We used the tangent difference identity to rewrite the LHS in terms of [tex]\( \tan x \) and \( \tan \frac{\pi}{4} \)[/tex], then simplified to match the RHS, confirming the identity.
To prove the identity [tex]\( \tan(x-\frac{\pi}{4}) = \frac{\tan x - 1}{\tan x + 1} \),[/tex] we'll start with the left-hand side (LHS) and manipulate it to match the right-hand side (RHS).
LHS: [tex]\( \tan(x-\frac{\pi}{4}) \)[/tex]
Using the tangent difference identity, [tex]\( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b} \)[/tex], we have:
[tex]\[ \tan(x-\frac{\pi}{4}) = \frac{\tan x - \tan \frac{\pi}{4}}{1 + \tan x \cdot \tan \frac{\pi}{4}} \][/tex]
Since [tex]\( \tan \frac{\pi}{4} = 1 \)[/tex], we can substitute:
[tex]\[ \tan(x-\frac{\pi}{4}) = \frac{\tan x - 1}{1 + \tan x \cdot 1} \][/tex]
[tex]\[ = \frac{\tan x - 1}{\tan x + 1} \][/tex]
This matches the RHS of the identity. Therefore, we have successfully proved the identity.
Which expression has a value of 23?
A.
7 – (–16)
B.
–3 – 20
C.
10 – 13
D.
–18 – (–5)
Which coordinate pair is in the solution set for y < 1 − 6x?
To find the coordinate pair in the solution set for y < 1 - 6x, substitute x and y values into the inequality and check if it is satisfied.
Explanation:The question asks for a coordinate pair that is in the solution set for the inequality y < 1 - 6x. To find the solution set, we need to find the values of x and y that satisfy the inequality. Let's pick a few coordinate pairs and substitute the values into the inequality to see if they satisfy it.
For example, let's try the coordinate pair (1, 0). Substituting x = 1 and y = 0 into the inequality, we get 0 < 1 - 6(1), which simplifies to 0 < 1 - 6, and further simplifies to 0 < -5. Since this statement is false, the coordinate pair (1, 0) is not in the solution set.
Similarly, we can try other coordinate pairs and substitute the values into the inequality to check if they satisfy it. The coordinate pair that satisfies the inequality will be in the solution set.
How do you write 317 in words?
Multiplying a trinomial by a trinomial follows the same steps as multiplying a binomial by a trinomial. Determine the degree and maximum possible number of terms for the product of these trinomials: (x2 + x + 2)(x2 – 2x + 3). Explain how you arrived at your answer.
The product of the given trinomials will have a degree of 4, and since there are 9 terms, the maximum possible number of terms for the product is 9.egree of 4 and a maximum possible number of terms of 9.
When multiplying two polynomials, the degree of the resulting polynomial is the sum of the degrees of the two polynomials being multiplied.
In this case, both trinomials have a degree of 2.
The maximum possible number of terms in the product of two polynomials is the product of the number of terms in each polynomial.
For the first trinomial, there are 3 terms, and for the second trinomial, there are also 3 terms.
Multiplying these together gives a maximum possible number of terms of 9.
To arrive at this conclusion, you can use the distributive property, where each term in the first trinomial is multiplied by each term in the second trinomial, resulting in a total of 9 possible term combinations.
Therefore, the product of the two trinomials will have a dTo multiply two trinomials, like [tex]\( (x^2 + x + 2)(x^2 - 2x + 3) \)[/tex], you apply the distributive property repeatedly, multiplying each term in the first trinomial by each term in the second trinomial and then summing up the products.
Each term in the first trinomial needs to be multiplied by each term in the second trinomial, resulting in a total of [tex]\( 3 \times 3 = 9 \)[/tex] products.
Now, regarding the degree, when you multiply two polynomials, you add the degrees of the polynomials.
Here, both polynomials are of degree 2, so the resulting polynomial will be of degree 2 + 2 = 4.
