Answers
SOLUTION:
Step 1:
In this question, we have that:
Step 2:
Part A:
We are meant to show that the equation:
[tex]5sinx=1+2cos^2x[/tex]can be written in the form
[tex]2sin^2\text{x + 5 sin x - 3=0}[/tex]Proof:
[tex]\begin{gathered} \text{5 sin x = 1 + 2 cos }^2x\text{ } \\ \text{But cos}^2x+sin^2x\text{ = 1} \\ \text{Then,} \\ \cos ^2x=1-sin^2x\text{ } \\ \text{Hence,} \\ 5sinx=1+2(1-sin^2x_{}) \\ 5sinx=1+2-2sin^2x \\ 5sinx=3-2sin^2x \end{gathered}[/tex]Re-arranging, we have that:
[tex]2sin^2x\text{ + 5 sin x - 3 = 0 }[/tex]Part B:
b) Hence, solve for x in the interval:
[tex]0\text{ }\leq\text{ x }\leq\text{ 2}\pi[/tex]
Related Questions
Help
Show work please
Answers
Answer:
check the attached files.
Given circle O with diameter AC, tangent AD, and the measure of arc BC is 74 degrees, find the measures of all other indicated angles.
Answers
We want to find the measure of the angles 1 to 8, given that the diameter is AC and the measure of the Arc BC is 74°.
The angle 5, ∡BOC is central and it is equal to the measure of the arc it intercepts, the arc BC. Thus the angle 5 is 74°.
The angle 4, ∡AOB also is central, and it is equal to the measure of the arc AB. As the line AC is the diameter of the circle O, the arc AC is equal to 180°, and thus, the sum of the angles 4 and 5 will be 180°:
[tex]\begin{gathered} \measuredangle4+\measuredangle5=180^{\circ} \\ \measuredangle4=180^{\circ}-\measuredangle5=180^{\circ}-74^{\circ}=106^{\circ} \end{gathered}[/tex]Thus, the angle 4 is 106°.
The angle 6 is an inscribed angle, and thus it is half of the arc it intersects, the arc AB. This means that the angle 6 is 106°/2=54°.
The angle 2 also is an inscribed angle, half of the arc BC, and thus, the angle 2 is 74°/2=37°.
Now, the triangle BOC has the angles 5, 6 and 7, and the sum of those angles is 180°. This means that:
[tex]\begin{gathered} \measuredangle5+\measuredangle6+\measuredangle7=180^{\circ} \\ 74^{\circ}+54^{\circ}+\measuredangle7=180^{\circ} \\ 128^{\circ}+\measuredangle7=180^{\circ} \\ \measuredangle7=180^{\circ}-128^{\circ}=52^{\circ} \end{gathered}[/tex]Thus, the angle 7 is 52°.
Following a same argument, we can get the angle 8, as being part of the triangle AOB.
[tex]\begin{gathered} \measuredangle2+\measuredangle4+\measuredangle8=180^{\circ} \\ \measuredangle8=180^{\circ}-37^{\circ}-106^{\circ}=37^{\circ} \end{gathered}[/tex]This means that the angle 8 is 37°.
As the line AD is tangent to the circle O, this means that the lines AC and AD are perpendicular, and thus, the angle 1 is 90°.
Lastly, as the angles 1, 2 and 3 are coplanar, their sum is 180°. This is:
[tex]\begin{gathered} \measuredangle1+\measuredangle2+\measuredangle3=180^{\circ} \\ \measuredangle3=180^{\circ}-\measuredangle1-\measuredangle2 \\ \measuredangle3=180^{\circ}-90^{\circ}-37^{\circ}=180^{\circ}-127^{\circ}=53^{\circ} \end{gathered}[/tex]Thus, the angle 3 is 53°.
There are 130 people in a sport centre.
76 people use the gym
60 people use the swimming pool.
32 people use the track.
23 people use the gym and the pool.
8 people use the pool and the track.
20 people use the gym and the track.
6 people use all three facilities.
