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3 years ago

What is the answer to this question. Please give me the answer ASAP

Mathematics

1 answer:

strojnjashka [21]3 years ago

6 0

still, if you have any doubt ask me!

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For 0 ≤ ϴ < 2π, how many solutions are there to tan(StartFraction theta Over 2 EndFraction) = sin(ϴ)? Note: Do not include va

Black_prince [1.1K]

Answer:

3 solutions:

$\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}}$

Step-by-step explanation:

So, first of all, we need to figure the angles that cannot be included in our answers out. The only function in the equation that isn't defined for some angles is $tan(\frac{\theta}{2})$ so let's focus on that part of the equation first.

We know that:

$tan(\frac{\theta}{2})=\frac{sin(\frac{\theta}{2})}{cos(\frac{\theta}{2})}$

therefore:

$cos(\frac{\theta}{2})\neq0$

so we need to find the angles that will make the cos function equal to zero. So we get:

$cos(\frac{\theta}{2})=0$

$\frac{\theta}{2}=cos^{-1}(0)$

$\frac{\theta}{2}=\frac{\pi}{2}+\pi n$

$\theta=\pi+2\pi n$

we can now start plugging values in for n:

$\theta=\pi+2\pi (0)=\pi$

if we plugged any value greater than 0, we would end up with an angle that is greater than $2\pi$ so, that's the only angle we cannot include in our answer set, so:

$\theta\neq \pi$

having said this, we can now start solving the equation:

$tan(\frac{\theta}{2})=sin(\theta)$

we can start solving this equation by using the half angle formula, such a formula tells us the following:

$tan(\frac{\theta}{2})=\frac{1-cos(\theta)}{sin(\theta)}$

so we can substitute it into our equation:

$\frac{1-cos(\theta)}{sin(\theta)}=sin(\theta)$

we can now multiply both sides of the equation by $sin(\theta)$

so we get:

$1-cos(\theta)=sin^{2}(\theta)$

we can use the pythagorean identity to rewrite $sin^{2}(\theta)$ in terms of cos:

$sin^{2}(\theta)=1-cos^{2}(\theta)$

so we get:

$1-cos(\theta)=1-cos^{2}(\theta)$

we can subtract a 1 from both sides of the equation so we end up with:

$-cos(\theta)=-cos^{2}(\theta)$

and we can now add $cos^{2}(\theta)$

to both sides of the equation so we get:

$cos^{2}(\theta)-cos(\theta)=0$

and we can solve this equation by factoring. We can factor $cos(\theta)$ to get:

$cos(\theta)(cos(\theta)-1)=0$

and we can use the zero product property to solve this, so we get two equations:

Equation 1:

$cos(\theta)=0$

$\theta=cos^{-1}(0)$

$\theta={\frac{\pi}{2}, \frac{3\pi}{2}}$

Equation 2:

$cos(\theta)-1=0$

we add a 1 to both sides of the equation so we get:

$cos(\theta)=1$

$\theta=cos^{-1}(1)$

$\theta=0$

so we end up with three answers to this equation:

$\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}}$

7 0

2 years ago

Look at this graph:

andreev551 [17]

Answer:

The slope is 2/1.

Step-by-step explanation:

y2 - y1 /x2 - x1

6 - 4/ 2 - 1

2/1

8 0

3 years ago

How do the particular characteristics of an ecosystem arise?

kumpel [21]

Answer:

Step-by-step explanation:

Ecosystems form in response to the unique but predictable climate of each geographic area. And since elevation and topography affect climate, ecosystems are different at different elevations. The life in any given ecosystem is the direct result of elevation, topography, and temperature, and rainfall patterns.

4 0

3 years ago

Pls hellpp! urgentt! please

Masja [62]

Answer:

Step-by-step explanation:

Perimeter of Square = 40 cm

4*side = 40

side = 40/4

side = 10 cm

Area of shaded region= Area of square - [Area of triangle 1 + Area of triangle 2]

$=[side*side]-[\frac{1}{2}*b*h+\frac{1}{2}*b*h]\\\\=[10*10]-[\frac{1}2}*3*10+\frac{1}{2}*10*6]\\\\=100-[3*5+5*6]\\\\=100-[15+30]\\\\=100-45\\\\=55cm^{2}\\$