/ three -pi/ four)= one / two
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Similar expressions
- Cos(x/3+pi/4)=1/2
Cos(x/3-pi/4)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ cos|- - --| = 1/2 \3 4 /
$$\cos{\left(\frac{x}{3} - \frac{\pi}{4} \right)} = \frac{1}{2}$$
Detail solution
Given the equation
$$\cos{\left(\frac{x}{3} - \frac{\pi}{4} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = 2 \pi n - \frac{\pi}{12}$$
$$\frac{x}{3} = 2 \pi n + \frac{7 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n - \frac{\pi}{4}$$
$$x_{2} = 6 \pi n + \frac{7 \pi}{4}$$
$$\cos{\left(\frac{x}{3} - \frac{\pi}{4} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{3} + \frac{\pi}{4} = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = 2 \pi n - \frac{\pi}{12}$$
$$\frac{x}{3} = 2 \pi n + \frac{7 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
we get the answer:
$$x_{1} = 6 \pi n - \frac{\pi}{4}$$
$$x_{2} = 6 \pi n + \frac{7 \pi}{4}$$
Rapid solution
[src]
-pi
x1 = ----
4
$$x_{1} = - \frac{\pi}{4}$$
7*pi
x2 = ----
4
$$x_{2} = \frac{7 \pi}{4}$$
x2 = 7*pi/4
Sum and product of roots
[src]
sum
pi 7*pi - -- + ---- 4 4
$$- \frac{\pi}{4} + \frac{7 \pi}{4}$$
=
3*pi ---- 2
$$\frac{3 \pi}{2}$$
product
-pi 7*pi ----*---- 4 4
$$- \frac{\pi}{4} \frac{7 \pi}{4}$$
=
2 -7*pi ------ 16
$$- \frac{7 \pi^{2}}{16}$$
-7*pi^2/16
Numerical answer
[src]
x1 = -51.0508806208341
x2 = -220.696883914683
x3 = 18.0641577581413
x4 = -57.3340659280137
x5 = -38.484510006475
x6 = -1339.10386859265
x7 = 62.0464549083984
x8 = 55.7632696012188
x9 = 43.1968989868597
x10 = 99.7455667514759
x11 = 608.683576633022
x12 = 307.090681888402
x13 = 93.4623814442964
x14 = -69.9004365423729
x15 = 36.9137136796801
x16 = -597.688002345458
x17 = -0.785398163397448
x18 = -208.130513300324
x19 = -19.6349540849362
x20 = 80.8960108299372
x21 = 24.3473430653209
x22 = -95.0331777710912
x23 = 118.595122673015
x24 = -76.1836218495525
x25 = -13.3517687777566
x26 = 5.49778714378214
x27 = 432.754388031994
x28 = -32.2013246992954
x29 = -88.7499924639117
x30 = 74.6128255227576
x30 = 74.6128255227576