Given that a randomly selected person
uses the gym and the track, what is
the probability they do not use the
swimming pool?
Answers
The probability is 0.57
What is meant by probability?
Probability is a discipline of mathematics that deals with appropriate units of how probable an event is to occur or how likely a statement is to be true. The probability of an occurrence is a number ranging from zero and 1, where 0 denotes the event's feasibility and 1 represents certainty. The greater the likelihood of an occurrence, the more probable it will occur. Tossing a fair (unbiased) coin is a basic example. Because the coin is fair, the two possibilities ("heads" and "tails") are equally likely; the chance of "heads" equals the probability of "tails," and because no other outcomes are possible, the probability of either "heads" or "tails" is 1/2.
Probability of using the pool = 97/225 = 0.43
Probability that they do not use the swimming pool = 1 - 0.43 = 0.57
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A manufacturing process has a 70% yield, meaning that 70% of the products
Answers
Answer:
are acceptable and 30% are defective.
Step-by-step explanation:
Factor the polynomial and use the factored form to find the zeros. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)
P(x) = x3 − 2x2 − 15x
x =
Answers
Answer:
-3, 0, 5
Step-by-step explanation:
You want the zeros of P(x) = x³ − 2x² − 15x using the factored form.
Factored formWe notice right away that x is a factor of every term. Factoring that out gives us a quadratic to factor:
P(x) = x(x² -2x -15)
To factor this, we need two factors of -15 that have a sum of -2. The factors -5 and +3 have those properties. That means our factored form is ...
P(x) = x(x +3)(x -5) . . . . factored form
ZerosThis product will be zero when any of its factors is zero. Considering them one at a time, we find the zeros of P(x) to be ...
x = 0
x +3 = 0 ⇒ x = -3
x -5 = 0 ⇒ x = 5
The zeros of P(x) are -3, 0, 5.
Determine the measures of angles x, y, and z: x = 75°95°105°° y = 75°95°105°° z = 75°95°105°°
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Consider the figure,
So, we have, Two angles are called complementary when their measures add to 90 degrees. Two angles are called supplementary when their measures add up to 180 degrees.
So, here, [tex]<\text{AHB}+z=180[/tex]
Therefore, z can be calculated as,
[tex]z=180-<\text{AHB}=180-105=75[/tex]Now, the angles DHC and
If the 10 letters are {aa,aa,aa,aa,bb,bb,cc,cc RR,RR} are available and all 10 of them are to be selected without replacement,what is the number of different permutations?
Answers
In order to calculate the number of permutations, first we start with the factorial of the number of letters.
There are 10 letters, so we start with the factorial of 10.
Then, we need to check the number of repetitions. Each repetition will be a factorial in the denominator:
[tex]x=\frac{10!}{a!\cdot b!\operatorname{\cdot}...}[/tex]We have four repetitions of aa, two repetitions of bb, two repetitions of cc and two repetitions of RR, therefore the final expression for the number of permutations is:
[tex]x=\frac{10!}{4!2!2!2!}[/tex]Calculating this expression, we have:
[tex]x=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5\operatorname{\cdot}4!}{4!\operatorname{\cdot}2\operatorname{\cdot}2\operatorname{\cdot}2}=\frac{10\operatorname{\cdot}9\operatorname{\cdot}8\operatorname{\cdot}7\operatorname{\cdot}6\operatorname{\cdot}5}{8}=18900[/tex]Therefore there are 18900 permutations.
Hello I'd like some help on my practice question I'd prefer if it's quick because I have other questions I need to solve thank you
Answers
f(x) = -2
the answer is the second option
The horizontal line at y = -2 which is parallel to x-axis
x Michael uses synthetic division to divide f(x) by g(x), his last line of work 0/3is shown. How would he write his answer of f(x) divided by g(x). *7 0 24 0 0
Answers
It's important to know that synthetic division gives a polynomial as a result.
Michael obtained 7 0 24 0 0. We just need to add variables to it. As you can observe, there are 5 terms, that means the polynomial is grade 4.
