A perfect square trinomial can be factored using the pattern x²+2ab+b²=(x+a)² or x²-2ab+b²=(x-a)². The term 'a' is the square root of the last term, and 'b' is half of the second term's coefficient.
Explanation:In mathematics, a perfect square trinomial is a type of second-order polynomial or quadratic function with a specific form. It is used in the factorization process, particularly when solving quadratic equations of the form ax² + bx + c.
A perfect square trinomial is a trinomial in the form x²+2ab+b² or x²-2ab+b². To factor such a trinomial, the pattern (x+a)² or (x-a)² is used where 'x' is the variable, 'a' is the square root of the third term, and 'b' is half of the coefficient of the second term. Hence, x²+2ab+b² factors to (x+a)², whereas x²-2ab+b² factors to (x-a)².
Take for example, the trinomial x²+6x+9. Here, 'a' is 3 (as √9=3) and 'b' is 3 (as half of 6 is 3). Both 'a' and 'b' are equal so we know the trinomial is perfect square and it factors to (x+3)².
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The factor a perfect square trinomial, use the pattern [tex](a+b)^2[/tex] = [tex]a^2[/tex] +2ab+ [tex]b ^{2}[/tex] and simplify .
The pattern for a perfect square trinomial is a helpful algebraic expression that represents the square of a binomial.
It takes the form [tex](a+b)^2[/tex] = [tex]a^2[/tex] +2ab+[tex]b ^{2}[/tex]
where a and b are variables. When faced with a trinomial that fits this pattern, you can use it to factor the trinomial efficiently.
To factor a perfect square trinomial, compare it with the pattern. If the trinomial can be expressed in the form [tex](a+b)^2[/tex] =[tex]a^2[/tex]+2ab+ [tex]b ^{2}[/tex] , then it is a perfect square trinomial.
Identify [tex]a^2[/tex] ,2ab and [tex]b ^{2}[/tex] terms in the trinomial.
Once identified, you can write the factored form as [tex](a+b)^2[/tex]. This means the trinomial can be factored as [tex](a+b)^2[/tex] where a is the square root of the first term, and b is the square root of the last term.
This process is essentially reverse-engineering the expansion of the perfect square trinomial pattern.
Factoring using the perfect square trinomial pattern can save time compared to other methods, making it a valuable tool in algebraic manipulations and simplifying expressions.
I need help I keep getting confuse
Answer:
Step-by-step explanation:
2(9+4)
According to the BODMAS rule:
B = BRACKET
O = OPEN
D= DIVISION
M = MULTIPLICATION
A = ADDITION
S= SUBTRACTION
First we will solve the bracket:
=2(9+4)
=2(13)
=26....
Answer:
A, B, and C
Step-by-step explanation:
= 2 (9 +4)
= 2 (13)
= 26
A
= 2(9) + 2 (4)
= 18 + 8
= 26
B
= 2(13)
= 26
C
= 18 + 8
= 26
D
= 22
D does not apply because it does not equal 26
For f (x) = 3x+1 and g(x) = x2 - 6, find (f.g)(x).
Answer:
So we have
[tex](fg)(x)=3x^3+x^2-18x-6[/tex]
[tex](f \circ g)(x)=3x^2-17[/tex]
Step-by-step explanation:
I'm going to do two problems just in case.
We are given [tex]f(x)=3x+1[/tex] and [tex]g(x)=x^2-6[/tex].
[tex](fg)(x)=f(x)g(x)=(3x+1)(x^2-6)[/tex]
Multiply out using foil!
First: 3x(x^2)=3x^3
Outer: 3x(-6)=-18x
Inner: 1(x^2)=x^2
Last: 1(-6)=-6
-------------------Add together:
[tex]3x^3+x^2-18x-6[/tex]
[tex](f \circ g)(x)=f(g(x))=f(x^2-6)=3(x^2-6)+1=3x^2-18+1=3x^2-17[/tex]
t + 3/4t - 7/2 how do I simplify the 3/4t fraction with t?
