asinus

Thanks for the correction. It's all right.

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evaluate cos x

Hello Guest!

\(tan\ x + sec\ x = 2\\ sec\ x=1/cos\ x\\ tan\ x=\dfrac{\sqrt{1-cos^2x}}{cos\ x}\\ \dfrac{\sqrt{1-cos^2x}}{cos\ x}+\dfrac{1}{cos\ x}=2\\ \sqrt{1-cos^2x}+1=2cos\ x \)

\(\sqrt{1-cos^2x}=2cos\ x -1\ |\ cos\ x=y\\ 1-y^2=(2y-1)^2\\ 1-y^2=4y^2-4y+1\\ 5y^2-4y=0\\ y\in \{0,0.8\}\ |\ y=cos\ x\\ \color{blue}cos\ x\in \{0,0.8\}\)

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Hello Qube73!

I checked my solution again and got 150. Please explain me your solution 144.

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 What is the value of a+b+c?

Hello Guest!

\( x^2 + 15x + 54=0\\ x=-7.5\pm \sqrt{56.25-54}\\ x\in \{-9,-6\}\\ (x+a)(x+b)=(x+6)(x+9)=0\\ a=6\\ b=9\)

\(x^2 - 17x + 72=0\\ x=8.5\pm \sqrt{72.25-72}\\ x\in \{8,9\} \\ (x-b)(x-c)=(x-9)(x-8)\\ b=9\\ c=8\\ \color{blue}a+b+c=6+9+8=23\)

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Find the center and radius of this circle.

Hello Guest!

\( x^2 - 2x + y^2 = 24\\ -y^2=x^2-2x-24\\ x=1\pm \sqrt{1+24}\\ x\in \{-4,6\}\)

The center of the circle is (1,0).

The radius of the circle is 5.

\(x^2-2x+y^2-24=0\\ (x-1)^2-1 +y^2=24\\ (x-1)^2+y^2=25\)

The center of the circle is (1,0).

The radius of the circle is 5.

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Compute a^3 + b^3.

Hello Guest!

\(2x^2 - 8x + 7 = x^2 - 6x + 1\\ x^2-2x+6=0\\ x=1\pm \sqrt{1-6}\\ \color{blue}x\in \{1-i\sqrt{5},1+i\sqrt{5}\}\\ a=1-i\sqrt{5}\\ b=1+i\sqrt{5}\\a^3=-14+2\sqrt{5}i\\ b^3=-14-2\sqrt{5}i\\ \color{blue}a^3+b^3=-28\)

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DX = 11

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 How many different chords of the circle contain and have integer lengths?

Hello Guest!

The largest chord is diameter 30. Your sections are 6 and 24.

The product of the respective chord sections is constant (tendon law).

The product of all chord segments is 144.

144 = 6 * 24,     6 + 24 = 30 This is the largest chord.

144 = 8 * 18,     8 + 18 = 26

144 = 9 * 16,     9 + 16 = 25

144 = 12 * 12, 12 + 12 = 24 This is the smallest chord.

Four different chords with integer length are possible.

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Find the area of triangle ACE.

Hello Qube73!

\(\overline{AE}=\sqrt{22^2+10^2}=24.166\\ \angle CAE=\ atan \frac{10}{22}=24.444^\circ\\ \overline{CE}=\sqrt{\overline{AE}^2+30^2-2\cdot 24.166\cdot 30\cdot cos(\angle\ CAE)}\\ \overline{CE}=\sqrt{24.166^2+30^2-2\cdot 24.166\cdot 30\cdot cos\ 24.444^\circ}\\ \overline{CE}=12.806\ \color{Unnecessary !}\)

\(A_{ACE}=\frac{1}{2}\cdot 30\cdot \overline{AE}\cdot sin\ \angle CAE\\ A_{ACE}=\frac{1}{2}\cdot 30\cdot 24.166\cdot sin\ 24.444^\circ\\ \color{blue}A_{ACE}=150\)

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(4,9*10^3)*(4*10^2)

Здравей приятелю!

\(​(4.9\cdot 10^3)\cdot (4\cdot 10^2)\\ =​4.9\cdot 10^3\cdot4\cdot 10^2\\ =4.9\cdot4\cdot 10^{3+2}\\ \color{blue}=19.6\cdot10^5=1.96\cdot 10^6=1\ 960\ 000\)

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