Answer:
3 solutions:
[tex]\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}} [/tex]
Step-by-step explanation:
So, first of all, we need to figure the angles that cannot be included in our answers out. The only function in the equation that isn't defined for some angles is [tex]tan(\frac{\theta}{2})[/tex] so let's focus on that part of the equation first.
We know that:
[tex]tan(\frac{\theta}{2})=\frac{sin(\frac{\theta}{2})}{cos(\frac{\theta}{2})}[/tex]
therefore:
[tex]cos(\frac{\theta}{2})\neq0[/tex]
so we need to find the angles that will make the cos function equal to zero. So we get:
[tex]cos(\frac{\theta}{2})=0[/tex]
[tex]\frac{\theta}{2}=cos^{-1}(0)[/tex]
[tex]\frac{\theta}{2}=\frac{\pi}{2}+\pi n[/tex]
or
[tex] \theta=\pi+2\pi n[/tex]
we can now start plugging values in for n:
[tex] \theta=\pi+2\pi (0)=\pi[/tex]
if we plugged any value greater than 0, we would end up with an angle that is greater than [tex]2\pi[/tex] so, that's the only angle we cannot include in our answer set, so:
[tex]\theta\neq \pi[/tex]
having said this, we can now start solving the equation:
[tex]tan(\frac{\theta}{2})=sin(\theta)[/tex]
we can start solving this equation by using the half angle formula, such a formula tells us the following:
[tex]tan(\frac{\theta}{2})=\frac{1-cos(\theta)}{sin(\theta)}[/tex]
so we can substitute it into our equation:
[tex]\frac{1-cos(\theta)}{sin(\theta)}=sin(\theta)[/tex]
we can now multiply both sides of the equation by [tex]sin(\theta)[/tex]
so we get:
[tex]1-cos(\theta)=sin^{2}(\theta)[/tex]
we can use the pythagorean identity to rewrite [tex]sin^{2}(\theta)[/tex] in terms of cos:
[tex]sin^{2}(\theta)=1-cos^{2}(\theta)[/tex]
so we get:
[tex]1-cos(\theta)=1-cos^{2}(\theta)[/tex]
we can subtract a 1 from both sides of the equation so we end up with:
[tex]-cos(\theta)=-cos^{2}(\theta)[/tex]
and we can now add [tex]cos^{2}(\theta)[/tex]
to both sides of the equation so we get:
[tex]cos^{2}(\theta)-cos(\theta)=0[/tex]
and we can solve this equation by factoring. We can factor [tex]cos(\theta)[/tex] to get:
[tex]cos(\theta)(cos(\theta)-1)=0[/tex]
and we can use the zero product property to solve this, so we get two equations:
Equation 1:
[tex]cos(\theta)=0[/tex]
[tex]\theta=cos^{-1}(0)[/tex]
[tex]\theta={\frac{\pi}{2}, \frac{3\pi}{2}}[/tex]
Equation 2:
[tex]cos(\theta)-1=0[/tex]
we add a 1 to both sides of the equation so we get:
[tex]cos(\theta)=1[/tex]
[tex]\theta=cos^{-1}(1)[/tex]
[tex]\theta=0[/tex]
so we end up with three answers to this equation:
[tex]\theta={0, \frac{\pi}{2}, \frac{3\pi}{2}} [/tex]
Answer:
its D. 3
Step-by-step explanation:
for speed runners
i need help im confused
f(-4) = -18, f(-2) = -10, f(6) = 12 and f(-1) = -6 using the piecewise function.
What are piecewise functions?A function defined by numerous sub-functions, where each sub-function applies to a distinct interval in the domain, is known in mathematics as a piecewise-defined function (also known as a piecewise function, a hybrid function, or definition by cases). Instead of being a property of the function itself, piecewise definition is a means to express the function.