Answer:
1 3/4 t -7/2
Step-by-step explanation:
t + 3/4t - 7/2
Get a common denominator for the t terms, which is 4
4/4t + 3/4t - 7/2
7/4 t -7/2
This is an improper fraction so change it to a mixed number
1 3/4 t -7/2
Cal is trying to raise his average weekly income to be at least $131. His first
two weekly paychecks were $128 and $135. What is the lowest amount on his next
paycheck that Cal must earn so that he can reach his goal?
a) $132
b) $131
c) $135
d) $130
Answer:
d) $130
Step-by-step explanation:
Average means you add up the numbers and divide by the number of numbers.
We want our average to be 131.
We have 128,135, and the next pay check amount.
So let's average the three numbers together.
[tex]\frac{128+135+x}{3}[/tex]
[tex]x[/tex] represent the amount on the next pay check.
We want this average to equal 131 so we have this equation to solve:
[tex]\frac{128+135+x}{3}=131[/tex]
First step: Add 128 and 135:
[tex]\frac{263+x}{3}=131[/tex]
Second step: Multiply both sides by 3:
[tex]263+x=3(131)[/tex]
Third step: Multiply 3 and 131:
[tex]263+x=393[/tex]
Fourth step: Subtract 263 on both sides:
[tex]x=393-263[/tex]
Fifth step: Subtract 393 and 263:
[tex]x=130[/tex]
d. 130
D). $130
Step-by-step explanation:In this question, we're trying to figure out what would be the lowest amount of money he earned on his next paycheck in order for his income to be an average of $131.
To do this, we would need to use some important information in the question.
Important information:
$128 paycheck$135 paycheckWith the information above, we can use that to get the answer to the question.
We know that he made $128 and $135 on his recent paychecks, but we need to find on how much he needs to make on his next paycheck in order to reach his goal average.
We would be solving for x:
128, 135, x
131, 131, 131
We would add the numbers:
128, 135, x = 263 + x
131, 131, 131 = 393
We will now solve.
[tex]393 = 263+x\\\\\text{Subtract 263 on both sides}\\\\130=x[/tex]
When you're don solving, you should get 130. This means that Cal needs to make $130 on his next paycheck in order to have an average of $131.
Checking to see if it's right:We can check to see if it's right by adding up all of the numbers and divide by how many there are (3).
[tex]128+130+135=393\\\\393\div3=131[/tex]
Now, we can confirm that D). $130 would be the correct answer.
I hope this helps you out.Good luck on your academics.Have a fantastic day!What are the solutions to the equation (x – 2)(x + 5) = 0?
Answer:
x = 2 or x = -5
Step-by-step explanation:
It s given that, (x – 2)(x + 5) = 0
To find the solution of given equation
Let (x – 2)(x + 5) = 0
From this we get
either (x – 2) = 0 or (x + 5) = 0
If (x - 2) = 0 then x = 2
If (x + 5) = 0 then x = -5
Therefore the solutions of given equation (x – 2)(x + 5) = 0 are,
x = 2 or x = -5
Answer: X = 2 or X = -5
Step-by-step explanation:
(X - 2) x ( X - 5) = 0
X - 2 = 0 or X - 5 = 0
Therefore X=2 or X=5
Fiona is serving iced tea and lemonade at a picnic. She has only 44 glasses in which to serve the drinks. If x represents the number of glasses of iced tea and y represents the number of glasses of lemonade, which equation represents the number of glasses of ice tea she can serve?
Answer:
x = 44 − y
Step-by-step explanation:
x is the number of glasses of iced tea, and y is the number of glasses of lemonade. The sum is 44, so:
x + y = 44
Solving for x:
x = 44 − y
Answer: 44-y represents the number of glasses of ice tea she can serve.
Step-by-step explanation:
Since we have given that
Let x be the number of glasses of iced tea.
Let y be the number of glasses of lemonade.
