I REALLY NEED HELP! ive tried this so many times and just cant get it! thank you!

zzz [600]

Answer:

$\theta = \frac{\pi}{3} , \pi, \frac{5\pi}{ 3}$

Step-by-step explanation:

we want to solve the following trigonometric equation:

$\displaystyle - 2 { \sin}^{2} (\theta )+ \cos( \theta) + 1 = 0, \: \theta \in [0,2\pi)$

The first step of solving trigonometric equation is to rewrite the equation in terms of one trigonometric function . With Pythagorean theorem, we know that sin²x=1-cos²x . It will be helpful to rewrite the equation in terms of one trig functions. Therefore, substitute 1-cos² $\theta$ in the place of sin² $\theta$ :

$\displaystyle - 2 (1 - \cos ^{2} ) + \cos( \theta) + 1 = 0, \: \theta \in [0,2\pi)$

simplify:

$\displaystyle\implies - 2 (1 - \cos ^{2} (\theta) + \cos( \theta) + 1 = 0, \: \theta \in [0,2\pi)\\\implies- 2 + \cos ^{2} (\theta)+ \cos( \theta) + 1 = 0, \: \theta \in [0,2\pi)\\\implies\cos ^{2} (\theta)+ \cos( \theta) -1 = 0$

Consider cos² $\theta\implies$ x. Thus,

$2 {x}^{2} + x - 1 = 0$

solving the quadratic equation yields:

${x}^{} = \frac{1}{2} \\ x = - 1$

back-substitute:

$\begin{cases} \cos( \theta) = \dfrac{1}{2} \\ \cos( \theta) = - 1 \end{cases} \theta \in[0,2\pi)$

take inverse trig in both sides

$\implies \begin{cases} \theta = \dfrac{\pi}{3} + 2n\pi\\\theta= \frac{5\pi}{3} + 2n\pi \\ \theta = \pi + 2n\pi\end{cases} \theta \in[0,2\pi) \\\\\implies\theta= \frac{\pi}{3}+\dfrac{2n\pi}{3},\theta\in [0,2\pi)\\\text{when n=0}\\ \implies \theta = \frac{\pi}{3} \\ \text{when n=1}\\ \theta= \pi\\ \text{when n=2}\\\theta=\frac{5\pi}{ 3}$

In conclusion,

$\displaystyle\theta = \frac{\pi}{3} , \pi, \frac{5\pi}{ 3}$

7 0

2 years ago

In The Figure Below, What is the reflection of point D across line x=0?

nignag [31]

X=0 is actually the y axis, so the y coordinate of the reflection doesn't change, while the x coordinate changes from a negative to a positive or from a positive to a negative.
the new confidante of D is (1,-1)

7 0

3 years ago

If the measure of angle 5 is (11X-14)º and x = 6, which expression could represent the measure of angle 2?

Sauron [17]

11(6)-14=56
The answer should be 9x+2 which would be 56–> 9(6)+2=56

5 0

3 years ago

100/100= <br><br>what's the answer

Nitella [24]

Answer:

Step-by-step explanation:

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3 0

3 years ago

A ball bearing is fired straight up at a speed of 40m/s. Calculate the time during which it is in the air. Ignore air resistance

ololo11 [35]

Answer:

t = 8.2 sec

Step-by-step explanation:

the ball will move with uniform deceleration due to gravity until it reaches the highest point at which v = 0 then will change its direction and continue with uniform acceleration until it hits the ground.

the time the ball takes to reach the highest point equals the time it takes to return to the ground again.

to calculate the total time, we calculate the first half then multiply it by 2

we use the first equation because we have all the variables except time

v = 0 , u = 40 , a = -9.8

0 = 40 - 9.8t

t = 4.1 sec

total time = 4.1 x 2 = 8.2 sec

3 0

4 years ago