To solve the equation cos2x - sqrt 2 sinx = 1, rewrite cos2x as 2cos^2x - 1. Use the quadratic formula to solve for cosx. Substitute the values back into the equation cos2x - sqrt 2 sinx = 1 and solve for x.
Explanation:To solve the equation cos2x - √2 sinx = 1, we can use trigonometric identities and equations. First, we can rewrite cos2x as 2cos^2x - 1. So, the equation becomes 2cos^2x - √2 sinx - 1 = 0. To solve this quadratic equation, let's set 2cos^2x - √2 sinx - 1 = 0 and solve for cosx.
Next, we can use the quadratic formula to solve for cosx. The quadratic formula states that x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 2, b = -√2 sinx, and c = -1. Plugging in these values, we can solve for cosx.
After solving for cosx, we can substitute the values back into the equation cos2x - √2 sinx = 1 and solve for x. The student's question is about solving the equation cos(2x) - √2 sin(x) = 1 for all solutions. We can use the trigonometric identities cos(2x) = 1 - 2sin2(x) or cos(2x) = 2cos2(x) - 1 to rewrite the equation. Since we have a sine term in the original equation, let's use the former identity:
cos(2x) - √2 sin(x) = 1
(1 - 2sin2(x)) - √2 sin(x) = 1
2sin2(x) + √2 sin(x) - 1 = 0
This is a quadratic equation in sin(x).
We can solve this quadratic equation for sin(x), then find x using inverse trigonometric functions. By factoring the quadratic or using the quadratic formula, we get solutions for sin(x). Then, we solve for x by considering all possible angles in the unit circle that correspond to the found sine values.
The equation cos(2x) - √2 sin(x) = 1 can be solved by using trigonometric identities to simplify and factor the equation, leading to solving for sin(x) using inverse operations.
Explanation:The original equation given is cos(2x) - √2 sin(x) = 1. To solve this equation, we can use trigonometric identities to simplify the cosine term. One such identity is cos(2x) = 1 - 2sin²(x), which allows us to rewrite the equation as 1 - 2sin²(x) - √2 sin(x) = 1. From there, we subtract 1 from both sides, thus isolating the sine terms on the left: - 2sin²(x) - √2 sin(x) = 0. Factoring out the common term sin(x), we get sin(x)(-2sin(x) - √2) = 0. Setting each factor equal to zero gives us two possible solutions: sin(x) = 0 and sin(x) = -√2/2. In the context of a right triangle, these solutions correspond to specific angles where the sine value is 0 and -√2/2, respectively. The solutions can be found using standard trigonometric unit circle values or by calculating the inverse sine for -√2/2.
Algebraic terms are separated by ______.
Choose all that apply.
1)=
2)+
3)x
4)÷
5)-
In algebra, terms are separated by addition (+), subtraction (-), multiplication (x), and division (÷) symbols. The equals (=) sign is used to show equality, not to separate terms.
Explanation:In the realm of mathematics, particularly in algebra, algebraic terms are separated by addition (+), subtraction (-), multiplication (x), and division (÷) symbols. These operations are the main components that allow us to manipulate and solve algebraic expressions and equations. For example, in the expression 3x - 2y + 5z, the terms (3x, 2y, 5z) are separated by subtraction (-) and addition (+) symbols. Please note that the equals (=) sign does not separate terms; rather, it is used to show the equality between two mathematical expressions.
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Final answer:
Algebraic terms are separated by plus (+) and minus (-) signs, which denote addition or subtraction between terms in an expression.
Explanation:
Algebraic terms are separated by plus (+) and minus (-) signs.
These signs denote addition or subtraction between terms in an algebraic expression.
While the multiplication sign (x) and division sign (÷) indicate operations between terms, they do not separate independent terms within an expression.
For instance, in the expression 2x + 3y - 5z, the plus and minus signs separate the terms 2x, 3y, and 5z.
What is 9/2 as a decimal?
The initial number of views for a reader board was 25. The number of views is growing exponentially at a rate of 18% per week. What is the number of views expected to be four weeks from now? Round to the nearest whole number Enter your answer in the box.
This is an exponential growth problem. Let's write the equation for exponential growth.
[tex]P=P_{0} (1+r)^{n}[/tex]
Where,
P is the final amount[tex]P_{0}[/tex] is the initial amountr is the rate at which increasingn is the timeFrom the problem, we know initial number of views, [tex]P_{0}[/tex], is 25. Rate of increase, r, is 0.18. Time, n, is 4 weeks. Plugging in these values in the equation and solving for P will give us the number of views expected in four weeks time.