[tex]7x^4+0x^3+24x^2+0x+0[/tex]Therefore, the resulting polynomial is
[tex]7x^4+24x^2[/tex]Why was math created
Answers
SOLUTION:
Step 1:
In this quesdtion, we are meant to explain the topic:
Why was Math created?"
Step 2:
The details of the solution are as follows:
1. Mathematics is a body of knowledge and knowledge and practice, that is derived from the contributions of thinkers throughout the ages and across the globe.
2. It gives us a way to understand patterns, to quantify relationships, and to predict the future.
3. Math helps us understand the world — and we use the world to understand math.
4. Excellent for your brain: Creative and analytical skills are highly desired by employers.
5. It has a lot of real-world applications.
6. It helps in better problem-solving skills.
7. Mathematics is needed in almost every career and profession.
8. Mathematics helps understand the world better.
9. Mathematics is a universal language.
10. Numbers help us understand the world, and Mathematics helps us understand numbers.
The real-life applications of Mathematics are endless.
We are surrounded by numbers, equations and algorithms – especially in this age of data science, with huge data sets that can only be understood through statistical models and analysis.
Answer:
Maths was invented to understand the world and to make measurements, do calculations etc. make and measure shapes, measure angles, and to use these things in real life. The Egyptians used the Pythagoras theorem to accurately make their pyramids.
An object moves in simple harmonic motion with period 6 seconds and amplitude 4cm. At time =t0 seconds, its displacement d from rest is 0cm, and initially it moves in a negative direction. Give the equation modeling the displacement d as a function of time t.
Answers
The general function for describing the displacement from the mean position in harmonic motion is:
[tex]d(t)=A\cdot\sin (\frac{2\pi}{T}\cdot t+\phi)\text{.}[/tex]Where:
• A is the amplitude,
,• T is the period,
,• φ is initial phase displacement.
From the statement, we know that:
• the amplitude is 4 cm,
,• at time t = 0 its displacement d from the rest is 0 → d(t = 0) = 0,
,• initially, it moves in a negative direction.
s
If Omar still needs 458How much does he need after saving for 5 weeks
Answers
(a) Setting A(w)=458, we get:
[tex]800-18w=458.[/tex]Subtracting 800 from the above equation we get:
[tex]\begin{gathered} 800-18w-800=458-800, \\ -18w=-342. \end{gathered}[/tex]Dividing the above equation by -18 we get:
[tex]\begin{gathered} \frac{-18w}{-18}=\frac{-342}{-18}, \\ w=19. \end{gathered}[/tex]Therefore Omar has been saving for 19 weeks.
(b) Recall that to evaluate a function at a given value, we substitute the variable by the given value.
Evaluating A(w) at w=5 we get:
[tex]A(5)=800-18\cdot5.[/tex]Simplifying the above result we get:
[tex]\begin{gathered} A(5)=800-90 \\ =710. \end{gathered}[/tex]Answer:
(a) 19.
(b) $710.
Graph the line y = 5x - 1, then name the slope and y-intercept by looking at the graph. What is m= and what is b= and how do I graph this what are the points ?
Answers
Answer:
Step-by-step explanation:
Slope-intercept form: y = mx + b
The 'm' in this formula means slope. The 'b' means the y-intercept.
y = 5x - 1
m = 5.
b = -1.
Now that we have identified the slope and the y-intercept, we can graph the equation.
When graphing these kinds of equations, always start at the y-intercept.
The y-intercept is -1, so we start from there and move up 5 and right 1 repeatedly.
Remember, slope = rise/run. We rise 5, and we run 1.