Total number of glasses = 44
According to question,
[tex]x+y=44[/tex]
So, the number of glasses of ice tea she can serve is given by
[tex]x=44-y[/tex]
Hence, 44-y represents the number of glasses of ice tea she can serve.
Will mark brainliest, please answer:)
Find the value of PQ . Round the answer to the nearest tenth. Explain
( Use Pythagorean Theorem 3D rule and Question is above)
Answer:
√135
Step-by-step explanation:
3^2+b^2=12^2
9+b^2=144
9-9+b^2=144-9
b^2=135
√135=b
In parallelogram EFGH, the measure of angle F is (3x − 10)° and the measure of angle G is (5x + 22)°. What is the measure of angle G?
Answer:
So angle G has measurement 127 degrees.
Step-by-step explanation:
E F
H G
I had to write it out the parallelogram so I could have a better visual.
F and G are consecutive angles in a parallelogram (not on opposite sides).
This means they add to be 180 degrees.
F+G=180
(3x-10)+(5x+22)=180
(3x+5x)+(-10+22)=180
8x +12=180
Subtract 12 on both sides:
8x =180-12
Simplify:
8x =168
Divide both sides by 8:
x =168/8
x =21
If x=21 and want the measurement of angle G, then
(5x+22)=(5*21+22)=127.
So angle G has measurement 127 degrees.
Answer: [tex]127^{\circ}[/tex]
Step-by-step explanation:
Given : In parallelogram EFGH, the measure of angle F is (3x − 10)° and the measure of angle G is (5x + 22)°.
We known that in a parallelogram , the sum of two adjacent angles is 180° .
Therefore , we have
[tex]3x -10+5x + 22=180\\\\\Rightarrow\ 8x+12=180\\\\\Rightarrow\ 8x=180-12\\\\\Rightarrow\8x=168\\\\\Rightarrow\ x=21[/tex]
Now, the measure of angle G =[tex](5x + 22)^{\circ}=(5(21)+22)^{\circ}=127^{\circ}[/tex]
Hence, the measure of angle G = [tex]127^{\circ}[/tex]
why can 0.825 be written as a fraction explain
Answer:
It can be written as 825/1000. You can simplify this to get the simplest form which would be 33/40. All decimals are out of one, they are a part. When you first get a decimal, put the numbers such as 825 on top. The last number is in the thousandths place, so it is out of 1000. 1000 is your denominator. Your fraction is then 825/1000. From here you can simplify if possible.
Hope this helps ^-^
Simplify (-3c^-3w^5)^3
A -9w^8
B. -27cw^8
C. w^15/27c^9
D.-27w^15/c^9
Answer:
[tex]\large\boxed{D.\ \dfrac{-27w^{15}}{c^9}}[/tex]
Step-by-step explanation:
[tex](-3c^{-3}w^5)^3\qquad\text{use}\ (ab)^n=a^nb^n\\\\=(-3)^3(c^{-3})^3(w^5)^3\qquad\text{use}\ (a^n)^m=a^{nm}\\\\=-27c^{-3\cdot3}w^{5\cdot3}=-27c^{-9}w^{15}\qquad\text{use}\ a^{-n}=\dfrac{1}{a^n}\\\\=-27\left(\dfrac{1}{c^9}\right)w^{15}=\dfrac{-27w^{15}}{c^9}[/tex]
Final answer:
To simplify (-3c⁻³w⁵)³, you must distribute the power of 3 to each factor inside the parenthesis and apply the rule for negative exponents to end up with -27w¹⁵/c⁹, which is answer choice D.
Explanation:
The expression to simplify is (-3c⁻³w⁵)³. We will apply the rule for exponents to simplify the expression. When applying this rule and the negative exponent rule which states that a⁻ⁿ = 1/aⁿ, we get:
Step 1: Apply the power to each term inside the parenthesis: (-3)3 * c⁻⁹ * w¹⁵.Step 2: Simplify each term: -27 * c⁻⁹* w¹⁵.Step 3: Apply the negative exponent rule to c⁻⁹: -27 * (1/c⁹) * w¹⁵.Step 4: Write the final simplified expression as a single fraction: -27w¹⁵/c⁹.The correct answer is D. -27w¹⁵/c⁹.
