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2 years ago

What are the exact values of sin(2π3radians) and cos(2π3radians)

1 answer:

8 0

We know that
2π/3 radians-------> convert to degrees-----> 2*180/3---> 120°
120°=90°+30°

Part a) Find sin(2π/3)
sin(2π/3)=sin (90°+30°)
we know that
sin (A+B)=sin A*cos B+cos A*sin B
so
sin (90°+30°)=sin 90*cos 30+cos 90*sin 30
sin 90=1
cos 30=√3/2
cos 90=0
sin 30=1/2
sin (90°+30°)=1*√3/2+0*1/2-----> √3/2

the answer part a) is
sin(2π/3)=√3/2

Part b) Find cos (2π/3)
cos (2π/3)=cos (90°+30°)
we know that
cos (A+B)=cos A*cos B-sin A*sin B
so
cos (90°+30°)=cos 90*cos 30-sin 90*sin 30
sin 90=1
cos 30=√3/2
cos 90=0
sin 30=1/2
cos (90°+30°)=0*√3/2-1*1/2----> -1/2

the answer part b) is
cos (2π/3)=-1/2

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Find the sum of the first 25 terms in this geometric series: 8 + 6 + 4.5...

Ksivusya [100]

Step-by-step explanation:

Given the geometric sequence

8 + 6 + 4.5...

A geometric sequence has a constant ratio and is defined by

$a_n=a_1\cdot r^{n-1}$

$\mathrm{Compute\:the\:ratios\:of\:all\:the\:adjacent\:terms}:\quad \:r=\frac{a_{n+1}}{a_n}$

$\frac{6}{8}=\frac{3}{4},\:\quad \frac{4.5}{6}=\frac{3}{4}$

$\mathrm{The\:ratio\:of\:all\:the\:adjacent\:terms\:is\:the\:same\:and\:equal\:to}$

$r=\frac{3}{4}$

$\mathrm{The\:first\:element\:of\:the\:sequence\:is}$

$a_1=8$

$\mathrm{Therefore,\:the\:}n\mathrm{th\:term\:is\:computed\:by}\:$

$a_n=8\left(\frac{3}{4}\right)^{n-1}$

$\mathrm{Geometric\:sequence\:sum\:formula:}$

$a_1\frac{1-r^n}{1-r}$

$\mathrm{Plug\:in\:the\:values:}$

$n=25,\:\spacea_1=8,\:\spacer=\frac{3}{4}$

$=8\cdot \frac{1-\left(\frac{3}{4}\right)^{25}}{1-\frac{3}{4}}$

$\mathrm{Multiply\:fractions}:\quad \:a\cdot \frac{b}{c}=\frac{a\:\cdot \:b}{c}$

$=\frac{\left(1-\left(\frac{3}{4}\right)^{25}\right)\cdot \:8}{1-\frac{3}{4}}$

$=\frac{8\left(-\left(\frac{3}{4}\right)^{25}+1\right)}{\frac{1}{4}}$

$\mathrm{Apply\:exponent\:rule}:\quad \left(\frac{a}{b}\right)^c=\frac{a^c}{b^c}$

$=\frac{8\left(-\frac{3^{25}}{4^{25}}+1\right)}{\frac{1}{4}}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a}{\frac{b}{c}}=\frac{a\cdot \:c}{b}$

$=\frac{\left(1-\frac{3^{25}}{4^{25}}\right)\cdot \:8\cdot \:4}{1}$

$\mathrm{Multiply\:the\:numbers:}\:8\cdot \:4=32$

$=\frac{32\left(-\frac{3^{25}}{4^{25}}+1\right)}{1}$

$=\frac{32\cdot \frac{4^{25}-3^{25}}{4^{25}}}{1}$ ∵ $\mathrm{Join}\:1-\frac{3^{25}}{4^{25}}:\quad \frac{4^{25}-3^{25}}{4^{25}}$