When the domain may be divided into intervals on which the property holds, the phrase "property holding piecewise for a function" is used, which is separate but related. This is a characteristic of the function itself, as opposed to the idea mentioned above.
f(-4) = 4(-4)-2 = -18 ∵-4≤-1
f(-2) = 4(-2)-2= -10 ∵-2≤-1
f(6) = [tex]\frac{1}{2}*6 + 9 = 12[/tex] ∵6>-1
f(-1) = 4(-1)-2 = -6 ∵-1≤-1
f(-4) = -18, f(-2) = -10, f(6) = 12 and f(-1) = -6 using the piecewise function.
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A rectangle piece of lawn i 55 m and 98 m long find the length of the fence around it
Step-by-step explanation:
draw a rectangle with width 98 and height 55 there are two of each side, so 98 + 98 + 55 +55 = 306
Answer:
306m
Step-by-step explanation:
You need to find the perimeter of the rectangle. To do that, you just add the side lenghts. So, this is your lawn with the side lengths:
55m
_____________
| |
| |
| |
98m | | 98m
| |
| |
| |
_____________
55m
You just add them together!
55 + 55 + 98 + 98 = 306
so you need 306 meters of fencing
What is the standard deviation of the data given below 10 28 13 18 29 30 22 23 25 and 32?
Hence, the standard deviation of the given data is 7 .
10 28 13 18 29 30 22 23 25 32
Here is the total number of observations n are 10
total= 10 +28+ 13+ 18+ 29+ 30+ 22+ 23+ 25+ 32 = 230
mean (μ) = total/n
= 230/10
= 23
Xi Xi-μ (Xi-μ)²
10 10-23=-13 169
28 28-23=5 25
13 13-23=-10 100
18 18-23=-5 25
29 29-23=6 36
30 30-23=7 49
22 22-23=-1 1
23 23-23=0 0
25 25-23=2 4
32 32-23=9 81
Use the formula to calculate the standard deviation.
√1/n*∑ (Xi-μ)² =√1/10*490
= √49
=7
Hence, the standard deviation of the given data is 7 .
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cm 2 What is the area of the rectangle? Enter the answer in the box. 9 cm 4 cm
Answer:
Let,
length of rectangle ( L) =9cm
breadth of rectangle (b) =4cm
Area of rectangle=L×b
=9×4
=36cm^2
Hi can someone answer this
Answer:
first one and second one
Step-by-step explanation:
x + y is lesser or equal to 20
20 being the max amount of hours he can work in one week and x and y being hours he works
second one because they can earn at least 300 dollars, not less than, but x+Y must still be less than 20 though
How can I calculate my height at 2?
By combining the mother's as well as father's heights in inches, you may determine the midparental height. For boys, measure 5 inches longer; for girls, measure 5 inches shorter. multiply by two.
How can you accurately anticipate a children's adult height?A person's height can be predicted by other factors besides genes. A child may occasionally be significantly taller above their relatives and parents.There is no reliable method for estimating a children's adult height.However, a number of calculations can offer a reliable approximation of a child's growth. Here is a common illustration:
In inches or millimeters, multiply the father's height by the mother's height.For boys, add 5 inches (13 centimeters), and for girls, take 5 inches (13 centimeters) away.Multiply by 2.Double with a boy's height around age 2 or even a girl's height from age 18 months to get an idea of a child's adult height.To know more about the children's adult height, here
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The Above Contour Map Is Given For A Function F(X,Y). Use It To Estimate The Value Of The Partial Derivatives Fx(2, 1) And Fy(2, 1).
We may deduce from the contour map that the estimates of the given functions are;
fx(2, 1) = 10
fy(2, 1) = 3
What exactly does the term "counter map" mean?The geographic mapping used as a form of activism against prevailing power systems is known as counter-mapping.
How Do You Interpret Contour Lines on a Map?Contour lines are described as the faint lines drawn on a map that link places of equal height above sea level. This indicates that even if you followed a contour line physically, the elevation (height of the land) would stay constant.
According to the contour map, the contour lines indicate that the coordinate (2, 1) in the x-direction has a value of 10.
