All categories

3 years ago

Evaluate : $\cos 12 \cos 24 \cos 36 \cos 48 \cos 72 \cos 84$

1 answer:

4 0

The answer in itself is 1/128 and here is the procedure to prove it:
cos(A)*cos(60+A)*cos(60-A) = cos(A)*(cos²60 - sin²A)

= cos(A)*{(1/4) - 1 + cos²A} = cos(A)*(cos²A - 3/4) 

= (1/4){4cos^3(A) - 3cos(A)} = (1/4)*cos(3A) 

Now we group applying what we see above

cos(12)*cos(48)*cos(72) = 
=cos(12)*cos(60-12)*cos(60+12) = (1/4)cos(36) 

Similarly, cos(24)*cos(36)*cos(84) = (1/4)cos(72) 

Now the given expression is: 

= (1/4)cos(36)*(1/4)*cos(72)*cos(60) = 

= (1/16)*(1/2)*{(√5 + 1)/4}*{(√5 - 1)/4} [cos(60) = 1/2; 
cos(36) = (√5 + 1)/4 and cos(72) = cos(90-18) = 
= sin(18) = (√5 - 1)/4] 
And we seimplify it and it goes: (1/512)*(5-1) = 1/128

You might be interested in

Can someone explain what geometric sequences are?

Fed [463]

Explanation:

A sequence is a list of numbers.

A geometric sequence is a list of numbers such that the ratio of each number to the one before it is the same. The common ratio can be any non-zero value.

Examples

1, 2, 4, 8, ... common ratio is 2
27, 9, 3, 1, ... common ratio is 1/3
6, -24, 96, -384, ... common ratio is -4

___

General Term

Terms of a sequence are numbered starting with 1. We sometimes use the symbol a(n) or an to refer to the n-th term. The general term of a geometric sequence, a(n), can be described by the formula ...

a(n) = a(1)×r^(n-1) . . . . . n-th term of a geometric sequence

where a(1) is the first term, and r is the common ratio. The above example sequences have the formulas ...

a(n) = 2^(n -1)
a(n) = 27×(1/3)^(n -1)
a(n) = 6×(-4)^(n -1)

You can see that these formulas are exponential in nature.

Sum of Terms

Another useful formula for geometric sequences is the formula for the sum of n terms.

S(n) = a(1)×(r^n -1)/(r -1) . . . . . sum of n terms of a geometric sequence

When |r| < 1, the sum converges as n approaches infinity. The infinite sum is ...

S = a(1)/(1-r)

8 0

3 years ago

Cranberry juice costs $6.30 per quart and apple juice costs $3.60 per quart. Terrence wants to know how many quarts q of cranber

riadik2000 [5.3K]

Q = The Ammount Of Cranberry Juice

(q+4)*4.50=q*6.30+4*3.60

4.5*q+18=6.30*q+14.4

6.30*q-4.5*q=18-14.4

1.8*q=3.6

q=3.6/1.8

It Would Be 2 Quarts Of Cranberry Juice

4 0

3 years ago

a ball is thrown straight up from the top of a 128 foot tall building with an initial speed of 32 feet per second. the height of

STatiana [176]

Factoring, you have
h(t) = -16(t^2 -2 -8) = -16(t -4)(t +2)

The ball hits the ground when h(t) = 0, so
0 = -16(t -4)(t +2)
t = 4 or -2

The positive solution is the one of interest.
It will take 4 seconds for the ball to hit the ground.

6 0

3 years ago

Each morning Rin rides 1.5 miles to school and then rides home in the afternoon. Later in the evening, she rides to the park and

brilliants [131]

Answer:

Last equation given in the list of possible answers:

5 ( 1.5 + 1.5 + x ) = 25

Step-by-step explanation:

We need to include in the total addition of miles ridden during the week:

a) 1.5 miles to the school

b) 1.5 miles from school back home

c) x miles for the evening ride

so for the miles ridden per day we have: "1.5 +1.5 + x"

Now, since per week she does 5 days like this, then we need to multiply the expression above by 5 in order to total the number of miles she rides weekly (25 miles)

5 ( 1.5 + 1.5 + x ) = 25

And we can use this equation to find the amount "x" that Rin rides in the evening.

7 0

3 years ago

Its

damaskus [11]

Answer:

$P(Green\ and\ Green) = \frac{25}{144}$