3*sin(x)+4*cos(x)=1 (уравнение)

Найду корень уравнения: 3*sin(x)+4*cos(x)=1

Вы ввели

3*sin(x) + 4*cos(x) = 1

$$3 \sin{\left(x \right)} + 4 \cos{\left(x \right)} = 1$$

Быстрый ответ

           /        ___\
           |3   2*\/ 6 |
x1 = 2*atan|- - -------|
           \5      5   /

$$x_{1} = 2 \operatorname{atan}{\left(\frac{3}{5} - \frac{2 \sqrt{6}}{5} \right)}$$

           /        ___\
           |3   2*\/ 6 |
x2 = 2*atan|- + -------|
           \5      5   /

$$x_{2} = 2 \operatorname{atan}{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}$$

Сумма и произведение корней

сумма

          /        ___\         /        ___\
          |3   2*\/ 6 |         |3   2*\/ 6 |
0 + 2*atan|- - -------| + 2*atan|- + -------|
          \5      5   /         \5      5   /

$$\left(2 \operatorname{atan}{\left(\frac{3}{5} - \frac{2 \sqrt{6}}{5} \right)} + 0\right) + 2 \operatorname{atan}{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}$$

=

      /        ___\         /        ___\
      |3   2*\/ 6 |         |3   2*\/ 6 |
2*atan|- - -------| + 2*atan|- + -------|
      \5      5   /         \5      5   /

$$2 \operatorname{atan}{\left(\frac{3}{5} - \frac{2 \sqrt{6}}{5} \right)} + 2 \operatorname{atan}{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}$$

произведение

        /        ___\       /        ___\
        |3   2*\/ 6 |       |3   2*\/ 6 |
1*2*atan|- - -------|*2*atan|- + -------|
        \5      5   /       \5      5   /

$$1 \cdot 2 \operatorname{atan}{\left(\frac{3}{5} - \frac{2 \sqrt{6}}{5} \right)} 2 \operatorname{atan}{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}$$

=

      /        ___\     /        ___\
      |3   2*\/ 6 |     |3   2*\/ 6 |
4*atan|- - -------|*atan|- + -------|
      \5      5   /     \5      5   /

$$4 \operatorname{atan}{\left(\frac{3}{5} - \frac{2 \sqrt{6}}{5} \right)} \operatorname{atan}{\left(\frac{3}{5} + \frac{2 \sqrt{6}}{5} \right)}$$

Численный ответ

x1 = 11.8404333171479

x2 = -16.8366164067409

x3 = 58.5616072794141

x4 = -94.9737169049051

x5 = -60.818913556998

x6 = -29.4029870211001

x7 = -25.8586785259296

x8 = -23.1198017139205

x9 = -38.4250491402888

x10 = 68.3891010817642

x11 = 39.7120513578754

x12 = -85.9516547857164

x13 = 33.4288660506958

x14 = 133.959830965569

x15 = -57.2746050618276

x16 = 24.4068039315071

x17 = -76.1241609833663

x18 = 93.5218423104825

x19 = 30.6899892386866

x20 = 190.508498730185

x21 = -4.27024579238174

x22 = -63.5577903690071

x23 = 49.5395451602254

x24 = -10.5534310995613

x25 = 62.1059157745846

x26 = 14.579310129157

x27 = 64.8447925865937

x28 = 36.9731745458662

x29 = 77.4111632009529

x30 = -32.1418638331092

x31 = -54.5357282498184

x32 = 5.5572480099683

x33 = 74.6722863889438

x34 = -50.991419754648

x35 = -13.2923079115705

x36 = -41.9693576354593

x37 = -7.00912260439087

x38 = -88.6905315977255

x39 = -44.7082344474684

x40 = -92.234840092896

x41 = 52.2784219722345

x42 = -82.4073462905459

x43 = -98.5180254000755

x44 = 55.822730467405

x45 = -69.8409756761867

x46 = 87.2386570033029

x47 = -79.6684694785368

x48 = 83.6943485081325

x49 = 20.8624954363366

x50 = 71.1279778937733

x51 = 27.1456807435162

x52 = -35.6861723282797

x53 = -19.57549321875

x54 = 18.1236186243275

x55 = 45.995236665055

x56 = -0.725937297211281

x57 = 8.29612482197744

x58 = -48.2525429426388

x59 = -67.1020988641776

x60 = -73.3852841713572

x61 = 2.01293951479785

x62 = 99.8050276176621

x63 = 96.2607191224917

x64 = 89.9775338153121

x65 = 80.9554716961233

x66 = 43.2563598530458

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