Thus, the product of the given trinomials will have a degree of 4, and since there are 9 terms, the maximum possible number of terms for the product is 9.egree of 4 and a maximum possible number of terms of 9.
Is 4√5 rational or irrational
How much would $500 invested at 6% invested compounded monthly be worth after 5 year? Round your answer to the nearest cent.
200x50 mathematics problem
Answer:
10,000
Step-by-step explanation:
2 x 5= 10 then add 3 more zeros
The ratio of peaches to plums is 3:8. If there are 99 fruits how many plums are there?
The equation y=ax describes the graph of a line. If the value of A is megative the line:
a) goes down and to the left
b) goes up and to the left
c) is vertical
d) is horizontal
Solve the given equation. (Enter your answers as a comma-separated list. Let k be any integer. Round terms to three decimal places where appropriate. If there is no solution, enter NO SOLUTION.)
2 sin^2 θ − sin θ − 1 = 0 ...?
Answer: 30°, 300° and 330°
Step-by-step explanation:
This is a quadratic equation in trigonometry format.
Given 2 sin^2 θ − sin θ − 1 = 0
Let a constant 'k' = sin θ...(1)
The equation becomes
2k²-k-1 =0
Factorizing the equation completely we have,
(2k²-2k)+(k-1) = 0
2k(k-1)+1(k-1)=0
(2k+1)(k-1)=0
2k+1=0 and k-1=0
2k = -1 and k=1
k=-1/2 and 1
Substituting the value of k into equation 1 to get θ
sin θ = 1
θ = arcsin1
θ = 90°
Similarly
sin θ = -1/2
θ = arcsin-1/2
θ = -30°
This angle is negative and falls in the 3rd and 4th quadrant
In the third quadrant, θ = 270 +30 = 300° and
in the 4th quadrant, θ = 360 - 30° = 330°
Therefore the values of θ are 30°, 300° and 330°
I hope you find this helpful?
Final answer:
The trigonometric equation is transformed into a quadratic equation by substituting sin θ with x. Then, the quadratic formula is applied to find x, which is then used to find the solutions for θ in radians, considering the periodic nature of the sine function. The solutions are then verified.
Explanation:
To solve the trigonometric equation 2 sin^2 θ − sin θ − 1 = 0, treat it as a quadratic equation by setting x = sin θ. The reformed equation is 2x^2 - x - 1 = 0. Now, factor this equation or use the quadratic formula to find the values of x, and subsequently the values of θ.
Using the quadratic formula:
x = [-(-1) ± √((-1)^2 - 4(2)(-1))]/(2*2)x = (1 ± √(1 + 8))/4x = (1 ± √9)/4x = (1 ± 3)/4Therefore, the solutions for x are:
x = 1x = -0.5Convert these back into solutions for θ by finding θ such that sin θ = x. Use units of radians for angles and remember to consider the periodic nature of the sine function.
Answer:
For x = 1: θ = ½π + 2πkFor x = -0.5: θ = −⅓π + 2πk or θ = −&frac43;π + 2πk, for all integers k.Check if the answers are reasonable by substituting back into the original equation and verifying that they produce true statements.
Given the function f(x) = −2x + 8, find x if f(x) = 14 ...?
The graph shows the processing speeds of computers as a function of time since 1980.
Which statements are true about the function?
Select each correct answer.
A. The range is y≥16 , so the processing speeds are always positive.
B. The point (0,16) represents an initial processing speed of 16 MHz.
C. The point (10,256) represents the maximum processing speed.
D. The domain is x>0 , so the processing speed increases over time.
E. The graph is ascending, indicating that the processing speed increases over time.
Answer:
A, B and E are correct.
Step-by-step explanation:
We are given a function representing the processing speed of computers with respect to time.
Let, x-axis = axis corresponding generation and y-axis = axis corresponding the processing speed.
We can see that the the starting point of the function is (0,16). This gives us that the range of the function is y≥16 and (0,16) represents the initial speed of the computer.