Therefore; fx(2, 1) = 10
Using the same idea, we can deduce from the attached contour map that the coordinate (2, 1) in the y-direction will have a value between 0 and 6. Therefore;
Average = (0 + 6) divided by 2 = 3
Thus; fy(2, 1) = 3
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The matrix below is named E. Create a matrix that is 5E. Now what is the element
at e12?
4 -6 -7
12 3 4
five plus eight times nine
What are the 3 methods to find the mean?
The three methods to find the mean are arithmetic, geometric and harmonic .
The arithmetic mean method: This method involves adding up all the values in the set and dividing by the total number of values. This gives the average value of the set.The geometric mean method: This method involves finding the product of all the values in the set and taking the square root of that product. This gives the average value of the set when the values represent quantities that are multiplied together (such as interest rates or growth rates).The harmonic mean method: This method involves taking the reciprocal of each value in the set (1/value), adding those reciprocals up, and taking the reciprocal of that sum. This gives the average value of the set when the values represent quantities that are inversely related (such as speeds or rates).To learn more about mean, visit:
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Is the equation 3x² 11x 10 0 a quadratic equation?
Yes , the equation 3x² + 11x + 10 = 0 is a quadratic equation and the roots are x = -5/3 and x = -2
What is Quadratic Equation?
A quadratic equation is a second-order polynomial equation in a single variable x , ax² + bx + c = 0. with a ≠ 0. Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has at least one solution. The solution may be real or complex.
The roots of the quadratic equations are
x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
where ( b² - 4ac ) is the discriminant
when ( b² - 4ac ) is positive, we get two real solutions
when discriminant is zero we get just one real solution (both answers are the same)
when discriminant is negative we get a pair of complex solutions
Given data ,
Let the equation be represented as A
Now , the value of A is
3x² + 11x + 10 = 0 be equation (1)
Now , the equation is of the form ax² + bx + c = 0 , where the value of a = 3 , b = 11 and c = 10
So , the coefficient a is has the variable x² and equation 3x² + 11x + 10 = 0 is a quadratic equation
The roots of the quadratic equations are
x = [ -b ± √ ( b² - 4ac ) ] / ( 2a )
Substituting the values in the equation , we get
x = [ -11 ± √ ( ( 11 )² - 4 ( 3 ) ( 10 ) ) ] / 2 ( 3 )
On simplifying the equation , we get
x = [ -11 ± √ ( 121 - 120 ) ] / 6
x = ( -11 ± 1 ) / 6
So , the two roots of the equations are
x = ( -11 - 1 ) / 6 and
x = ( -11 + 1 ) / 6
x = -2 and
x = -5/3
Therefore , the values of x are -5/3 and -2
Hence , the roots of the quadratic equations are -5/3 and -2
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Is base times height divided by 2?
The base is the length of one of the triangle's sides, and the height is the distance from that side to the triangle's opposite corner.
What is triangle?A triangle is a three-edged polygon with three vertices. It is a fundamental form in geometry. Triangle ABC denotes a triangle with vertices A, B, and C. In Euclidean geometry, any three non-collinear points define a unique triangle and, by extension, a unique plane. Triangles are three-sided polygons with three vertices. The angles of the triangle are formed by connecting the three sides end to end at a point. The total of the triangle's three angles equals 180 degrees.
Here,
The base of the triangle is the length of one of its sides, and the height is the distance from that side to the triangle's opposite corner.
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How do you find the factor of a root?
Answer:
Step-by-step explanation:
you can use the zero-product property to find the roots of a factored equation. After you factor a polynomial into its different pieces, you can set each piece equal to zero to solve for the roots with the zero-product property.
Suppoe a merry-go-round i pinning once every 3 econd. If a point on the outide edge ha a peed of 12. 56 feet per econd, what i the radiu of the merry-go-round? (Ue 3. 14 for
The value of radius of the merry-go-round will be 6 feet.
What is a circle?A circle is a geometrical figure which becomes by plotting a point around a fixed point by keeping a constant distance.
In our daily life, we always see circular objects for example our bike wheel.
The longest line which can be drawn inside the circle will be the diameter.
The perimeter of the circle = 2πr where r is the radius of the circle.