5 can also be represented as a fraction: [tex]\frac{5}{1}[/tex]
Let me know if you have any questions.
help meee pleaseeee pleasee
Answers
So, let’s evaluate:
f(0)=-(0)^3+7(0)^2-2(0)+12
All the terms with the variable x are accompanied by a coefficient and exponents, which both, by definition, are forms of multiplication. So, we know that a•0=0, so any term with 0 is canceled to 0. We get:
f(0)=0+12
f(0)=12
can you please give me any examples on how to do this
Answers
we can take two numbers of the sequence and subtract them to see the difference
so
[tex]1.9-1.2=0.7[/tex]the sequence adds 0.7 each step
the next 3 terms are
[tex]3.3+0.7=4[/tex][tex]4+0.7=4.7[/tex][tex]4.7+0.7=5.4[/tex]The price of acorn squash is proportional to the weight in pounds. You pay $6.36 for 4 pounds of acorn squash. How much does3 pounds of acorn squash cost?
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In order to calculate the cost for 3 pounds of acorn squash, we can write the following rule of three:
[tex]\begin{gathered} \text{weight}\to\text{cost} \\ 4\text{ pounds}\to6.36 \\ 3\text{ pounds}\to x \end{gathered}[/tex]Then, we can write the following equation and solve it for x:
[tex]\begin{gathered} \frac{4}{3}=\frac{6.36}{x} \\ 4\cdot x=3\cdot6.36 \\ 4x=19.08 \\ x=\frac{19.08}{4} \\ x=4.77 \end{gathered}[/tex]Therefore the cost of 3 pounds is $4.77.
1 + c + 1.4 = c + 2.4I need help
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1 + c + 1.4 = c + 2.4
c + 2.4 = c + 2.4
c = c + 2.4 - 24
c = c
7^2 × 7^8. 7^a------------ = -------- = 7^b7^4 7^4
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We have to find the values of a and b:
[tex]\frac{7^2\cdot7^8}{7^4}=\frac{7^a}{7^4}=7^b[/tex]We can use the laws of exponents to write:
[tex]\begin{gathered} 7^2\cdot7^8=7^a \\ 7^{2+8}=7^a \\ 7^{10}=7^a \\ 10=a \end{gathered}[/tex]Then, we can solve for b as:
[tex]\begin{gathered} \frac{7^a}{7^4}=7^b \\ 7^{a-4}=7^b \\ a-4=b \\ 10-4=b \\ 6=b \end{gathered}[/tex]Answer: a=10 and b=6
What is the solution to the equation below? 6x= x + 20 O A. x = 4 B. X = 20 C. x = 5 D. No Solutions
Answers
Simplify the equation 6x = x +20 to obtain the value of x.
[tex]\begin{gathered} 6x=x+20 \\ 6x-x=20 \\ 5x=20 \\ x=\frac{20}{5} \\ =4 \end{gathered}[/tex]So answer is x = 4
Option A is correct.
Write the slope-intercept form of the equation of the line graphed on the coordinate plane.
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The slope-intercept form is:
[tex]y\text{ = mx + b}[/tex]We have to find these coefficients. To do that we have to choose two points in the graph and apply the following formula. I will use (0,1) and (-1,-1). The formula is:
[tex]y-yo\text{ = m(x-xo)}[/tex]The formula of the coefficient 'm' is:
[tex]m\text{ = }\frac{y2-y1}{x2-x1}[/tex]Let's substitute the points into the formula above to find the value of m. Then we use one of the points to find the slope-intercept form of the equation:
[tex]m\text{ = }\frac{-1-1}{-1-0}=2[/tex]Applying it to the second equation using the point (0,1):
[tex]y-1=2(x-0)[/tex][tex]y=2x+1[/tex]Answer: The slope-intercept form of the equation will be 2x+1.
Alexa lives 4 kilometers away from school. She leaves home and rides her bicycle toward school at a speed of 0.25 kilometer per minute.Enter the function f(x) that represents Alexa's distance in kilometers from school after x minutes. I NEED ANSWERS SO I DONT FAIL THIS SCHOOL YEAR PLEASE
Answers
Alexa lives 4 kilometers away from school.