2 Points
Which shows the equation below written in standard form?
9 - 7x = (4x-3)2 + 5
O A. 16x2 - 17x- 5 = 0
O B. 16x2-31x+ 5 = 0
O C. 16x2 - 31x- 5 = 0
O D. 16x2 - 17x+ 5 = 0
Answer:
D
Step-by-step explanation:
Given
9 - 7x = (4x - 3)² + 5 ← expand the squared factor
9 - 7x = 16x² - 24x + 9 + 5, that is
9 - 7x = 16x² - 24x + 14 ( subtract 9 - 7x from both sides )
0 = 16x² - 17x + 5, that is
16x² - 17x + 5 = 0 ← in standard form → D
What is the solution to the system of equations?
y = 5x + 2
3x = -y + 10
(4, -18)
(-18, -4)
(7,1)
(1,7)
Answer:
(1, 7)Step-by-step explanation:
[tex]\left\{\begin{array}{ccc}y=5x+2&(1)\\3x=-y+10&(2)\end{array}\right\\\\\text{substitute (1) to (2):}\\\\3x=-(5x+2)+10\\3x=-5x-2+10\qquad\text{add 5x to both sides}\\8x=8\qquad\text{divide both sides by 8}\\x=1\\\\\text{put the value of x to (1):}\\\\y=5(1)+2\\y=5+2\\y=7[/tex]
Tiles spelling the word “Restaurant” are placed on a sign above a building. If one of the letters falls down at random, what is the probability that the letter is not a vowel? JUSTIFY
Answer:
3/5
Step-by-step explanation:
The word "Restaurant" contains a total of 10 letters, out of which 4 of the letters are vowels and 6 of the letters are consonants. The probability of selecting a vowel out of this word is:
P(selected letter is a vowel) = number of vowels/number of letters.
P(selected letter is a vowel) = 4/10 = 2/5.
Similarly, probability of selecting a non-vowel out of this word is:
P(selected letter is not a vowel) = number of non-vowels/number of letters.
P(selected letter is not a vowel) = 6/10 = 3/5.
Given that one of the letters randomly falls down, and assuming that the probabilities of each letter falling down is uniform and independent from each other, then:
P(a non-vowel falls down) = non-vowels/total = 6/10 = 3/5.
So the correct answer is 3/5!!!
Kite ABCD has an area of 48 ft2. Calculate the length of AC.
A. 6 ft
B. 12 ft
C. 18 ft
Answer:
B. 12ft
Step-by-step explanation:
Area of kite is 2 times the area of triangle
= 2[(1/2)(4)AC] = 48, ie., AC = 12 ft
Write the parametric equation of the line 10x-4y=20
Answer:
x = 1 - t and y = -2.5 - 2.5t.
Step-by-step explanation:
Parametric equations are the equations in which the all the variables of the equation are written in terms of a single variable. For example in 2-D plane, the equation of the line is given by y=mx+c, there x is the independent variable, y is the dependent variable, m is the slope, and c is the y-intercept. The equation of the given line is 10x - 4y = 20. The goal is to convert the variables x and y in terms of a single variable t. First of all, take two points which lie on the line. By taking x=1, y comes out to be -2.5 and by taking x=0, y comes out to be -5. The general form of the straight line is given by:
(x, y) = (x0, y0) + t(x1-x0, y1-y0), where (x, y) is the general point, (x0, y0) is the fixed point, t is the parametric variable, and (x1-x0, y1-y0) is the slope.
Let (x0, y0) = (1, -2.5) and (x1, y1) = (0, -5). Substituting in the general equation gives:
(x, y) = (1, -2.5) + t(-1, -2.5). This implies that x = 1 - t and y = -2.5 - 2.5t!!!