The Merry cover on the round in 3 seconds thus, to cover the 2πr path it takes 3 seconds.
2πr/3 = 12.56
2πr = 37.68
r = 5.996 ≈ 6 feet.
Hence "The value of the radius of the merry-go-round will be 6 feet".
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What is the x and y intercept of y=3x-15 and is (2,-9) a solution
The x and y intercept of y=3x-15 is X intercept is 5, Y interceptis -15
How to find X intercept and Y intercept?To find X intercept substitute y=0
y=0
0=3x-15
3x=15
x=15/3
x=5
X intercept is 5 line cuts at (5,0)
To find Y intercept substitute x=0
y=3(0)-15
y=-15
Y interceptis -15 line cuts at (0,-15)
The distance on the axis from the origin where a line or a curve intersects the coordinate axis of the graph is known as the X and Y-intercept. The distance from the origin of a line or curve that cuts the x-axis at a point is known as the x-intercept, and the distance from the origin of a line or curve that cuts the y-axis at a point is known as the y-intercept.-c/m for the x-interceptc = y-interceptA line's equation has the general form axe + by + c = 0. Here, we change y to equal 0 and solve for x to obtain the x-intercept of the line. We then change x to equal 0 and solve for y in the same equation.To learn more about intercepts refer to:
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How do you prove SSS and SAS?
The triangles are congruent if all three sets of comparable sides are present. The side-side-side shortcut is a congruence technique (SSS).
Triangles are considered congruent by SAS if their two sides match up with their corresponding two sides on the other triangle, and if their included angles match up in both triangles.
We can prove two triangles, either by SSS or SAS by simply taking two triangles and checking whether the sides and the angles of the corresponding triangles are congruent or not.
By simply comparing it on the basis of their sides and angles we can easily prove the corresponding triangles by SSS or SAS.
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How do you solve 3 to the power of 1?
To answer, let's use the exponent rules to solve the power of 1.
What is exponent rule?Exponents, often known as powers, specify how many times the base number must be multiplied.
Exponent rules are principles that are used to simplify expressions using exponents. These principles are also useful in simplifying formulas with exponents that are decimals, fractions, irrational numbers, or negative integers.
Negative exponents Rule: [tex]a^{-m} = 1/a^m[/tex]
Product Rule: [tex]a^m * a^n = a^{(m+n)}[/tex]
Quotient Rule: [tex]a^m/a^n = a^{(m-n)}[/tex]
Power of a Power Rule: [tex](a^m)^n = a^{mn}[/tex]
Here, given that, 3 to the power of 1
The number 3¹ may be expressed as 3 to the power of 1, according to the exponent rule.
The base number in this formula is 3, while the power number is 1, and we know that any integer increased to that same power of that one equals the original number.
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Show all your work which is the simplest form of the expression below
Answer: 2³
Step-by-step explanation:
We will use the order of operations to help us solve. This means, in this case, starting with exponents and then multiplying. We will also use the exponent rules.
Given:
[tex]\displaystyle (2^2)^4\;*\;2^{-5}[/tex]
Power rule of exponents:
[tex]\displaystyle (2^2)^4\;*\;2^{-5}=(2^{2*4})\;*\;2^{-5}[/tex]
[tex](2^{8})\;*\;2^{-5}[/tex]
Product rule of exponents:
[tex]\displaystyle (2^{8})\;*\;2^{-5} = (2^{8+-5})= (2^{8-5})[/tex]
[tex](2^{3})[/tex]
The simplified form of the expression, from the given options, is 2³.
Are isosceles triangles equal to 180?
Yes, the isosceles triangles are equal to 180 degrees.
We know that isosceles triangles are triangles which have two equal sides. Since the sides are equal, they subtend equal angles as well.
However, the two equal angles subtended by the equal sides and one more angle which is subtended by the third side of the isosceles triangle all add up to a total of 180 degree as we know that the sum of all the angles of a triangle is always equal to 180 degrees.
Hence we can conclude that the isosceles triangles are also equal to 180 degrees like every other triangle.
