All categories

3 years ago

Prove the identity: cos(3t)=cos^3(t)-3sin^2(t)cos(t)

1 answer:

3 0

Solutions

To solve the given equation lets use double-angle theorem. Double-angle formulas allow the expression of trigonometric functions of angles equal to 2α in terms of α, which can simplify the functions and make it easier to perform more complex calculations, such as integration, on them.

cos(3t)=cos^3(t)-3sin^2(t)cos(t)

cos(A+B) = cosAcosB - sinAsinB

Now cos(3t) = cos(2t+t) 

⇒cos(2t)cos(t) - sin(2t)sin(t) 

⇒(2cos^2(t)-1)cos(t) - 2sin(t)cos(t)sin(t) 

⇒ 2cos^(3)t - cos(t) - 2sin^2(t)cos(t) 

⇒ 2cos^3(t) - cos(t) - 2(1-cos^2(t))cos(t) 

⇒2cos^3(t) - cos(t) - 2cos(t) + 2cos^3(t) 

⇒4cos^3(t) - 3cos(t)

You might be interested in

What is the slope of the line that passes through the points (0,5) (2,10)

Nadya [2.5K]

Answer:

m=5/2

Step-by-step explanation:

the equation of slope is (y2-y1)/(x2-x1)

here are the points given: (0,5), (2,10)

let's label the points:

x1=0

y1=5

x2=2

y2=10

now substitute into the equation (m is the slope)

m=(10-5)/(2-0)

subtract

m=5/2

the slope is 5/2

8 0

3 years ago

A dolphin swims 665,280 feet in 3 hours. Use the formula dart, where d represents distance, r represents rate, and t represents

barxatty [35]

Step-by-step explanation:

a) r=d/t ( same as formula of speed)

b) r= 665280/3=221760 feet/hr

c) distance in miles = 665280/5280= 126

so r=126/3 = 42 miles/hr

d) i think using miles makes more sense cz using feet will result in greater number of values so there is a greater chance of human error

5 0

3 years ago

Read 2 more answers

What is 1.63 x 0.98 =

yarga [219]

Answer:

1.5974

Step-by-step explanation:

8 0

3 years ago

Read 2 more answers

The Washington, DC, region has one of the fastest-growing foreclosure rates in the nation, as 15,613 homes went into foreclosure

Ilia_Sergeevich [38]

Answer:

a) 0.6226 = 62.26% probability that in a given year, fewer than 2 out of 100 houses in the Washington, DC area will go up for foreclosure.

b) 0.7837 = 78.37% probability that in a given year, fewer than 2 out of 100 houses in the nation will go up for foreclosure.

c) The proportion of foreclosures in the Nation is lower than in Washington, which means that with a sample size of 100, it is likely to have a small number(fewer than 2) of foreclosures than Washington DC.

Step-by-step explanation:

For each home, there are only two possible outcomes. Either it goes into foreclosure, or it does not. The probability of a home going into foreclosure is independent of other homes. This means that we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

In which $C_{n,x}$ is the number of different combinations of x objects from a set of n elements, given by the following formula.

$C_{n,x} = \frac{n!}{x!(n-x)!}$

And p is the probability of X happening.

a. What is the probability that in a given year, fewer than 2 out of 100 houses in the Washington, DC area will go up for foreclosure?

The foreclosure rate is 1.31% for the Washington, DC area, which means that $p = 0.0131$

We wanto to find, with $n = 100$ :

$P(X < 2) = P(X = 0) + P(X = 1)$

In which

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{100,0}.(0.0131)^{0}.(0.9869)^{100} = 0.2675$

$P(X = 1) = C_{100,1}.(0.0131)^{1}.(0.9869)^{99} = 0.3551$

$P(X < 2) = P(X = 0) + P(X = 1) = 0.2675 + 0.3551 = 0.6226$

0.6226 = 62.26% probability that in a given year, fewer than 2 out of 100 houses in the Washington, DC area will go up for foreclosure.

b. What is the probability that in a given year, fewer than 2 out of 100 houses in the nation will go up for foreclosure?

Foreclosure rate of 0.87% for the nation, which means that $p = 0.0087$ . So

$P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}$

$P(X = 0) = C_{100,0}.(0.0087)^{0}.(0.9913)^{100} = 0.4174$

$P(X = 1) = C_{100,1}.(0.0087)^{1}.(0.9913)^{99} = 0.3663$

$P(X < 2) = P(X = 0) + P(X = 1) = 0.4174 + 0.3663 = 0.7837$

0.7837 = 78.37% probability that in a given year, fewer than 2 out of 100 houses in the nation will go up for foreclosure.

c. Comment on the above findings.

The proportion of foreclosures in the Nation is lower than in Washington, which means that with a sample size of 100, it is likely to have a small number(fewer than 2) of foreclosures than Washington DC.

7 0

3 years ago

Please solve this , i will mark brainliest

erastova [34]

C thank me later thanks

6 0

3 years ago

Read 2 more answers