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3 years ago

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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Answer:21 $

Step-by-step explanation:

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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 [0,2π) 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

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-25x is the slope
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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.)

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?

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3 years ago