What point on the unit circle corresponds to an angle of 173º

nikdorinn [45]

45.40 is your number

5 0

3 years ago

I'm having trouble with #2. I've got it down to the part where it would be the integral of 5cos^3(pheta)/sin(pheta). I'm not sur

Butoxors [25]

$\displaystyle\int\frac{\sqrt{25-x^2}}x\,\mathrm dx$

Setting $x=5\sin\theta$ , you have $\mathrm dx=5\cos\theta\,\mathrm d\theta$ . Then the integral becomes

$\displaystyle\int\frac{\sqrt{25-(5\sin\theta)^2}}{5\sin\theta}5\cos\theta\,\mathrm d\theta$
$\displaystyle\int\sqrt{25-25\sin^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\sqrt{1-\sin^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\sqrt{\cos^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$

Now, $\sqrt{x^2}=|x|$ in general. But since we want our substitution $x=5\sin\theta$ to be invertible, we are tacitly assuming that we're working over a restricted domain. In particular, this means $\theta=\sin^{-1}\dfrac x5$ , which implies that $\left|\dfrac x5\right|\le1$ , or equivalently that $|\theta|\le\dfrac\pi2$ . Over this domain, $\cos\theta\ge0$ , so $\sqrt{\cos^2\theta}=|\cos\theta|=\cos\theta$ .

Long story short, this allows us to go from

$\displaystyle5\int\sqrt{\cos^2\theta}\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$

to

$\displaystyle5\int\cos\theta\dfrac{\cos\theta}{\sin\theta}\,\mathrm d\theta$
$\displaystyle5\int\dfrac{\cos^2\theta}{\sin\theta}\,\mathrm d\theta$

Computing the remaining integral isn't difficult. Expand the numerator with the Pythagorean identity to get

$\dfrac{\cos^2\theta}{\sin\theta}=\dfrac{1-\sin^2\theta}{\sin\theta}=\csc\theta-\sin\theta$

Then integrate term-by-term to get

$\displaystyle5\left(\int\csc\theta\,\mathrm d\theta-\int\sin\theta\,\mathrm d\theta\right)$
$=-5\ln|\csc\theta+\cot\theta|+\cos\theta+C$

Now undo the substitution to get the antiderivative back in terms of $x$ .

$=-5\ln\left|\csc\left(\sin^{-1}\dfrac x5\right)+\cot\left(\sin^{-1}\dfrac x5\right)\right|+\cos\left(\sin^{-1}\dfrac x5\right)+C$

and using basic trigonometric properties (e.g. Pythagorean theorem) this reduces to

$=-5\ln\left|\dfrac{5+\sqrt{25-x^2}}x\right|+\sqrt{25-x^2}+C$

4 0

2 years ago

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Cheyenne took out a 30-year loan for $350,000 at 2.9% interest, compounded monthly. If her monthly payment on the loan will rema

Leto [7]

The answer is d.

Explanation:

1456.80 per month
1456.80 times 12 = how much per year
17481.6 = how much per year
17481.6 times 30 years = 524448
$524448 - the original loan(350,000) = $174,448

6 0

3 years ago

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The temperature in Scottsdale increased by 2 degrees in 2 days. If the temperature changed by the same

Vsevolod [243]

Answer:

yniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh olleh

Step-by-step explanation:yniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era csacasccacwoh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh ollehyniar hcum oot sti ereh pleh esaelp retaw eht rednu ma i uoy era woh olleh

5 0

2 years ago

Graph the function: y=x-5

ANEK [815]

Answer:

Step-by-step explanation:

3 0

3 years ago

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