[tex]P=(25)(1+0.18)^{4}\\=(25)(1.18)^{4}\\=48.47[/tex]
Rounding to nearest whole number, it is 48.
ANSWER: 48
Determine the sum -46.38 (-24.6)
Find the equation of the tangent line to the curve y = 2sinx at the point (pi/6,1). The equation of this tangent line can be written in the form y = mx+b. Compute m and b
The equation of the tangent line to the curve y = 2sinx at the point (pi/6,1) is y = sqrt(3)x + (1 - sqrt(3)(pi/6)).
Explanation:To find the equation of the tangent line to the curve y = 2sinx at the point (pi/6, 1), we need to find the slope of the curve at that point. The slope of a curve at a point is equal to the derivative of the function at that point.
Taking the derivative of y = 2sinx, we have dy/dx = 2cosx. Evaluating this derivative at x = pi/6, we get dy/dx = 2cos(pi/6) = sqrt(3). Therefore, the slope of the tangent line is sqrt(3).
Now we have the slope and a point on the line, (pi/6, 1). We can use the point-slope form of a line, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope.
Plugging in the values, we have y - 1 = sqrt(3)(x - pi/6).
Simplifying this equation, we get y = sqrt(3)x - sqrt(3)(pi/6) + 1. Finally, we can rewrite the equation in the form y = mx + b by simplifying the y-intercept, giving us y = sqrt(3)x + (1 - sqrt(3)(pi/6)).
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What is the factor of 40?
the lines below are parallel. if the slope of the green line is -3, what is the slope of the red line
Write the area as a product and as a sum.
The area of the square as a product is a multiplication of the two sides.
What is the area of the square?The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the square in a two-dimensional plane is called the area of the square.
The square is a quadrilateral having all the sides to be equal and all the angles at 90 degrees. The area of the square will be calculated as,
Area = side x side
Therefore, the area of the square as a product is a multiplication of the two sides.
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"Sarah is renting a car for her weekend trip to the mountains. The total cost of the rental, y, as it relates to the number of miles driven, x, is shown in the graph below.
Fill in the blanks so that the functions below, written to represent this situation, are correct. If necessary, the answer in terms of a decimal, rounded to the nearest hundredth.
f(x)= ______x+________"
Answer:
[tex]f(x)=\dfrac{1}{4}x+25[/tex]
Step-by-step explanation:
Clearly from the graph we could see that it passes through two points (0,25) and (20,30) and also as the graph is linear so with the help of these two points we can find the relation between f(x) and x.
We know that the equation of a line passing through two points (a,b) and (c,d) is given by:
[tex]y-b=\dfrac{d-b}{c-a}\times (x-a)[/tex]
Here we have:
y=f(x) , (a,b)=(0,25) and (c,d)=(20,30).
Hence the equation of this graph is given as:
[tex]f(x)-25=\dfrac{30-25}{20-0}\times (x-0)\\\\f(x)-25=\dfrac{5}{20}\times x\\\\f(x)-25=\dfrac{1}{4}\times x\\\\f(x)=\dfrac{1}{4}x+25[/tex]
Hence,
[tex]f(x)=\dfrac{1}{4}x+25[/tex]
I do not understand how to do this problem please help!
6. Below is a graph that represents the total profits for a third month. Write the equation of the line that represents this graph. Show your work or explain how you determined the equations.
The first month they made 1200 and the second month they made 1500...
graph a line using the slope-intercept form:
Manipulate the equation into slope-intercept form, y = mx + b.
Identify and plot the y-intercept of the line. Remember, the y-intercept is b. Don't forget: The sign goes with the number.
Identify and use the slope of the line to find a second point. Remember, the slope is m. Don't forget: Starting at the y-intercept, the numerator tells you the rise (count up if positive and down if negative), the denominator tells you the run (count right.)
Draw a straight line through the two points to complete the graph.
Linear Functions:
A linear function is a function that is defined by a linear equation.
To write a linear function with function notation, first write it in slope-intercept form, and then replace y with f(x).
The graph of a linear function is all points (x, f(x)), where x is in the domain of the function.
Please complete z2+9z-90=(z-6)(z+?) ...?
Answer:
Answer would be 15.
Step-by-step explanation:
We have to complete the expression given as
[tex]z^{2}+9z-90=(z-6)(z+?))[/tex]
We will try to factorize the expression given in the left side.