Answer:
C. x=2t, y=5t-5
Step-by-step explanation:
Ben sold his small online business for $100,000. The purchaser will pay him $20,000 today, then $20,000 every year for the next four years. Assume
that Ben could invest a lump-sum payment today in an account yielding an interest rate of 4% annually. Find the total present value of all five
payments
A.
$87,096
B
$88,384
c. $92,598
D. $93,964
The answer is c $92,598
Answer:
$92,598
Step-by-step explanation:
The purchaser pays Ben $20,000 today and then $20,000 every year for the next 4 years.
The interest rate is 4% per annum.
So the net present value of all the payments is :
20000 + 20000/1.04 + 20000/(1.04^2) + 20000/(1.04^3) + 20000/(1.04^4)
= 20000 + 19230.77 + 18491.12 + 17779.73 + 17096.08
= 92597.7
= 92,598 (approx)
So the net present value of all the payments made to Ben is $92,598.
a house increased in value by 36% since it was purchased. The current value is $306,000. What was the value when purchased?
Answer:
$225,000
Step-by-step explanation:
To calculate the percentage change we will apply the formula:
p= N-O/O *100
p is the increased percentage
N is the current value
O is the old value.
Substitute the values in the formula:
36 = 306,000 - O/O *100
Divide both the sides by 100
36/100 = 306,000 - O/O *100/100
36/100 = 306,000 - O/O
Now multiply O at both sides
36/100 * O = 306,000-O/O * O
At R.H.S O will be cancelled by O
At L.H.S 36/ 100 = 0.36
0.36 O= 306,000-O
Combine the like terms:
0.36 O+O =306,000
1.36 O = 306,000
Divide both the terms by 1.36
1.36 O/ 1.36 = 306,000/1.36
O= $225,000
Therefore when the house was purchased its value was $225,000....
In a graph, x represents the number of months since a
business opened, and y represents the total amount of
money the business has earned. The following three
points are from the graph:
(2, 1990) (5, 4225) (9, 7205)
Find the slope and y-intercept. Explain what each
represents.
Answer:
The slope is 745 and the y-intercept is 500
The slope means The amount of money increases by $745 per month
The y-intercept means the business opened with initial amount $500
Step-by-step explanation:
* Lets explain how to solve the question
- The graph represents the relation between the number of months
since the business opened and the total amount of money the
business has earned
- The x-axis represents the number of month
- The y-axis represents the amount of money
- In the line the slopes from any two points on the line are equal
- The slope of the line whose end-points are (x1 , y1) and (x2 , y2)
is [tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]
- The equation of the line is y = mx + c ,where m is the slope of the line
and c is the y-intercept
* Lets check is the relation between x and y is linear by calculating the
slopes between each to points
∵ (2 , 1990) , (5 , 4225) , (9 , 7205) are points from the graph
- m1 is the slope of the points (2 , 1990) and (5 , 4225) , m2 is the slope
of the points (5 , 42250) and (9 , 7205) , m3 is the slope of the points
(2 , 1990) and (9 , 7205)
∵ [tex]m_{1}=\frac{4225-1990}{5-2}=745[/tex]
∵ [tex]m_{2}=\frac{7205-4225}{9-5}=745[/tex]
∵ [tex]m_{3=\frac{7205-1990}{9-2}}=745[/tex]
∴ m1 = m2 = m3 = 745
∴ The relation between the number of months and the amount of
money is linear
* The slope is 745
∵ The form of the linear equation is y = mx + c
∵ m = 745
∴ y = 745 x + c
- The y-intercept means the line intersect the y-axis at point (0 , c),
then to find c substitute x and y of the equation by the coordinates
of any point on the line
∵ x = 2 , y = 1990
∴ 1990 = 745(2) + c
∴ 1990 = 1490 + c ⇒ subtract 1490 from both sides
∴ c = 500
∵ c is the y-intercept
* The y-intercept is 500
* The slope represents the rate of increasing of money per month
∴ The amount of money increases by $745 per month
* The y-intercept represents the initial amount of money when the
business opened
∴ The business opened with initial amount $500
Answer:
The slope is 745 and the y-intercept is 500
The slope means The amount of money increases by $745 per month
The y-intercept means the business opened with initial amount $500
Step-by-step explanation:
please please please help and try to explain
Answer:
-3 , 21
Step-by-step explanation:
RS = R + S = 12
S lies on 9
There are 12 spaces in between R & S, so you can add and subtract 12 from S:
9 - 12 = -3
9 + 12 = 21
R can be located on either -3 or 21.