So, option A and B are correct.
As, we can see that the function is going upwards after passing the point (10,256). This gives us that, this point does not the represent the maximum processing speed.
So, option C is not correct.
Also, only the domain being x>0 does not imply that the processing speed increases.
So, option D is not correct
Further, as the graph is ascending, so we see that as time increases, the processing speed increases.
So, option E is correct.
Answer:
The correct statements are in bold.
A. The range is y≥16 , so the processing speeds are always positive. (As the initial point is 16 and the graph is going upwards from there hence,the range y≥16 shows positive speed growth. Below this range, the speed will fall.)
B. The point (0,16) represents an initial processing speed of 16 MHz. (This is clearly visible in the graph.)
C. is not right as the graph is going up from 10256.
D. This is also not right. The given domain is not specific of speed.
E. The graph is ascending, indicating that the processing speed increases over time. (We can see as the time increases, the processing speed increases.)
Javier drove 45 miles. this represents 60% of his entire rode trip. what is the total number of miles in javier's trip?
What is 8.5 divided by 520?
the quantities s and t are positive and are related by the equation s=k/t, where k is a constant. if the value of S increased by 50 percent, then the value of T decreases by what percent??? ...?
Which of the sets of ordered pairs represents a function?
A = {(−4, 5), (1, −1), (2, −2), (2, 3)}
B = {(2, 2), (3, −2), (9, 3), (9, −3)}
1. Only A
2. Only B
3. Both A and B
4. Neither A nor B
Casey is making a flower arrangement with roses(r) and carnations(c). The cost of each rose is $0.50 and the cost of each carnation is $0.10. The arrangement has a total of 80 flowers and the flower cost was $20. How many of each flower did Casey put in her arrangement?
Boys and girls, we love to teach. The students love it. Teaches a student is to be grateful (pleasing). We love to teach the disciples of the history of the Roman letter ("by means of Literature", abl. Of means). In the game a lot (many) are students. * * To mature to learn the use of Latin in the disciples
Factor and simplify: 4a^2c^2(a^2-b^2+c^2)^2
A rectangle is inscribed in the upper half of the circle x2 y2 = a2 as shown at right. calculate the area of the largest such rectangle.
Hydra are small freshwater animals. They can double in number every two days. Suppose there is an initial population of 60 hydra, when will there be more than 5000 hydra?
What is the value of y in terms of x in the following equation? x - 8y = 1/4
triangle ABC has vertices A (0,10) B (4,10) and C (-2,4) find the orthocenter of triangle ABC
Answer:
Hence the orthocenter is (-2,12)
Step-by-step explanation:
We need to find the Orthocenter of ΔABC with vertices A(0,10) , B(4,10) and C(-2,4).
" Orthocenter of a triangle is a point of intersection, where three altitudes of a triangle connect ".
Step 1 : Find the perpendicular slopes of any two sides of the triangle.
Step 2 : Then by using point slope form, calculate the equation for those two altitudes with their respective coordinates.
Step 1 : Given coordinates are: A(0,10) , B(4,10) and C(-2,4)
Slope of BC = [tex]\dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}= \dfrac{4-10}{-2-4}=\dfrac{-6}{-6}=1[/tex]
Perpendicular Slope of BC = -1
( since for two perpendicular lines the slope is given as: [tex]m_{1}\times m_{2}=-1[/tex]
where [tex]m_{1},m_{2}[/tex] are the slope of the two lines. )
Slope of AC = [tex]\dfrac{y_2-y_1}{x_2-x_1}=\dfrac{4-10}{-2-0}=\dfrac{-6}{-2}=3[/tex]
Perpendicular Slope of AC= [tex]\dfrac{-1}{3}[/tex]
Step 2 : Equation of AD, slope(m) = -1 and point A = (0,10)
[tex]y - y_{1} = m\times (x-x_{1})[/tex]
[tex]y - 10 = -1(x - 0)\\y - 10 = -x \\x + y = 10---------------(1)[/tex]
Equation of BE, slope(m) =\dfrac{-1}{3} and point B = (4,10)
[tex]y - y_1 = m(x - x_1)[/tex]
[tex]y - 10= \dfrac{-1}{3} \times (x - 4)[/tex]
[tex]3y-30=-x+4[/tex]
[tex]x+3y =34[/tex]----------(2)
Solving equations (1) and (2), we get
(x, y) = (-2,12)
Hence, the orthocenter is (-2,12).