$=32\cdot \frac{4^{25}-3^{25}}{4^{25}}$

$=\frac{\left(4^{25}-3^{25}\right)\cdot \:32}{4^{25}}$

$=\frac{2^5\left(4^{25}-3^{25}\right)}{2^{50}}$ ∵ $\mathrm{Factor}\:32:\ 2^5$ , $\mathrm{Factor}\:4^{25}:\ 2^{50}$

$=\frac{4^{25}-3^{25}}{2^{45}}$ ∵ $\mathrm{Cancel\:}\frac{\left(4^{25}-3^{25}\right)\cdot \:2^5}{2^{50}}:\quad \frac{4^{25}-3^{25}}{2^{45}}$

$\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{a\pm \:b}{c}=\frac{a}{c}\pm \frac{b}{c}$

$=\frac{4^{25}}{2^{45}}-\frac{3^{25}}{2^{45}}$

$=32-\frac{3^{25}}{2^{45}}$ ∵ $\frac{4^{25}}{2^{45}}=32$

$=32-0.024$ ∵ $\frac{3^{25}}{2^{45}}=0.024$

$=31.98$

Therefore, the sum of the first 25 terms in this geometric series: 31.98

3 0

3 years ago

Prime factorazation of 13

Neporo4naja [7]

Answer:

13 = 13

Step-by-step explanation:

13 is a prime, so 13 is its only prime factor.

4 0

2 years ago

Please Help Me! I need to pass.

Andru [333]

42 is the answer

Explanation:
50% of 80 is half of 80 which is 40

Then you would add 40 with 5% of 40

What you would do then is 5% times 40 then you would change the 5% into a decimal which is 0.05 and multiply it by 40 , which would get u 2

Then you would add the price with tax
40+2 = 42

6 0

2 years ago

pls help will give brainliest Rena used the steps below to evaluate the expression (StartFraction (x Superscript negative 3 Base

kaheart [24]

Answer:

First "order of operations" mistake: step 2

First arithmetic mistake: step 4

Step-by-step explanation:

As we understand Rena's work, she wants to simplify ...

$\left(\dfrac{x^{-3}y^{-2}}{2x^4y^{-4}}\right)^{-3}$

for x = -1 and y = 2.

Her work seems to be ...

Step 1

$\text{Substitute $x=-1$ and $y=2$ into the expression}\\\\\left(\dfrac{(-1)^{-3}2^{-2}}{2(-1)^42^{-4}}\right)^{-3}\qquad\text{no error}$

Step 2

$\text{Simplify the parentheses}\\\\\left(\dfrac{2^4}{2(-1)^4(-1)^32^2}\right)^{-3}=\left(\dfrac{2^2}{2(-1)^7}\right)^{-3}\qquad\text{order of operations error}$

Step 3

$\text{Evaluate the power to a power}\\\\\dfrac{2^{-6}}{2^{-3}(-1)^{21}}\qquad\text{no error}$

Step 4

$\text{Use reciprocals and find the value}\\\\\dfrac{1}{2^32^6(-1)^{21}}=\dfrac{1}{8\cdot 64\cdot (-1)}=\dfrac{-1}{512}\qquad\text{error: $2^3$ is used instead of $2^{-3}$}$

_____

So, the first arithmetic error is in Step 4. However, the order of operations requires exponents be evaluated first. Doing that makes step 2 look like ...

$\left(\dfrac{-\dfrac{1}{4}}{2(1)\dfrac{1}{16}}\right)^{-3}=(-2)^{-3}\qquad\text{proper Step 2}$

We expect your answer is supposed to be Step 4.

7 0

3 years ago

1 2/12 - 2/12 - 1/12=

aleksley [76]

Answer:

11/12

Step-by-step explanation:

1 2/12 = 14/12

14/12 - 2/12 - 1/12

since they all have 12, it doesn't matter until the end

14 - 2 = 12 - 1 = 11

put the /12 behind the 11 and you have an answer

7 0

3 years ago