~
Find the second, fifth, and ninth terms of the sequence.
an = -7 + (n - 1) 4
Answer:
- 3, 9, 25
Step-by-step explanation:
To find the required terms of the sequence substitute n = 2, 5, 9 into the given rule, that is
[tex]a_{2}[/tex] = - 7 + (2 - 1)4 = - 7 + (1 × 4) = - 7 + 4 = - 3
[tex]a_{5}[/tex] = - 7 + (5 - 1)4 = - 7 + (4 × 4) = - 7 + 16 = 9
[tex]a_{9}[/tex] = - 7 + (9 - 1)4 = - 7 + (8 × 4) = - 7 + 32 = 25
The second, fifth, and ninth terms of the sequence are -3, 9, and 25, respectively.
The second, fifth, and ninth terms of the sequence defined by the formula an = -7 + (n - 1) * 4 are -3, 9, and 25, respectively.
To find the second, fifth, and ninth terms of the sequence given by the formula an = -7 + (n - 1) * 4, we can simply plug the corresponding values of n into the formula.
For the second term (a2), where n=2:
a2 = -7 + (2 - 1) * 4
a2 = -7 + 1 * 4
a2 = -7 + 4
a2 = -3
For the fifth term (a5), where n=5:
a5 = -7 + (5 - 1) * 4
a5 = -7 + 4 * 4
a5 = -7 + 16
a5 = 9
For the ninth term (a9), where n=9:
a9 = -7 + (9 - 1) * 4
a9 = -7 + 8 * 4
a9 = -7 + 32
a9 = 25
Therefore, the second, fifth, and ninth terms of the sequence are -3, 9, and 25, respectively.
Find the distance between these points.
R(-1,0), S(8,6)
V(26)
V(85)
3V(13)
Answer:
The distance is equal to [tex]3\sqrt{13}\ units[/tex]
Step-by-step explanation:
we know that
the formula to calculate the distance between two points is equal to
[tex]d=\sqrt{(y2-y1)^{2}+(x2-x1)^{2}}[/tex]
we have
[tex]R(-1,0)\\S(8,6)[/tex]
substitute the values
[tex]d=\sqrt{(6-0)^{2}+(8+1)^{2}}[/tex]
[tex]d=\sqrt{(6)^{2}+(9)^{2}}[/tex]
[tex]d=\sqrt{36+81}[/tex]
[tex]d=\sqrt{117}\ units[/tex]
Simplify
[tex]d=3\sqrt{13}\ units[/tex]
In △ABC, m∠A=72°, c=61, and m∠B=16°. Find the perimeter of the triangle.
Answer:
136
Step-by-step explanation:
The perimeter of the triangle is about 136
Further explanationFirstly , let us learn about trigonometry in mathematics.
Suppose the ΔABC is a right triangle and ∠A is 90°.
sin ∠A = opposite / hypotenusecos ∠A = adjacent / hypotenusetan ∠A = opposite / adjacentThere are several trigonometric identities that need to be recalled, i.e.
[tex]cosec ~ A = \frac{1}{sin ~ A}[/tex]
[tex]sec ~ A = \frac{1}{cos ~ A}[/tex]
[tex]cot ~ A = \frac{1}{tan ~ A}[/tex]
[tex]tan ~ A = \frac{sin ~ A}{cos ~ A}[/tex]
Let us now tackle the problem!
This problem is about Sine Rule.
