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3 years ago

Sin\theta +cos\theta =1

Mathematics

1 answer:

Mamont248 [21]3 years ago

4 0

Answer:

θ = 2 π n_1 + π/2 for n_1 element Z or θ = 2 π n_2 for n_2 element Z

Step-by-step explanation:

Solve for θ:

cos(θ) + sin(θ) = 1

cos(θ) + sin(θ) = sqrt(2) (cos(θ)/sqrt(2) + sin(θ)/sqrt(2)) = sqrt(2) (sin(π/4) cos(θ) + cos(π/4) sin(θ)) = sqrt(2) sin(θ + π/4):

sqrt(2) sin(θ + π/4) = 1

Divide both sides by sqrt(2):

sin(θ + π/4) = 1/sqrt(2)

Take the inverse sine of both sides:

θ + π/4 = 2 π n_1 + (3 π)/4 for n_1 element Z

or θ + π/4 = 2 π n_2 + π/4 for n_2 element Z

Subtract π/4 from both sides:

θ = 2 π n_1 + π/2 for n_1 element Z

or θ + π/4 = 2 π n_2 + π/4 for n_2 element Z

Subtract π/4 from both sides:

Answer: θ = 2 π n_1 + π/2 for n_1 element Z

or θ = 2 π n_2 for n_2 element Z

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Two different radioactive isotopes decay to 10% of their respective original amounts. Isotope A does this in 33 days, while isot

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Answer:

The approximate difference in the half-lives of the isotopes is 66 days.

Step-by-step explanation:

The decay of an isotope is represented by the following differential equation:

$\frac{dm}{dt} = -\frac{t}{\tau}$

Where:

$m$ - Current mass of the isotope, measured in kilograms.

$t$ - Time, measured in days.

$\tau$ - Time constant, measured in days.

The solution of the differential equation is:

$m(t) = m_{o}\cdot e^{-\frac{t}{\tau} }$

Where $m_{o}$ is the initial mass of the isotope, measure in kilograms.

Now, the time constant is cleared:

$\ln \frac{m(t)}{m_{o}} = -\frac{t}{\tau}$

$\tau = -\frac{t}{\ln \frac{m(t)}{m_{o}} }$

The half-life of a isotope ( $t_{1/2}$ ) as a function of time constant is:

$t_{1/2} = \tau \cdot \ln2$

$t_{1/2} = -\left(\frac{t}{\ln\frac{m(t)}{m_{o}} }\right) \cdot \ln 2$

The half-life difference between isotope B and isotope A is:

$\Delta t_{1/2} = \left| -\left(\frac{t_{A}}{\ln \frac{m_{A}(t)}{m_{o,A}} } \right)\cdot \ln 2+\left(\frac{t_{B}}{\ln \frac{m_{B}(t)}{m_{o,B}} } \right)\cdot \ln 2\right|$

If $\frac{m_{A}(t)}{m_{o,A}} = \frac{m_{B}(t)}{m_{o,B}} = 0.9$ , $t_{A} = 33\,days$ and $t_{B} = 43\,days$ , the difference in the half-lives of the isotopes is: