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Delvig [45]
1 year ago
14

Use the given information to find the exact value of the trigonometric function

Mathematics
1 answer:
eimsori [14]1 year ago
7 0
\begin{gathered} \csc \theta=-\frac{6}{5} \\ \tan \theta>0 \\ \cos \frac{\theta}{2}=\text{?} \end{gathered}

Half Angle Formula

\cos \frac{\theta}{2}=\pm\sqrt[\square]{\frac{1+\cos\theta}{2}}\tan \theta>0\text{ and csc}\theta\text{ is negative in the third quadrant}\begin{gathered} \csc \theta=-\frac{6}{5}=\frac{r}{y} \\ x^2+y^2=r^2 \\ x=\pm\sqrt[\square]{r^2-y^2} \\ x=\pm\sqrt[\square]{6^2-(-5)^2} \\ x=\pm\sqrt[\square]{36-25} \\ x=\pm\sqrt[\square]{11} \\ \text{x is negative since the angle is on the 3rd quadrant} \end{gathered}\begin{gathered} \cos \theta=\frac{x}{r}=\frac{-\sqrt[\square]{11}}{6} \\ \cos \frac{\theta}{2}=\pm\sqrt[\square]{\frac{1+\cos\theta}{2}} \\ \cos \frac{\theta}{2}is\text{ also negative in the 3rd quadrant} \\ \cos \frac{\theta}{2}=-\sqrt[\square]{\frac{1+\frac{-\sqrt[\square]{11}}{6}}{2}} \\ \cos \frac{\theta}{2}=-\sqrt[\square]{\frac{\frac{6-\sqrt[\square]{11}}{6}}{2}} \\ \cos \frac{\theta}{2}=-\sqrt[\square]{\frac{6-\sqrt[\square]{11}}{12}} \\  \\  \end{gathered}

Answer:

\cos \frac{\theta}{2}=-\sqrt[\square]{\frac{6-\sqrt[\square]{11}}{12}}

Checking:

\begin{gathered} \frac{\theta}{2}=\cos ^{-1}(-\sqrt[\square]{\frac{6-\sqrt[\square]{11}}{12}}) \\ \frac{\theta}{2}=118.22^{\circ} \\ \theta=236.44^{\circ}\text{  (3rd quadrant)} \end{gathered}

Also,

\csc \theta=\frac{1}{\sin\theta}=\frac{1}{\sin (236.44)}=-\frac{6}{5}\text{ QED}

The answer is none of the choices

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3 years ago
Someone help ASAP!!!!!!
Alchen [17]

Answer:

Length, l = 11 ft.

Width, w = 9 ft.

Step-by-step explanation:

From the given data, the area of the rectangle = 99 ft².

Area of the rectangle = Length, l X Width, w

Here, Length, l = 7 more than twice the width

⇒       Length, l = 7 + 2w

Therefore, Area, A = 99 = (7 + 2w)w

⇒ 99 = 7w + 2w²

⇒ 2w² + 7w - 99 = 0

Solve the Quadratic equation using the formula: x = $ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ for the quadratic equation ax² + bx + c = 0.

Therefore, w = $ \frac{-7 \pm \sqrt{7^2 -4(2)(-99)}}{2(2)} $

$ = \frac{-7 \pm \sqrt{49 + 8(99)}}{4} $

$ = \frac{-7 \pm + \sqrt{841}}{4} $

Since, $ \sqrt{841} = 29 $ we get:

$ w = \frac{-7 \pm 29}{4} $

This gives two values of 'w', viz., w = $ \frac{-7 - 29}{4} $, $ \frac{-7 + 29}{4} $

$ \implies w = \frac{22}{4}, \frac{-36}{4} $

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If w = -9, then l = 2(-9) + 7

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3 years ago
It is monday. : there is chicken for lunch. write the inverse in if-then form. if it is monday, then there is chicken for lunch.
ololo11 [35]
If Monday, then Chicken
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3 0
3 years ago
Read 2 more answers
Simplify using log properties: −log1/100 <br> -10 <br> 2 <br> -2 <br> 1/2
Andreyy89

The answer is 2 .

simplify log(down)10 (1/100) : -2

apply log rule : -log(down)10 (100)

rewrite 100 in power-base form : -log(down)10 (10^2)

apply log rule again : -2log(down)10 (10)

apply one more time : -2*1

Multiply : -2

refine from -(-2) = 2

I hope this helps





5 0
3 years ago
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