First of all, we will calculate the ∠C :
∠A + ∠B + ∠C = 180°
72° + 16° + ∠C = 180°
∠C = 180° - 72° - 16°
∠C = 92°
Next, we will use the Sine Rule to find the length of the other side of the triangle.
[tex]\frac{c}{\sin \angle C} = \frac{b}{\sin \angle B}[/tex]
[tex]\frac{61}{\sin 92^o} = \frac{b}{\sin 16^o}[/tex]
[tex]b \approx \boxed {16.82}[/tex]
[tex]\frac{c}{\sin \angle C} = \frac{a}{\sin \angle A}[/tex]
[tex]\frac{61}{\sin 92^o} = \frac{a}{\sin 72^o}[/tex]
[tex]a \approx \boxed {58.05}[/tex]
Finally, we can find the perimeter of a triangle with the following formula
[tex]\text{Perimeter of the triangle} = a + b + c[/tex]
[tex]\text{Perimeter of the triangle} = 58.05 + 16.82 + 61[/tex]
[tex]\text{Perimeter of the triangle} \approx \boxed {136}[/tex]
Learn moreCalculate Angle in Triangle : https://brainly.com/question/12438587Periodic Functions and Trigonometry : https://brainly.com/question/9718382Trigonometry Formula : https://brainly.com/question/12668178Answer detailsGrade: College
Subject: Mathematics
Chapter: Trigonometry
Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse
6x-2y=-6 find the slope and the y intercept of the line
Answer:
y-int:3
slope:-3
Step-by-step explanation:
6x-2y=-6
change to y=mx+b: -2y=-6x-6
divide by -2
y=3x+3
Find the exact value of tan ^-1 (-root of 3)
Write your answer in radians in terms of n.
To find the exact value of tan^-1(-sqrt(3)), we first rewrite the equation using the definition of the arctangent function. Next, we use the trigonometric identity sin^2(x) + cos^2(x) = 1 to simplify the equation and solve for sin(x). We find that sin(x) = sqrt(3)/2. By looking at the unit circle, we determine that the angle whose sine is sqrt(3)/2 is pi/3 radians. Therefore, the exact value of tan^-1(-sqrt(3)) in radians is -pi/3 + 2*pi*n, where n is an integer.
To find the exact value of tan^-1(-sqrt(3)), we need to recall the definition of the arctangent function. The arctangent function returns an angle whose tangent is equal to a given number. In this case, we are looking for an angle whose tangent is -sqrt(3). Since tan(x) = sin(x)/cos(x), we can rewrite the equation as -sqrt(3) = sin(x)/cos(x).
Next, we can use the fact that sin^2(x) + cos^2(x) = 1 to rewrite the equation as -sqrt(3) = sin(x)/sqrt(1 - sin^2(x)). Cross-multiplying and rearranging, we get -sqrt(3)*sqrt(1 - sin^2(x)) = sin(x).
Now, we can square both sides and simplify the equation to get -3*(1 - sin^2(x)) = sin^2(x). Expanding and rearranging, we have -3 + 3sin^2(x) = sin^2(x). Combining like terms and isolating sin^2(x), we get sin^2(x) = 3/4. Taking the square root of both sides, we find sin(x) = sqrt(3)/2.
Finally, we can find the angle whose sine is sqrt(3)/2 by looking at the unit circle. The angle is pi/3 in radians. Therefore, the exact value of tan^-1(-sqrt(3)) in radians is -pi/3 + 2*pi*n, where n is an integer.
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Simplify (6^-4)^6
Please help me
Answer:
6^ -24
Step-by-step explanation:
We know that a^b^c = a^ (b*c)
(6^-4)^6 = 6^ (-4*6) = 6^ -24
We just found that a = -0.1 in this system of equations: 3 = 10a + b 2= 20a + b. Find the value of b.