What are 3 numbers that round to 54.5 when rounded to nearest tenth
Three numbers that round to 54.5 when rounded to the nearest tenth are 54.45, 54.55, and 54.55.
Explanation:When rounding to the nearest tenth, we look at the digit in the hundredths place. If the digit is 5 or greater, we round up. If the digit is less than 5, we round down. In this case, we want to find three numbers that round to 54.5 when rounded to the nearest tenth. One possible set of numbers is 54.45, 54.55, and 54.55. When rounded to the nearest tenth, all of these numbers round to 54.5.
The shortest distance from the curve xy=4 to the origin is... ...?
The shortest distance [tex]x + y = 4[/tex] that is closest to the origin is [tex]\boxed{2\sqrt 2 {\text{ units}}}.[/tex]
Further explanation:
The formula for distance between the two points can be expressed as follows,
[tex]\boxed{{\text{Distance}} = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} }[/tex]
Given:
The line is [tex]x + y = 4.[/tex]
Explanation:
The coordinate of the origin is [tex]\left( {0,0} \right).[/tex]
The first point is [tex]\left( {x,y} \right)[/tex] and the second point is [tex]\left( {0,0} \right).[/tex]
The distance between the two points can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}}&= \sqrt {{{\left( {x - 0} \right)}^2} + {{\left( {y - 0} \right)}^2}}\\&= \sqrt {{x^2} + {{\left( {4 - x} \right)}^2}}\\&= \sqrt {{x^2} + {x^2} - 8x + 16}\\&= \sqrt {2{x^2} - 8x + 16}\\\end{aligned}[/tex]
Differentiate the above equation with respect to [tex]x[/tex].
Substitute the first derivative equal to zero.
[tex]\begin{aligned}\frac{d}{{dx}}\left( {{\text{Distance}}} \right) &= 0\\\frac{{\left( {2x - 4} \right)}}{{\sqrt {2{x^2} - 8x + 16} }} &= 0\\2x - 4 &= 0\\2x &= 4\\x&= 2\\\end{aligned}[/tex]
Substitute [tex]x = 2[/tex] in equation [tex]x + y = 4[/tex] to obtain the value of [tex]y[/tex].
[tex]\begin{aligned}2 + y &= 4\\y&= 4 - 2\\y&= 2\\\end{aligned}[/tex]
The point is [tex]\left( {2,2} \right).[/tex]
The shortest distance can be obtained as follows,
[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{2^2} + {2^2}}\\&= \sqrt {4 + 4}\\&= 2\sqrt 2\\\end{aligned}[/tex]
The shortest distance [tex]x + y = 4[/tex] that is closest to the origin is [tex]\boxed{2\sqrt 2 {\text{ units}}}.[/tex]
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of derivatives
Keywords: Derivative, shortest distance, curve, origin, attains, maximum, value of x, function, differentiate, minimum value, closest point, line, y+x = 4.
Is a line is represented by a straight narrow segment with arrow signs at the end? ...?
Four out of fifteen people surveyed say they plan to vote yes on Measure 2. Based on this sample, how many people out of 210 would you expect to vote yes?
Answer:
56
Step-by-step explanation:
The answer is 56 just clarifying the top answer :))
Enzo begins playing video games at 7:10. he logs off at 7:25.
