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3 years ago

Expand the following expression by using the distributive method. 6(3b-4c)

Mathematics

1 answer:

sergeinik [125]3 years ago

8 0

18b-24c. U take the six times the 3b and then the 6 times the -4c. U “distribute” the six with both values

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Hello! Verify the identity. Please show your work! Use trigonometric identities to verify each expression is equal.

RSB [31]

Answer:

See Below.

Step-by-step explanation:

We want to verify the identity:

$\displaystyle \csc^2 x -2\csc x \cot x +\cot ^2 x = \tan^2\left(\frac{x}{2}\right)$

Note that the left-hand side is a perfect square trinomial pattern. Namely:

$a^2-2ab+b^2=(a-b)^2$

If we let a = csc(x) and b = cot(x), we can factor it as such:

$\displaystyle (\csc x - \cot x)^2 = \tan^2\left(\frac{x}{2}\right)$

Let csc(x) = 1 / sin(x) and cot(x) = cos(x) / sin(x):

$\displaystyle \left(\frac{1}{\sin x}-\frac{\cos x }{\sin x}\right)^2=\tan^2\left(\frac{x}{2}\right)$

Combine fractions:

$\displaystyle \left(\frac{1-\cos x}{\sin x}\right)^2=\tan^2\left(\frac{x}{2}\right)$

Square (but do not simplify yet):

$\displaystyle \frac{(1-\cos x)^2}{\sin ^2x}=\tan^2\left(\frac{x}{2}\right)$

Now, we can make a substitution. Let u = x / 2. So, x = 2u. Substitute:

$\displaystyle \frac{(1-\cos 2u)^2}{\sin ^22u}=\tan^2u$

Recall that cos(2u) = 1 - sin²(u). Hence:

$\displaystyle \frac{(1-(1-2\sin^2u))^2}{\sin ^2 2u}=\tan^2u$

Simplify:

$\displaystyle \frac{4\sin^4 u}{\sin ^2 2u}=\tan^2 u$

Recall that sin(2u) = 2sin(u)cos(u). Hence:

$\displaystyle \frac{4\sin^4 u}{(2\sin u\cos u)^2}=\tan^2 u$

Square:

$\displaystyle \frac{4\sin^4 u}{4\sin^2 u\cos ^2u}=\tan^2 u$

Cancel:

$\displaystyle \frac{\sin ^2 u}{\cos ^2 u}=\tan ^2 u$

Since sin(u) / cos(u) = tan(u):

$\displaystyle \left(\frac{\sin u}{\cos u}\right)^2=\tan^2u=\tan^2u$

We can substitute u back for x / 2:

$\displaystyle \tan^2\left(\frac{x}{2}\right)= \tan^2\left(\frac{x}{2}\right)$

Hence proven.

3 0

3 years ago

A circle and a regular pentagon can intersect in

lbvjy [14]

Answer:

I think it is option D,

hope it helps

5 0

3 years ago

Name all the factors for 1083

kow [346]

Answer:

1, 3, 19, 57, 361, and 1083.

Step-by-step explanation:

3 0

3 years ago

What is the ratio of 86 and 128

IRINA_888 [86]

Answer:67.1875%

Step-by-step explanation:

Convert ratio into fraction 86 / 128

6 0

3 years ago

Re-write the quadratic function below in Standard Form y= -9(x+3)2+2

disa [49]

Answer:

18x+y=-52

Step-by-step explanation:

we all know and love(being 100% sarcastic here)the standard form as

ax+by=c

y=-9(x+3)*2+2----This is weird and not so nice

so lets make it nice by distributing the 9::::

y=(-9x-27)2+2

and now the 2:

y=-18x-54+2

y=-18x-52

oops this is NOT in S.F we got to switch the -52 and y

52=-18x-y

-18 cant be neg. so we have to divide by -1 on both sides

-52=18x+y

18x+y=-52

HOPE THIS HELPS

7 0

3 years ago

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