Answer:
b = 4Step-by-step explanation:
Put a = -0.1 to the first an second equation and find velue of b:
3 = 10a + b
3 = 10(-0.1) + b
3 = -1 + b add 1 to both sides
4 = b → b = 4
2 = 20a + b
2 = 20(-0.1) + b
2 = -2 + b add 2 to both sides
4 = b → b = 4
CORRECT
Which double angle or half angle identity would you use to verify the following: csc x sec x = 2 csc 2x
Answer:
b
Step-by-step explanation:
I would use b.
Why?
[tex]2 \csc(2x)[/tex]
[tex]2 \frac{1}{\sin(2x)}[/tex]
[tex]\frac{2}{\sin(2x)}[/tex]
[tex]\frac{2}{2\sin(x)\cos(x)}[/tex]
[tex]\frac{1}{\sin(x)\cos(x)}{/tex]
[tex]\frac{1}{\sin(x)\frac{1}{\cos(x)}[/tex]
[tex]\csc(x) \sec(x)[/tex]
I applied the identity sin(2x)=2sin(x)cos(x) in line 3 to 4.
Answer: OPTION B.
Step-by-step explanation:
It is important to remember these identities:
[tex]csc(x)=\frac{1}{sin(x)}\\\\sec(x)=\frac{1}{cos(x)}[/tex]
Knowing this, we can say that:
[tex]csc(x) sec(x)=\frac{1}{sin(x)}*\frac{1}{cos(x)}=\frac{1}{sin(x)*cos(x)}[/tex]
Now we need to use the following Double angle identity :
[tex]sin(2x)=2sin(x)cos(x)[/tex]
And solve for [tex]sin(x)cos(x)[/tex]:
[tex]\frac{sin(2x)}{2}=sin(x)cos(x)[/tex]
The next step is to make the substitution into [tex]\frac{1}{sin(x)*cos(x)}[/tex] and finally simplify:
[tex]\frac{1}{\frac{sin(2x)}{2}}=\frac{\frac{1}{1}}{\frac{sin(2x)}{2}}=\frac{2}{sin(2x)}=2csc(2x)[/tex]
Which set of ordered pairs could be generated by an exponential function?
(0,0), (1, 1), (2, 8), (3, 27)
(0, 1), (1, 2), (2, 5), (3, 10)
(0,0), (1, 3), (2, 6), (3, 9)
(0, 1), (1, 3), (2, 9), (3, 27)
ANSWER
(0, 1), (1, 3), (2, 9), (3, 27)
EXPLANATION
The first and third options are completely out because the y-value of an exponential function is never zero.
For the second option the y-values has no geometric pattern or common ratio.
For the last option, we can observe the following pattern
[tex]1 = {3}^{0} [/tex]
[tex] {3}^{1} = 3[/tex]
[tex] {3}^{2} = 9[/tex]
[tex] {3}^{3} = 27[/tex]
:
:
[tex] {3}^{x} = y [/tex]
The correct choice is the last option
Answer:
D
Step-by-step explanation:
There are 3,280.84 feet in a kilometer. There are 5,280 feet in a mile. To the nearest hundredth, how many kilometers are in a mile?
Answer:
1.609344 kilometers
Step-by-step explanation:
If a triangle has sides of lengths 5, 8 and 12, it is a right triangle. true or false
Answer:
False.
Step-by-step explanation:
To see if these sides can form a right triangle, all we need to do is see if the following equation holds [tex]a^2+b^2=c^2[/tex] where [tex]c[/tex] is the larger measurement. [tex]a \text{ and } b[/tex] it doesn't really matter which you assign as 5 or 8.
So I'm choosing the following [tex]a=5,b=8,c=12[/tex].
[tex]c[/tex] has to be 12 because 12 is the largest.
Now we got to see if [tex]a^2+b^2=c^2[/tex] holds.
That is, we need to see if [tex]5^2+8^2=12^2[/tex] holds.
[tex]5^2+8^2=12^2[/tex]
[tex]25+64=144[/tex]
[tex]89=144[/tex]
That's totally false. 89 is definitely not 144 so 5,8, and 12 cannot be put together to form a right triangle.