Rita's company reimburses her expenses on food, lodging, and conveyance during business trips. The company pays $55 a day for food and lodging and $0.45 for each mile traveled. Rita drove 300 miles and was reimbursed $2,335. Part A: Create an equation that will determine the number of days x on the trip. (3 points) Part B: Solve this equation justifying each step with an algebraic property of equality. (6 points) Part C: How many days did Rita spend on this trip? (1 point)
Part A)
m = miles traveled
x = days on trip
y = reimbursement
55x + .45m = y
Part B)
55x + .45(300)= 2335
55x+ 135 = 2335
subtract 135 on both sides
55x= 2200
divide by 55 on both sides
x= 40
Part C) Rita Spent 40 Days on This Trip
List 5 numbers with 6 in the tens place
The list of 5 numbers with 6 in the tens place are [tex]\boxed{265,{\text{ 67, 163, 969, 2165, 31462}}}.[/tex]
Further explanation:
Explanation:
The whole numbers is the series of numbers that starts from zero.
The natural numbers are those numbers that start from one.
The place values are only natural numbers.
The base ten systems represent the position of a place value.
After decimal the first place is the tenth place, the second place is the hundredth and the third place is the thousandth.
Consider a number as [tex]265.[/tex]
Here, number 5 is on the ones place, number 6 is on the tens place and 2 is on the hundreds place.
Hence, the list of 5 numbers with 6 in the tens place are [tex]\boxed{265,{\text{ 67, 163, 969, 2165, 31462}}}.[/tex]
Learn more:
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Answer details:
Grade: Middle School
Subject: Mathematics
Chapter: Decimals
Keywords: list of 5 numbers, numbers, hundreds place, tens place, whole number, decimal, compare, contrast, methods, base ten system, place value, decimal expansion, natural numbers, real numbers.
Final answer:
Five numbers with 6 in the tens place are 60, 61, 62, 63, and 64. Each has the digit 6 immediately to the left of the ones place.
Explanation:
To list 5 numbers with 6 in the tens place means that each number must have its second digit from the right as a 6. Here are five numbers that meet this criterion:
60
61
62
63
64
Each of these numbers has a 6 in the tens place, which we can confirm by looking at the digit immediately to the left of the ones place.
Multivariable: Assume there is an opaque ball of radius 1 centered at the origin.
Suppose that you stand at the point (2,3,0) and look in the direction of a point that is not visible because it is behind the ball. You will then be looking at a point on the sphere.
Find the point on the sphere at which you look if you are looking in the direction of (−2,−3,2).
To find the point on the sphere at which you are looking in the direction of (-2, -3, 2), we need to find the intersection of the line passing through (2, 3, 0) and the direction (-2, -3, 2) with the sphere of radius 1 centered at the origin. By substituting the parametric equations of the line into the equation of the sphere and solving for t, we can find the two points on the sphere where you will be looking.
Explanation:To find the point on the sphere at which you look when you are looking in the direction of (-2, -3, 2), we need to find the intersection of the line passing through (2, 3, 0) and the direction (-2, -3, 2) with the sphere of radius 1 centered at the origin.
The equation of the line passing through (2, 3, 0) and in the direction (-2, -3, 2) can be expressed as:
x = 2 - 2t
y = 3 - 3t
z = 2t
Substitute these equations into the equation of the sphere, we get:
(2 - 2t)2 + (3 - 3t)2 + (2t)2 = 1
Simplifying the equation:
17t2 - 14t + 2 = 0
Using the quadratic formula:
t = (-b ± √(b2 - 4ac))/(2a)
Substituting the values, we get two possible values of t: t ≈ 0.23 and t ≈ 0.1.
Substituting these values back into the parametric equations of the line:
For t ≈ 0.23:
x ≈ 1.54
y ≈ 1.31
z ≈ 0.46
For t ≈ 0.1:
x ≈ 1.8
y ≈ 2.1
z ≈ 0.2
So, the points on the sphere at which you look when you are looking in the direction of (-2, -3, 2) are approximately (1.54, 1.31, 0.46) and (1.8, 2.1, 0.2).
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