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The table shows how a leading automobile dealer plans to spend its advertising dollars for this year. If this company plans to increase its budget by 5% next year, approximately how much will it spend on social media ads?
Advertising Budget: $81,000
48.5% Television
23% Magazines
20% Newspapers
7% Radio
1% Billboards
0.5% Social Media
A.
Using the concept of percentage, social media budget will cost $425.25
PercentageThis is the expression of a ratio or fraction in percentage form. This is done by multiplying a ratio by 100.
The original advertising budget is $81,000 and they plan to increase it by 5%.
Let's calculate the 5% increase in the budget.
5% of 81000 =
5 / 100 = x / 81000
0.05 = x / 81000
x = 81000 * 0.05
x = 4050
The budget for next year will 81000 + 4050 = 85050
In the breakdown for the advertising budget, social media budget got 0.5%
Let's find 0.5% of 85050
0.5% * 85050
0.5 / 100 = x / 85050
0.005 = x / 85050
x = 0.005 * 85050
x = 425.25
The social media budget is $425.25
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a tennis player computes her win ratio by dividing the number of matches she has won by the total number of matches she has played. at the start of a weekend, her win ratio is exactly $0.500$. during the weekend, she plays four matches, winning three and losing one. at the end of the weekend, her win ratio is greater than $0.503$. what's the largest number of matches she could've won before the weekend began?
Using the ratio we know that before the weekend started, she could have won as many as 164 games.
What is the ratio?An ordered pair of numbers a and b, represented as a / b, is a ratio if b is not equal to 0.
A proportion is an equation that sets two ratios at the same value.
For instance, you could express the ratio as follows: 1: 3 if there is 1 boy and 3 girls (for every one boy there are 3 girls) There are 1 in 4 boys and 3 in 4 girls.
So, let n be the number of victories so that:
n/2n = 1/2 and n+3/2n+4 > 503/1000
Now, cross multiply as follows:
1000n + 3000 > 1006n + 2012
n < 988/6 = 164 4/6 = 164 2/3
Therefore, using the ratio we know that before the weekend started, she could have won as many as 164 games.
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Solve the system of linear equations by substitution.
x=17-4y
y=x-2
Answer:
Step-by-step explanation
In the first equation
x=17-4y
In the second equation
We clear "x"
y=x-2
x=y+2
We ecuate "x"
x=17-4y=y+2
17-4y=y+2
15=5y
y=3
we substitute to find "x"
x=17-4y=17-4(3)=5
How do you write 6 more than a number?
“6 more than twice m” can be written in the mathematical form as 2m+6.
The question is saying that we have to write “6 more than twice m” as an algebraic expression or into a mathematical expression.
As in the question, it is mentioned that we have to take the number m as a variable.
So, we are going to use this ‘m’ for solving this question.
Let the number be m.
Then, twice of m will be similar to 2 times m or m multiplied by 2.
Therefore, twice m will be 2m.
Now, the question is saying that 6 more than twice m.
“6 more than” means 6 is added to that specific number.
Hence, 6 more than that number which is twice of m will be 2m+6.
Hence, “6 more than twice m” can be written in the mathematical form as 2m+6.
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How many integers are there between 100 and 999?
898 integers in that range excluding 100 and 999
There are 900 integers between 100 and 999, inclusive.
Numbers between 999 and 100 inclusive: 999 - 99 = 900
If you want to know the number of integers between 100 and 999, but not including either 100 or 999, then there are 898 integers in that range.
What is an integers number:
An integer, which can comprise both positive and negative integers, including zero, is a number without a decimal or fractional portion. Integers include things like -5, 0, 1, 5, 8, 97, and 3,043. Z is a set of integers that consists of the following:
Positive Numbers: If a number is greater than zero, it is considered positive. For instance, 1, 2, 3, etc.
Negative numbers are those that are less than zero. For instance: 1, 2, 3, etc.
Zero is described as being neither a positive nor a negative number. A whole number, that is.
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What is the slope y =- 5x 3?
