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denpristay [2]
2 years ago
10

What is the exact value of cos (67.5°)?

Mathematics
1 answer:
babymother [125]2 years ago
8 0

first off, make sure you have a Unit Circle, if you don't do get one, you'll need it, you can find many online.

let's double up 67.5°, that way we can use the half-angle identity for the cosine of it, so hmmm twice 67.5 is simply 135°, keeping in mind that 135° is really 90° + 45°, and that whilst 135° is on the 2nd Quadrant and its cosine is negative 67.5° is on the 1st Quadrant where cosine is positive, so

cos(\alpha + \beta)= cos(\alpha)cos(\beta)- sin(\alpha)sin(\beta) \\\\\\ cos\left(\cfrac{\theta}{2}\right)=\pm \sqrt{\cfrac{1+cos(\theta)}{2}} \\\\[-0.35em] ~\dotfill\\\\ cos(135^o)\implies cos(90^o+45^o)\implies cos(90^o)cos(45^o)~~ - ~~sin(90^o)sin(45^o) \\\\\\ \left( 0 \right)\left( \cfrac{\sqrt{2}}{2} \right)~~ - ~~\left( 1\right)\left( \cfrac{\sqrt{2}}{2} \right)\implies -\cfrac{\sqrt{2}}{2} \\\\[-0.35em] ~\dotfill

cos(67.5^o)\implies cos\left( \frac{135^o}{2} \right)\implies \pm \sqrt{\cfrac{ ~~ 1-\frac{\sqrt{2} ~~ }{2}}{2}}\implies \stackrel{I~Quadrant}{+\sqrt{\cfrac{ ~~ 1-\frac{\sqrt{2} ~~ }{2}}{2}}} \\\\\\ \sqrt{\cfrac{ ~~ \frac{2-\sqrt{2}}{2} ~~ }{2}}\implies \sqrt{\cfrac{2-\sqrt{2}}{4}}\implies \cfrac{\sqrt{2-\sqrt{2}}}{\sqrt{4}}\implies \cfrac{\sqrt{2-\sqrt{2}}}{2}

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Serhud [2]

If Sasha wants her account to be worth $1,000 in 5 years, then she must deposit $833.34.

<h3>What is simple interest?</h3>

Simple interest is a method of calculating interest on an amount for n period of time with a rate of interest of r. It is calculated with the help of the formula,

SI = PRT

where SI is the simple interest, P is the principal amount, R is the rate of interest, and T is the time period.

Sasha's bank account earns 4% simple interest. Therefore, if she wants it to be worth $1,000 in 5 years then the principal amount must be,

Account balance = PRT + P

$1,000 = P[(0.04×5)+1]

$1,000 = 1.2 P

P = $833.34

Hence, if Sasha wants her account to be worth $1,000 in 5 years, then she must deposit $833.34.

Learn more about Simple Interest:

brainly.com/question/2793278

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5 0
2 years ago
ites A &amp; Bare 5 inches apart on a map. Cities &amp; Dare 4 inches apart. If C&amp;D are 900 miles apart how many miles apart
lisabon 2012 [21]

Answer:

<u>Cities A and B are 1,125 miles apart</u>

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

Distance between A and B in the map = 5 inches

Distance between C and D in the map = 4 inches

Actual distance between C and D = 900 miles

2. How many miles apart are cities A and B?

We will solve the question, using Direct Rule of Three, as follows:

Distance in the map      Actual distance

A - B      5 inches                    x

C - D      4 inches                900 miles

***************************************************

4x = 900 * 5

4x = 4,500

x = 4,500/4

x = 1,125 miles

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3 years ago
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