[tex]z^{2}+9z-90=z^{2}+15z-6z-90[/tex]
= z(z+15)-6(z-15)
=[tex](z-6)(z-15)[/tex]
Now we will compare this factorized form with the right side of the expression.
(z-6) (z-15) (= (z-6) (z+?)
We find question mark is 15.
Therefore, the answer is 15.
Which set of ordered pairs represents y as a function of x?
{(5, 4), (2, 3), (1, 1), (2, 4)}
{(-9, 2), (0, 6), (1, -2), (0, 6)}
{(3, 2), (-4, -2), (3, 1), (-4, 1)}
{(-1, 0), (4, 3), (-7, -3), (-1, -8)}
HELP PLEASE?! I'm very bad at math and I need help with this question please.
What is the value of y in the equation 6 + y = −3?
A.) −9
B.) −3
C.) 3
D.) 9
factor the common factor out of 8x^3+12x
Sandra rode her bike 5 times as many miles as Barbara. If b, the distance Barbara rode, equals 3.4 miles, what is the correct expression and distance Sandra rode?
a) b+5; when b=3.4, the distance Sandra rode is 17 miles.
b) 5b; when b=3.4, the distance Sandra rode is 17 miles.
c) 5b; when b=3.4, the distance Sandra rode is 8.4 miles.
d) b+5; when b=3.4, the distance Sandra rode is 8.4 miles.
Answer:
Option b is correct
5b; when b = 3.4 miles , the distance Sandra rode is 17 miles
Step-by-step explanation:
Here, b represents the distance Barbara rode.
As per the statement:
Sandra rode her bike 5 times as many miles as Barbara.
⇒[tex]\text{Distance Sandra rode} = 5b[/tex] miles ....[1]
It is also given that:
the distance Barbara rode, equals 3.4 miles
⇒[tex]b = 3.4[/tex] miles
Substitute in [1] we have;
[tex]\text{Distance Sandra rode} = 5(3.4)=17[/tex] miles
Therefore, the correct expression and distance Sandra rode is
5b ; when b = 3.4 miles , the distance Sandra rode is 17 miles
Assuming x > 0, which of these expressions is equivalent to 11 times the square root of 245 x to the third plus 9 times the square root of 45 x to the third? ?
5 x times the square root of 104 x
20 times the square root of 290 x to the sixth
20 x times the square root of 290 x
104 x times the square root of 5 x
A(n) ______ is a letter or symbol that represents some unknown value.
A. term
B. equation
C. variable
D. expression
Answer:
It is a variable
Step-by-step explanation:
A variable is a characteristic, number, or quantity that increases or decreases over time or takes different values in different situations. Hope this helps.
The population of a city increases by 10,000 per year. The starting population is 800,000.
Let P be the ending population.
Which shows an equation for the total population after x years?
Given D = 5, E = 10 and F = 4, evaluate
D*E/F
...?
Factor of 3x^2 - 6x 3
What is the probability of getting 3 tails when tossing a coin 4 times?
The probability of getting 3 tails, when tossing a coin 4 times is 25%
To find the probability of getting 3 tails when tossing a coin 4 times, we can use the binomial probability formula:
[tex]\[ P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k} \][/tex]
Where:
- P(X = k) is the probability of getting exactly k successes (in this case, 3 tails).
- n is the number of trials (coin tosses).
- k is the number of successes we are interested in (in this case, 3 tails).
- p is the probability of success on each trial (in this case, the probability of getting tails).
- [tex]\( (1 - p) \)[/tex] is the probability of failure on each trial (in this case, the probability of getting heads).
Since the coin is fair, the probability of getting tails p is [tex]\( \frac{1}{2} \)[/tex], and the probability of getting heads [tex](\( 1 - p \))[/tex] is also [tex]\( \frac{1}{2} \)[/tex].
Substituting these values into the formula:
[tex]\[ P(X = 3) = \binom{4}{3} \times \left(\frac{1}{2}\right)^3 \times \left(1 - \frac{1}{2}\right)^{4 - 3} \][/tex]
[tex]\[ P(X = 3) = \binom{4}{3} \times \left(\frac{1}{2}\right)^3 \times \left(\frac{1}{2}\right)^1 \][/tex]
[tex]\[ P(X = 3) = 4 \times \left(\frac{1}{8}\right) \times \left(\frac{1}{2}\right) \][/tex]
[tex]\[ P(X = 3) = \frac{4}{16} \][/tex]
[tex]\[ P(X = 3) = \frac{1}{4} \][/tex]
To express the probability as a percentage, we multiply by 100:
[tex]\[ P(X = 3) = \frac{1}{4} \times 100\% = 25\% \][/tex]
Therefore, the probability of getting 3 tails when tossing a coin 4 times is 25%.