Parallel lines are two straight lines that never cross. For this to happen, the lines have to have the same slope.
In this case, the equation given is in what's called "slope-intersect" form. The general form of the slope intercept formula is y=mx + b where m is the slope of the line and b is the point on the Y-axis where the line crosses.
The equation given is y=5x+3 and since the 5 is in the place of m from the general equation, 5 is the slope. Since parallel lines have the same slope, 5 is the answer.
Slope-Intercept Form:
When we're given a linear equation we can immediately see if it's in the slope-intercept form if it follows the format y=mx+b. As long as y does not have a coefficient the slope and y-intercept are the values of m and b.
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How do you draw a SSS triangle?
We need to draw all sides of the triangle equal in order to make a SSS triangle because we know that the SSS triangle has all 3 sides equal.
What is the SSS triangle condition?Two triangles are said to be congruent if all three corresponding sides and all three corresponding angles have the same size.
We u can move, flip, twist, and turn these triangles to produce the same effect.
When relocated, they are parallel to one another.
Two triangles are deemed similar by the SSS test for triangle resemblance if their third sides are proportionate to one another.
Therefore, we need to draw all sides of the triangle equal in order to make a SSS triangle because we know that the SSS triangle has all 3 sides equal.
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How old are Grade 7 students?
Students are making lemonade from a p
owdered lemon drink mix. Rahul mixes 14 cups of water and 5 cups of powdered lemon mix. Aisha mixes 11 cups of water and 4 cups of powdered lemon mix. Use Rahul and Aisha’s percent of powdered lemon mix to determine whose mix will be more lemony.
Answer:
Aisha mix will be more lemony
Step-by-step explanation:
Find Rahul's percentage of the powdered lemon mix by
Finding the total number of cups
14 + 5 = 19 total cups
Then take
5 divided by 19, times 100% = 26.3%
Find Aisha's percentage of the powdered lemon mix by
Finding the total number of cups
11 + 4 = 15 total cups
Then take
4 divided by 15, times 100% = 26.6% or 27%
Answer: Ariana uses 85% drink mix, and Andres uses 90% drink mix. Therefore, Andres's drink will taste more lemony.
Step-by-step explanation:
Ariana uses 17 cups of mix out of the 20 total cups which is 85%
Andres uses 10 cups of mix out of the 11 total cups which is around 90%
Step-by-step explanation:
A car travels 210 miles in 3 hours and 45 minutes. How many miles does it travel per hour?
Answer: 56 miles per hour
Step-by-step explanation:
45 minutes = 0.75 hour
3 hours and 45 minutes = 3.75 hours
Takes 210 divided by 3.75 = 56 miles per hour
What is the final step in solving the inequality 2/5 4x 6x 4?
The final step is x < 3
What is an inequality?
In mathematics, "inequality" refers to a relationship between two expressions or values that are not equal to one another.
Less than (<), greater than (>), less than or equal (≤), greater than or equal (≥), and not equal (≠) are the five inequality symbols.
Given inequality is
-2(5 - 4x) < 6x - 4
The inequality sign is less than.
Applying distributive property on the left side
-10 + 8x < 6x -4
Subtract 6x from both sides
-10 + 8x - 6x < 6x -6x - 4
-10 + 2x < - 4
Add 10 on both sides:
-10 + 2x + 10 < - 4 + 10
2x < 6
Divide both sides by 2:
2x/3 < 6/3
x < 3
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If PB = 6, what is SB?
Answer:
SB = 4
Step-by-step explanation:
The figure is the intersection of the medians of this triangle. The medians of a triangle bisect the sides of the triangle. The intersection of the medians is called the centroid. In this case, the centroid of this figure would be S.
In fact, the distance from a vertex to the centroid of the triangle is 2/3 of the distance from said vertex to the vertex's opposite side.
Create the equation SB = 2/3(PB).
.....................................SB = 2/3(6)
....................................SB = 12/3
...................................∴SB = 4
Answer:
SB = 4
Step-by-step explanation:
SB = 2/3 BP
SB = 2/3 (6)
SB = 4