Using double angle formula find the exact value

, given secθ=4,3pi/2<θ<2pi 
sin2θ= 
cos2θ= 
tan2θ=

, 


 

 

i.e. we are in quadrant IV 

means... 

 
 


 


We forgo using tangent identities since we don't need them 

If you got 6 out of 25 questions wrong what percent did you answer correctly?

correct: 25-6=19 
percent of total: 

(19/25)100%= 76%

Solve: 16x^(3)-100x=0

x=0
4x=10, x=5/2
4x=-10, x=-5/2


:)

how much pure acid should be mixed with 3 gallons of 20% acid solution to get a 40% acid solution?

unknown amt: x
20% acid: 3(0.2)
new mixture: (x+3)(0.4)


x=1




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A purse contains 15 coins worth $1.35. If there are only dimes and nickels in the purse, how many coins of each kind are there?

informally this problem only requires a working knowledge of counting money
12 dimes: $1.20
3 nickels:$0.15
buck thirty five!

formally:
coins: n+d=15
cash value: 0.05n+0.10d=$1.35 (which we multiply by 20 for convenience)
n+2d=27
then
(n+2d)-(n+d)=27-15

d=12
so n=3



:)

Verify the basic identity. What is the domain of validity? cotθ = cosθcscθ

and it has the domain





:)

Simplify: -1 + ((x - b)( x - c ))/((a-b)(a-c)) +( (x - a)(x - b))/((c-a)(c-b)) +( (x - a)(x - c))/((b-a)(b-c))

The simplification is relatively lengthy so I will outline it: 
First off we set aside the -1 term and simplify the rational expression part. 

the lcd is 

(a-b)(a-c)(b-c) 

After some labor we arrive at these results: 
x^2 terms: 0 
ax terms: abx-acx 
bx terms: bcx-abx 
cx terms: acx-bcx 
bc terms: bc(b-c) 
ab terms: ab(a-b) 
ac terms: -ac(a-c) 

After collecting like terms we find that all x-terms add out (cancel each other to get a sum of 0) 

Leaving us with 

ab(a-b)-ac(a-c)+bc(b-c) 

which is equal to (a-b)(a-c)(b-c), the denominator! 

Therefore our rational expression simplifies to 1 so that the entire expression evaluates to 0 










:) 

Calculate the slope of the line between the points (3, -4) and (3, 5) Please help. Thank you.

the x-coordinate does not change so slope is undefined for this pair of points 

:)

In what quadrant is (–2 + 5i) ?

-2 is two units left of the origin
5i is 5 units above the origin 
that puts -2+5i in quad II 

:)

rewrite this expression in terms of cos(x), and then simplify it: cos^4(x)-sin^4(x)+sin^2(x)

 


which when simplified gives us... 


 


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Stan has 15 coins worth $3.55. If the coins are all quarters and nickels, how many quarters and nickels does he have?

number of coins: n+q=15
amt. of money: 0.05n+0.25q=3.55
for convenience we multiply by 20:
20(0.05n+0.25q)=20(3.55)
n+5q=71
Now we subtract the coins equation from this one...
(n+5q)-(n+q)=71-15
4q=56
q=14
therefore there is 1 nickel


:) 

Write an equation of a circle that has a center (-1,-3) and is tangent to the line 3x+y= -1

L: 3x+y+1=0 and P: (-1,-3) 
The perpendicular distance from L to P is to be the radius: 

 

From the equation of a circle: 

 

:)

assume that power set of A equal to power set of B.then A=B.

This is not true: 
A= {1, 2} 
B= {4, t} 
have 

 

but 

 

(two sets are equal if they have exactly same elements) 

:)

A regular heptagon has sides of 9in, find the perimeter

a regular heptagon has 7 congruent sides so it's perimeter is 
P=7s 
therefore 

P=7*9=63 in

Keisha Drove 300 Miles On 14 Gallons Of Gas. At This Rate, How Many Gallons Of Gas Would She Need To Drive 345 Miles?

We can solve by using proportions: 
 

so x=16.1 gallons 


:)

Evaluate: (4/5+1/10)^(2)/(3/2)^(2)

what is the inverse of f(x)= 2x-3

 


 



 


so... 

Solve 7+5|c|≤1-3|c|

 


 

 

W can stop immediately because the absolute value can never be negative so this inequality has no solution 

:) 

what is the value of x in 6/x = 7/(x+3)

 
then... 



We cross multiply because this is a proportion:
 




 

Solve: abs(2x-5)<=8

 


which means 
 


Now we simplify... 



 


 




 


:) 

Find all points on the y-axis that are 6 units from (4, -3).

We'll use this form: 

 

 



 


 
or 

 



:) 

Write xy=12 into polar form

 

 


 


 



:)

Solve |2x|<=|x-3|

We start by finding points from which we will find the boundaries for the intervals 

2x=x-3 or 2x=-x+3
x=-3 or x=1 

Next we test all intervals 

On x<-3 we test x=-4 and see 8<7 (false) 

On -3 On x>1 we test x=2 and see 4<1 (false)

Our solution interval: [-3,1]


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Find the slope of the line perpendicular to the line that passes through the following points (4,11) and (-2, 1)

 

Now recall that 

 


:)

find all sets of three consecutive positive even integers with a sum no greater than 36

the set of solutions is infinite: 

 

leads to n=11, but n must be even so n=10 

Our list of consecutive even integers with a sum less than or equal to 36 ends with {10, 12, 14} but it has no beginning. 

:)