Clark wants to figure out how many pens to order for the office there 48 workers and he needs to order one pen for each worker he knows that for every six people who prefer red door 6 prefer blue how many blue pens should he order
Given that the preference for red and blue pens is equal and that there are 48 workers, Clark should order 24 blue pens for the office.
Explanation:The student is interested in figuring out how many blue pens to order for 48 workers in an office knowing that the preferences for red and blue pens are equal. This means that for every 6 people who prefer red, the same number of people will prefer blue.
To determine the number of blue pens needed, we can use a simple proportion. Since 6 people prefer red and the same number prefer blue, we have a 1:1 ratio of red to blue preferences.
Therefore, the total number of blue pens needed would be half of the total number of workers:
Total workers = 48Number of blue pens = Total workers / 2 = 48 / 2Number of blue pens = 24A rectangular prism has a length of 4.2 cm, a width of 5.8 cm, and a height of 9.6 cm. A similar prism has a length of 14.7 cm, a width of 20.3 cm, and a height of 33.6 cm. The dimensions of the smaller prism are each multiplied by what factor to produce the corresponding dimensions of the larger prism?
Answer:
its (c) 3 1/2
Step-by-step explanation:
What is the standard form for y + 5 = 0?
Find the value of tan 39°. Round to the nearest ten-thousandth.
The value of the given trigonometric ratio tan 39° is approximately 0.8098.
Use the concept of trigonometric ratio defined as:
Trigonometric Identities are equality statements that hold true for all values of the variables in the equation and that use trigonometry functions.
There are several distinctive trigonometric identities that relate to a triangle's side length and angle. Only the right-angle triangle is consistent with the trigonometric identities.
The given trigonometric ratio is: tan 39°
Since,
[tex]\text{tan }\theta = \dfrac{\text{sin }\theta}{\text{cos }\theta}[/tex]
The value of the sine of 39° is approximately 0.6293.
The value of the cosine of 39° is approximately 0.7771.
Therefore,
tan 39° = 0.6293 / 0.7771
tan 39°≈ 0.8098 (After rounding to four decimal places).
Hence,
The required value tan 39° is approximately 0.8098.
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Which lines are parallel if m4 = m5? Justify your answer.
Answer:
r and s are parellel
Step-by-step explanation:
The correct answer is that lines 4 and 5 are parallel if [tex]\( m_4 = m_5 \).[/tex]
To justify this answer, we must understand the concept of slope in the context of lines in a plane. The slope of a line, denoted by m is a measure of its steepness and is defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line. Mathematically, if two points on a line have coordinates [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \),[/tex] then the slope m is given by: [tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex] For two lines to be parallel, they must have the same slope. This is because parallel lines have the same steepness and never intersect. Therefore, if the slope of line 4, [tex]\( m_4 \)[/tex], is equal to the slope of line 5, [tex]\( m_5 \)[/tex], then lines 4 and 5 are parallel. In mathematical notation, if [tex]\( m_4 = m_5 \),[/tex] then line 4 and line 5 are parallel, which can be written as: [tex]\[ m_4 = m_5 \Rightarrow \text{Line 4} \parallel \text{Line 5} \][/tex] This conclusion is based on the fundamental properties of slopes and parallel lines in Euclidean geometry. If the slopes of two lines are equal, it means that they rise and run at the same rate, and thus, they will maintain a constant distance from each other, never intersecting, which is the definition of parallel lines.
Kamal bought 7 packs of paper clips for a project. then he bought 8 more packs. there are 25 paper clips in each pack. how many paper clips did he buy?
Answer:
He bought 375 paper clips.
Step-by-step explanation:
Given,
The initial number of packs of paper clips = 7,
Also, the additional number of packs of paper clips = 8,
So, the total number of packs of paper clips = 7 + 8 = 15,
Now, the number of paper clips in each pack = 25,
Hence, the total number of paper clips = number of paper clips in each pack × total packs
= 25 × 15
= 375
p² 13p=-30 menentukan nilai p
Round 0.978 to the nearest tenth
On sunday,sheldon bought 4 and 1 half kg of plant food. he used 1 and 2 thirds kg on his strawberry plants and used 1 fourth kg for his tomato plants. how many kilograms of plant food did sheldon have left write one or more to show how u reached your answer.