Distance between Alexa house and school = 4km
She leaves home and rides her bicycle toward school at a speed of 0.25 kilometer per minute
Speed of Alexa = 0.25km/min
The relation between speed, distance and time is express as:
[tex]\text{Speed}=\frac{Dis\tan ce}{Time}[/tex]Let time = x minutes
Substitute the value, time = x, Distance = f(x), speed = 0.25km/min
[tex]\begin{gathered} \text{Speed}=\frac{Dis\tan ce}{Time} \\ 0.25=\frac{f(x)}{x} \\ f(x)\text{ = 0.25x} \end{gathered}[/tex]Answer: f(x) = 0.25x
15. Given f (n)=3( 12), what is the value off (8) ?
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We have some function f(x) and want to evaluate the function for some value of x, in this case for x=8.
Evaluate a function means replace the x for the value you want to evaluate, in this case for 8, so:
[tex]\begin{gathered} f(x)=3(1-x) \\ f(8)=f(x=8)=3(1-8) \\ f(8)=3\cdot(-7)=-21 \end{gathered}[/tex]why you can always solve a right triangle if you know the measures of one side and one acute angle.
Answers
In a right triangle, one angle is always 90.
If you know one acute angle, you automatically know the other (3rd) angle.
3 angles are solved.
Now, comes the sides.
If you already know 1 side, you can easily know another side by using the basic trig identities SIN, COS, or TAN.
When you know 2 sides, the 3rd side can always be find using:
• pythagorean theorem, or
,• again, trigonometric ratios (sin, cos, tan).
A point is chosen at random in the square shown below. Find the probability that the point is in the shaded circular region. Each side of the square is 6in, and the radius of the circle is 3in.Use the value 3.14 for π. Round your answer to the nearest hundredth.
Answers
We will have the following:
First, we determine the area of the square and of the shaded region, that is:
[tex]\begin{gathered} A_s=6in^2\Rightarrow A_s=36in^2 \\ \\ A_c=\pi(3)^2\Rightarrow A_c=9\pi in^2 \end{gathered}[/tex]Now, we will have that the probability will be of:
[tex]P=\frac{9\pi}{36}\Rightarrow P=\frac{\pi}{4}\Rightarrow P\approx0.79[/tex]So, the probability is approximately 79%.
The Oldest rocks on Earth are about 4 x 10^9 years old. For which of these ages could this be an approximation?
A. 3,862,100,000 years
B. 3.849999999x10^9 years
C. 0.000000004 years
D.4,149,000,000 years
E.3.45x10^9 years
Answers
The graph shows the proportional relationship between the number of gems collected and the number of levels that have been completed in a video game.
Graph with x axis labeled game levels and y axis labeled gems collected. A line begins at 0 comma 0 and goes through points 6 comma 420 and 8 comma 560.
Determine the constant of proportionality for the relationship.
p = 70
p = 140
p equals 2 over 140
p = 0.0143
Answers
Answer: P = 70
Step-by-step explanation:
p = 70 because on the graph everytime the y number is the x number multiplied by 70.
70 x 2 = 140
70 x 4 = 280
70 x 6 = 420
70 x 8 = 560
70 x 10 = 700.
Heres the chart for proof
The constant of proportionality (p) for this proportional relationship is equal to: A. p = 70.
How to determine the constant of proportionality?In Mathematics, the graph of any proportional relationship is characterized by a straight line because as the values on the x-axis increases or decreases, the values on the y-axis increases or decreases simultaneously.
Mathematically, a proportional relationship can be represented by the following equation:
y = px
Where:
p is the constant of proportionality.y represents the gems collected.x represents the game levels.Next, we would determine the constant of proportionality (p) for the data points on this graph as follows:
p = y/x
p = 420/6 = 560/8
p = 70.
