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1 year ago

Prove that sinA-sin3A+sin5A-sin7A/cosA-cos3A-cos5A+cos7A= cot2A

1 answer:

4 0

Write the left side of the given expression as N/D, where
N = sinA - sin3A + sin5A - sin7A
D = cosA - cos3A - cos5A + cos7A
Therefore we want to show that N/D = cot2A.

We shall use these identities:
sin x - sin y = 2cos((x+y)/2)*sin((x-y)/2)
cos x - cos y = -2sin((x+y)/2)*sin((x-y)2)

N = -(sin7A - sinA) + sin5A - sin3A
= -2cos4A*sin3A + 2cos4A*sinA
= 2cos4A(sinA - sin3A)
= 2cos4A*2cos(2A)sin(-A)
= -4cos4A*cos2A*sinA

D = cos7A + cosA - (cos5A + cos3A)
= 2cos4A*cos3A - 2cos4A*cosA
= 2cos4A(cos3A - cosA)
= 2cos4A*(-2)sin2A*sinA
= -4cos4A*sin2A*sinA

Therefore
N/D = [-4cos4A*cos2A*sinA]/[-4cos4A*sin2A*sinA]
= cos2A/sin2A
= cot2A

This verifies the identity.

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Company A charges $331.35 per week for a compact car with unlimited miles. Company B charges $175 per week plus $0.53 per mile,

Vladimir [108]

Answer:

Company A is a better deal than Company B for the number of miles greater than 295 miles

Step-by-step explanation:

Let

y ----> the charge per week in dollars

x ----> the number of miles

we have

Company A

$y=331.35$ -----> equation A

Company B

$y=0.53x+175$ -----> equation B

Solve the system by substitution

Equate equation A and equation B and solve for x

$331.35=0.53x+175$

$0.53x=331.35-175\\0.53x=156.35\\x=295\ mi$

For x=295 miles the charge in Company A and Company B is the same

therefore

Company A is a better deal than Company B for the number of miles greater than 295 miles

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2 years ago

A toy factory makes 5,000 teddy bears per day. The supervisor randomly selects 10 teddy bears from all 5,000 teddy bears and use

tankabanditka [31]

Answer:

Step-by-step explanation:

Degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. Degrees of freedom are related to sample size and calculated by n-1 where n is the sample size.

The Supervisor selects 10 teddy bears as sample from 5000 teddy bears produced daily. Therefore in this situation, there are 10-1=9 degrees of freedom

3 0

1 year ago

Tyler poured a total of 15 cups of water into 2 equal-sized bottles and filled each bottle. How much water was in each bottle?

Nitella [24]

Step-by-step explanation:

So 2 bottles and 15 cups

15 ÷ 2 = 7.5

He poured 7.5 cups of water in each bottle

4 0

2 years ago

The subjects of a study by Dugoff et al. (A-5) were 10 obstetrics and gynecology interns at the University of Colorado Health Sc

astra-53 [7]

Answer:

The 95 percent confidence interval for the mean of the population from which the study subjects may be presumed to have been drawn is (19.1269, 32.6730).

Step-by-step explanation:

Intern No. of Breast

Number Exams Performed X²

1 30 900

2 40 1600

3 8 64

4 20 400

5 26 676

6 35 1225

7 35 1225

8 20 400

9 25 625

10 20 400

 ∑ 259 ∑ 7515

Mean= X`= ∑x/n= 259/10= 25.9

Variance = s²= 1/n-1[∑X²- (∑x)²/n]

= 1/0[7515- (259)²/10]= 1/9[7515- 6708.1]

= 806.9/9=89.655= 89.66

Standard Deviation= √89.655= 9.4687

Hence

The value of t with significance level alpha= 0.05 and 9 degrees of freedom is t(0.025,9)= 2.262

The 95 % Confidence interval is given by

x`±t(∝,n-1) s/√n

So Putting the values

25.9± 2.262( 9.4687/√10)

= 25.9 ±2.262 (2.9943)

= 25.9 ± 6.7730

= 25.9 +6.7730=32.6730

25.9 -6.7730= 19.1269

= 19.1269, 32.6730

The 95 percent confidence interval for the mean of the population from which the study subjects may be presumed to have been drawn is (19.1269, 32.6730).

6 0

2 years ago

ΔABC is dilated using a scale factor of 12 to produce ΔA'B'C'. Select all of the statements that apply to the transformation.

GalinKa [24]

Answer:

Options (3) and (6)

Step-by-step explanation:

ΔABC is a dilated using a scale factor of $\frac{1}{2}$ to produce image triangle ΔA'B'C'.

Since, dilation is a rigid transformation,

Angles of both the triangles will be unchanged or congruent.

m∠A = m∠A' and m∠B = m∠B'

Since, sides of ΔA'B'C' = $\frac{1}{2}$ of the sides of ΔABC

Area of ΔA'B'C' = $\frac{1}{2}(\text{Area of triangle ABC})$

Area of ΔABC > Area of ΔA'B'C'

Since, angles of ΔABC and ΔA'B'C' are congruent, both the triangles will be similar.

ΔABC ~ ΔA'B'C'

Therefore, Option (3) and Option (6) are the correct options.

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1 year ago