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2 years ago

Simplify the trigonometric expression sin(4x)+2 sin(2x) using Double-Angle

Mathematics

1 answer:

drek231 [11]2 years ago

8 0

Answer:

${ \bf{ = \sin(4x) + 2 \sin(2x) }} \\ = { \bf{2 \sin(2x) \cos(2x) + 4 \sin(x) \cos(x) }} \\ { \bf{ = 4 \sin(x) \cos(x) . ({ \cos }^{2} x - { \sin}^{2} }x) + 4 \sin(x) \cos(x) } \\ = { \bf{4 \sin(x) \cos(x) ( { \cos }^{2}x - { \sin }^{2}x + 1) }} \\ = { \bf{4 \sin(x) \cos(x) }(2 { \cos }^{2} x)} \\ = { \bf{8 \sin(x) { \cos}^{3}x }}$

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What digit is in the ten-thousands place in the number 81,473

Ugo [173]

Answer:

Explanation:

8 - Ten-Thousands

1 - Thousands

4 - Hundreds

7 - Tens

3 - Ones

3 0

3 years ago

Read 2 more answers

Este curso nuestro instituto tiene 473 alumnos. Si nos dicen que respecto al curso anterior se ha producido un aumento de inscri

tester [92]

Answer:

En el curso anterior había 430 alumnos.

Step-by-step explanation:

El curso tiene 473 alumnos. Nos dicen que respecto al curso anterior se ha producido un aumento de inscripciones del 10 %. Entonces, siendo x la cantidad de alumnos que había en el curso anterior, se puede plantear la ecuación:

x + 0.1*x= 473

Resolviendo se obtiene:

1.1*x=473

x= 473 ÷1.1

x= 430

En el curso anterior había 430 alumnos.

7 0

3 years ago

Write a statement that correctly describes the relationship between these two sequences; 2,4,6,8,10 and 1,2,3,4,5

marta [7]

Answer: Any number in the first sequence is two times the second.

Step-by-step explanation: Mark me as brainiest!

4 0

2 years ago

Suppose a geyser has a mean time between irruption’s of 75 minutes. If the interval of time between the eruption is normally dis

lesya [120]

Answer:

(a) The probability that a randomly selected Time interval between irruption is longer than 84 minutes is 0.3264.

(b) The probability that a random sample of 13 time intervals between irruption has a mean longer than 84 minutes is 0.0526.

(c) The probability that a random sample of 20 time intervals between irruption has a mean longer than 84 minutes is 0.0222.

(d) The probability decreases because the variability in the sample mean decreases as we increase the sample size

(e) The population mean may be larger than 75 minutes between irruption.

Step-by-step explanation:

We are given that a geyser has a mean time between irruption of 75 minutes. Also, the interval of time between the eruption is normally distributed with a standard deviation of 20 minutes.

(a) Let X = the interval of time between the eruption

So, X ~ Normal( $\mu=75, \sigma^{2} =20$ )

The z-score probability distribution for the normal distribution is given by;

Z = $\frac{X-\mu}{\sigma}$ ~ N(0,1)

where, $\mu$ = population mean time between irruption = 75 minutes

$\sigma$ = standard deviation = 20 minutes

Now, the probability that a randomly selected Time interval between irruption is longer than 84 minutes is given by = P(X > 84 min)

P(X > 84 min) = P( $\frac{X-\mu}{\sigma}$ > $\frac{84-75}{20}$ ) = P(Z > 0.45) = 1 - P(Z $\leq$ 0.45)

= 1 - 0.6736 = 0.3264

The above probability is calculated by looking at the value of x = 0.45 in the z table which has an area of 0.6736.

(b) Let $\bar X$ = sample time intervals between the eruption

The z-score probability distribution for the sample mean is given by;

Z = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\mu$ = population mean time between irruption = 75 minutes

$\sigma$ = standard deviation = 20 minutes

n = sample of time intervals = 13

Now, the probability that a random sample of 13 time intervals between irruption has a mean longer than 84 minutes is given by = P( $\bar X$ > 84 min)

P( $\bar X$ > 84 min) = P( $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ > $\frac{84-75}{\frac{20}{\sqrt{13} } }$ ) = P(Z > 1.62) = 1 - P(Z $\leq$ 1.62)

= 1 - 0.9474 = 0.0526

The above probability is calculated by looking at the value of x = 1.62 in the z table which has an area of 0.9474.

(c) Let $\bar X$ = sample time intervals between the eruption

The z-score probability distribution for the sample mean is given by;

Z = $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

where, $\mu$ = population mean time between irruption = 75 minutes

$\sigma$ = standard deviation = 20 minutes

n = sample of time intervals = 20

Now, the probability that a random sample of 20 time intervals between irruption has a mean longer than 84 minutes is given by = P( $\bar X$ > 84 min)

P( $\bar X$ > 84 min) = P( $\frac{\bar X-\mu}{\frac{\sigma}{\sqrt{n} } }$ > $\frac{84-75}{\frac{20}{\sqrt{20} } }$ ) = P(Z > 2.01) = 1 - P(Z $\leq$ 2.01)

= 1 - 0.9778 = 0.0222

The above probability is calculated by looking at the value of x = 2.01 in the z table which has an area of 0.9778.

(d) When increasing the sample size, the probability decreases because the variability in the sample mean decreases as we increase the sample size which we can clearly see in part (b) and (c) of the question.

(e) Since it is clear that the probability that a random sample of 20 time intervals between irruption has a mean longer than 84 minutes is very slow(less than 5%0 which means that this is an unusual event. So, we can conclude that the population mean may be larger than 75 minutes between irruption.

8 0

2 years ago

A signal light is green for 4 minutes, yellow for 10 seconds, and red for 3 minutes. If you drive up to this light, what is the

posledela

Answer:

0.56 is the required probability.

Step-by-step explanation:

Time for which signal shows green light = 4 minutes

Time for which signal shows yellow light = 10 seconds

Time for which signal shows red light = 3 minutes

To find:

Probability that the signal will show green light when you reach the destination = ?

Solution:

First of all, let us convert each time to same unit before doing any calculations.

Time for which signal shows green light = 4 minutes = 4 $\times$ 60 seconds = 240 seconds

Time for which signal shows yellow light = 10 seconds

Time for which signal shows red light = 3 minutes = 3 $\times$ 60 seconds = 180 seconds

Now, let us have a look at the formula for probability of an event E:

$P(E) = \dfrac{\text{Number of favorable cases}}{\text {Total number of cases}}$

Here, E is the event that green light is shown by the signal.

Number of favorable cases mean the time for which green light is shown and Total number of cases is the total time (Time for which green light is shown + Time for which Yellow light is shown + Time for which red light is shown)

So, the required probability is:

$P(E) = \dfrac{240}{240+10+180}\\\Rightarrow P(E) = \dfrac{240}{430}\\\Rightarrow \bold{P(E) \approx 0.56 }$

5 0

2 years ago