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which point lies on the wall with point slope equation y+5=2(x+8)
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The slope intercept form of equation is given as
(y - y1) = m(x - x1)
Where m = slope
From the equation: y + 5 = 2(x + 8)
Equate y + 5 = 0 and x + 8 = 0
y + 5 = 0
y = 0- 5
y = -5
For x + 8 = 0
x + 8 = 0
x = 0 - 8
x = -8
Hence, the point is (-8, -5)
The answer is (-8, -5)
The graph below and to the left shows the time of sunsets occurring every other day during September in a certain town. The graph at the lower right shows the time of sunsets on either the 21st or 22nd day of each month for an entire year in the same town. The vertical axis is scaled to reflect hours after midnight. Round to 4 decimal places. a) Find a linear model for the data in the graph at the left. Include units to your variables. b) Find a cosine model for the data in the graph to the right. Include units to your variables,
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A) Given the points (1,18.35) and (29,17.5), we can find the linear model with the following formulas:
[tex]\begin{gathered} \text{slope:} \\ m=\frac{y_2-y_1}{x_2-x_1}=\frac{71.5-18.35}{29-1}=\frac{-0.85}{28}=-0.03 \\ \text{equation of the line:} \\ y-y_1=m(x-x_1) \\ \Rightarrow y-18.35=-0.03(x-1)=-0.03x+0.03 \\ \Rightarrow y=-0.03x+0.03+18.35=-0.03x+18.38 \\ y=-0.03x+18.38 \end{gathered}[/tex]therefore, the linear model is y = -0.03x+18.38
B)We have the general cosine model:
[tex]y(t)=A+B\cos (\omega(t-\phi))[/tex]Where A is the vertical shift, B is the amplitude, w is the frequency and phi is the phase shift.
First, we can find the vertical shift with the following formula:
[tex]A=\frac{y_{\max }+y_{\min }}{2}[/tex]in this case, we have that the maximum value for y is 19.47 and the minimum value for y is16.18, then:
[tex]A=\frac{19.47+16.18}{2}=17.825[/tex]next, we can find the amplitud with the following formula:
[tex]B=y_{\max }-A[/tex]We have then:
[tex]B=19.47-17.825=1.645[/tex]Now, notice that the graph will repeat every 356 values for t, then, for the frequency we have the following expression:
[tex]\omega=\frac{2\pi}{356}=\frac{\pi}{178}[/tex]To find the phase shift, notice that for the point (172,19.47), we have the following:
[tex]\begin{gathered} y(172)=19.47 \\ \Rightarrow17.825+1.645\cos (\frac{\pi}{178}(172-\phi))=19.47 \\ \Rightarrow1.645\cos (\frac{\pi}{178}(172-\phi))=1.645 \\ \Rightarrow\cos (\frac{\pi}{178}(172-\phi))=1 \end{gathered}[/tex]notice that if the cosine equals 1, then its argument must equal to 0, then, we have:
[tex]\begin{gathered} \frac{\pi}{178}(172-\phi)=0 \\ \Rightarrow172-\phi=0 \\ \Rightarrow\phi=172 \end{gathered}[/tex]we have that the phase shift is phi = 172, then, the final cosine model is:
[tex]y(x)=17.825+1.465\cos (\frac{\pi}{178}(x-172))[/tex]need help, what's the answer for the x and y?
Answers
Line equation in slope and y-intercept form:
y = mx + b
To calculate the slope, we use the first two points: (24,-15) and (28, -17)
m = (y2 - y1)/(x2 - x1)
m = (-17 - (-15))/(28 - 24)
m = (-17 + 15)/(4
m = -2/4 = -1/2
To find b we use the first point: (24, -15)
y = mx + b
b = y - mx = -15 - (-1/2)(24) = -15 + 12 = -3
b = -3
Answer:
y = (-1/2) x - 3
Write the coordinates of the vertices after a reflection over the line y=-x
Answers
The coordinates of the vertices after a reflection over the line y=-x are (y, -x)
How to determine the coordinates?From the question, the transformation rule is given as
Reflection over the line y=-x
There are four types of transformation,
These transformations are
DilationRotationReflectionTranslationEach of these transformations have their rule, and they are represented as
Reflection: reflection across linesDilation: k(x, y)Rotation: rotation by anglesTranslation: (x + h, y + k)So, we have
Reflection over the line y=-x
When represented as a coordinate, the coordinate is
(x, y) = (y, -x)
Hence, the coordinates are (y, -x)
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