PLS HELP (CLICK ON HERE TO SEE QUESTION)

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:

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

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

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

or

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

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

What is the equation of the following line? Be sure to scroll down first to see

dybincka [34]

Answer:

What is the equation of the following line? Be sure to scroll down first to see all answer options.

Step-by-step explanation:

y= 1/3x

6 0

3 years ago

Simple interest = Pxrxt

Zepler [3.9K]

Answer:

$18.Hope it helps.....

6 0

2 years ago

James is at a gift sale where everything costs $4 (tax included).

pantera1 [17]

Answer:

5 gifts

Step-by-step explanation:

Given information:

Cost of one gift = $4
Total money = $50
Money left after purchasing gifts = at least $28

Let x = greatest number of gifts James can buy

Create an inequality with the given information:

⇒ 50 - 4x ≥ 28

Solve the inequality

Add 4x to both sides:

⇒ 50 - 4x + 4x ≥ 28 + 4x

⇒ 50 ≥ 28 + 4x

Subtract 28 from both sides:

⇒ 50 - 28 ≥ 28 + 4x - 28

⇒ 22 ≥ 4x

Divide both sides by 4

⇒ 22 ÷ 4 ≥ 4x ÷ 4

⇒ 5.5 ≥ x

⇒ x ≤ 5.5

We must round down to 5 as if James buys 6 gifts, he will only have $26 remaining. Therefore, the greatest number of gifts James can buy is 5 gifts.

3 0

2 years ago

Read 2 more answers

In Mona’s new video game she earns 30 points for each fish caught and 80 points for each octopus captured, but she loses 15 poin

beks73 [17]

Answer:

Third one

Step-by-step explanation:

30x would be 30 times the amount of fish (x)

+ 80y would be 80 times the amount of octopus captured (y)

- 15z would be subtracting 15 times the amount of crabs she captures

6 0

2 years ago