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schepotkina [342]
3 years ago
14

Simplify (csc x)/(sec x)

Mathematics
1 answer:
Daniel [21]3 years ago
4 0
\dfrac{\csc x}{\sec x}=\dfrac{\frac{1}{\sin x}}{\frac{1}{\cos x}}=\dfrac{\cos x}{\sin x}=\boxed{\cot x}
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3 years ago
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Solve the trigonometric equation on the interval of [0,2π) below:
lara [203]

Answer:

\displaystyle  \theta =  \frac{3\pi}{4},\frac{11\pi}{12},  \frac{7\pi}{4}, \frac{23\pi}{12}

Step-by-step explanation:

we would like to solve the following trigonometric equation on interval of [0,2π)

\displaystyle  \cos \left(2 \theta +  \frac{\pi}{3} \right )  =  \frac{ \sqrt{3} }{2}

remember that,

\displaystyle  \cos(t)  =  \cos(2\pi - t)

so the equation has two solutions

\begin{cases} \displaystyle  \cos \left(2 \theta +  \frac{\pi}{3} \right )  =  \frac{ \sqrt{3} }{2}  \\  \cos \left(2\pi - (2 \theta +  \frac{\pi}{3} )\right )  =  \frac{ \sqrt{3} }{2}  \end{cases}

take inverse trig both sides which yields:

\begin{cases} \displaystyle  2 \theta +  \frac{\pi}{3}  =   \frac{\pi}{6}   \\   2\pi - 2 \theta -  \frac{\pi}{3}  =   \frac{\pi}{6}  \end{cases}

simplify the second equation:

\begin{cases} \displaystyle  2 \theta +  \frac{\pi}{3}  =   \frac{\pi}{6}   \\    \frac{5\pi}{3} - 2 \theta  =   \frac{\pi}{6}  \end{cases}

add period of 2kπ:

\begin{cases} \displaystyle  2 \theta +  \frac{\pi}{3}  =   \frac{\pi}{6} + 2k\pi   \\    \frac{5\pi}{3} - 2 \theta  =   \frac{\pi}{6}   + 2k\pi\end{cases}

By making theta subject of the equation we acquire:

\begin{cases} \displaystyle   \theta  =   \frac{11\pi}{12}  + k\pi   \\    \theta  =   \frac{3\pi}{4}    -  k\pi\end{cases}

since k\in\mathbb{Z} we get:

\begin{cases} \displaystyle   \theta  =   \frac{11\pi}{12}  + k\pi   \\    \theta  =   \frac{3\pi}{4}     +  k\pi\end{cases}

as we want to Solve the trigonometric equation on the interval of <u>[</u><u>0</u><u>,</u><u>2</u><u>π</u><u>)</u><u> </u>we get

\begin{cases} \displaystyle   \theta  =   \frac{11\pi}{12}   \\    \theta  =   \frac{3\pi}{4}    \end{cases}   \text{and} \begin{cases} \displaystyle   \theta  =   \frac{11\pi}{12}  + \pi   \\    \theta  =   \frac{3\pi}{4}     +  \pi\end{cases}

by simplifying we acquire:

\begin{cases} \displaystyle   \theta  =   \frac{11\pi}{12}   \\    \theta  =   \frac{3\pi}{4}    \end{cases}   \text{and} \begin{cases} \displaystyle   \theta  =   \frac{23\pi}{12}   \\    \theta  =   \frac{7\pi}{4}    \end{cases}

and we are done!

hence,

\displaystyle  \theta =  \frac{3\pi}{4},\frac{11\pi}{12},  \frac{7\pi}{4}, \frac{23\pi}{12}

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3 years ago
What is the slope of the graph y = -25x - 42?​
sweet [91]
-25x is the slope
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5 0
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a bookstore owner asks every fifth customer on a single day if he or she owns a e-book reader . select yes or no for each questi
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Yes, it is random because it isn't convenient for him.
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4 years ago
if the volume of the cube is increased by the factor of 8, what is the change in the length of the sides of the cube?
lorasvet [3.4K]

The new vol = 8 times the old vol.  Therefore the sides of the cube have been made 2 times larger.  (the cube root of 8 is 2.)

<span>Doubled: Volume increased 8 times 
Tripled: Volume increased 27 times 
Quadrupled: Volume increased 64 times 

Pretty simple. If you're doubling the lengths, you're increasing them by 2 times. Take 2 and raise it to the 3rd power. If you're tripling the lengths, you're increasing them 3 times over. Take 3 and raise it to the 3rd power. And so on. 

Doubled = 2^3 = 8 
Tripled = 3^3 = 27 
Quadrupled = 4^3 = 64 

Does that make sense?</span>

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3 years ago
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