The number of mice needed is given as follows:
a) 10 snakes: 23 mice.
b) 6 snakes: 15 mice.
c) x snakes: (3 + 2x) mice.
How to define the linear function?The slope-intercept definition of a linear function is given as follows:
y = mx + b.
In which:
m is the slope, representing the rate of change.b is the intercept, representing the value of y when x = 0.For every feeding, she needs three mice and an additional two mice for each snake, hence the slope and the intercept of the function are given as follows:
m = 2, b = 3.
The function is then defined as follows:
y = 2x + 3.
For 10 snakes, the number of mice is given as follows:
y = 2(10) + 3
y = 23 mice.
For 6 snakes, the number of mice is given as follows:
y = 2(6) + 3
y = 15 mice.
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Max organized a community toy drive for the holidays. He asked each family to donate a total of 2 gifts. Eighteen families participated in the toy drive. Write and solve an equation that represents how many gifts were donated.
g + 2 = 18; g = 16 gifts
g − 2 = 18; g = 20 gifts
g divided by 18 equals 2; g = 36 gifts
2g = 18; g = 9 gifts
Answer: 36
Step-by-step explanation:
g divided by 18 equals 2; g = 36 gifts.
The correct option is (3).
What is Arithmetic?The fundamental ideas in number theory, measurement, and calculation are frequently referred to as "arithmetic" (which is derived from the Greek word "arithmos," "number"). (that is, the processes of addition, subtraction, multiplication, division, raising to powers, and extraction of roots).
As per the given data:
A community toy drive for holidays was organized my Max.
Each family has to donate a total of 2 gifts.
Total number of families were equal to 18.
Let's assume the total number of gifts from all the families is g.
For finding the equation to find the total number of gifts given by all the families:
g = Gifts given by each family × Number of families
g = 18 × 2
∴ g divided by 18 equals 2; g = 36 gifts.
Hence, g divided by 18 equals 2; g = 36 gifts.
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Select all of the ways to describe the number 182.
Answer choices
A rational number
B integer
C whole number
D none of these are correct
Answer:
A , B and C
Step-by-step explanation:
182 is a rational number as it can be written in p/q form. (182/1)
182 is a positive integer. {Integers: ........ -4 , -3 , -2 , -1 , 0 , 1 , 2 , 3 .......}
182 is a whole number. {0, 1 , 2 , 3 ,...............}
If the numerator of a fraction is multiplied by 4 and the denominator is reduced by 2,the result is 4 . If the numerator of the fraction is increased by 15 and 2 subtracted from the double of the denominator, the result is 5/4. Find the fraction.
The fraction is given as follows:
8.3/10.3.
How to obtain the fraction?The fraction is modeled as follows:
x/y.
In which:
x is the numerator.y is the denominator.From the features of the problem, a system of equations is built to identify the variables.
If the numerator of a fraction is multiplied by 4 and the denominator is reduced by 2,the result is 4, hence the first equation is of:
4x/(y - 2) = 4.
Hence:
4x = 4(y - 2)
4x = 4y - 8
x = y - 2.
If the numerator of the fraction is increased by 15 and 2 subtracted from the double of the denominator, the result is 5/4, hence the second equation is of:
(x + 15)/(2y - 2) = 5/4.
We know that x = y - 2, hence the value of y is obtained as follows:
(y + 13)/(2y - 2) = 5/4.
5(2y - 2) = 4(y + 13)
10y - 10 = 4y + 52
6y = 62
y = 62/6
y = 10.3.
Then the value of x is of:
x = y - 2
x = 10.3 - 2
x = 8.3.
Hence the fraction is:
8.3/10.3.
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Evaluate. x =2 and y=-1
Can someone help please,I also need explanation of not much to ask
Answer:
= 16.
Step-by-step explanation:
x= 2
y = -1
[tex] {(2xy)}^{2} \\ = {(2(2)( - 1)) \\ }^{2} \\ = {(2( - 2)) \\ }^{2} \\ = {( - 4)}^{2} \\ = - 4 \times - 4 \\ = 16[/tex]
7. The ratio of dahlias to sunflowers is the same in all of the
flower arrangements at a banquet.
A. MTR Use Structure Complete the double number line to
show the ratios of dahlias to sunflowers in all the flower
arrangements at the banquet.
12
Dahlias 0
4
Sunflowers 0
8
16
6
a
15 18
B. How many dahlias will there be when there are 33 sunflowers?
C. What is the ratio of dahlias to all flowers when there are 21 total
flowers?
D. What is the ratio of sunflowers to all flowers in a bouquet with
70 total flowers?
20 24
12
21
32
8 dahlias
24
6 sunflo
7A) The completion of the number line, based on the ratio of Dahlias to Sunflowers, is as follows:
Dahilas 0 4 8 12 16 10 24 28 32
Sunflowers 0 3 6 9 12 15 18 21 24
7B) When there are 33 sunflowers, there will be 44 Dahlias.
7C) The ratio of Dahlias to all flowers when there are 21 total flowers is 0.57:1 or 12:21.
7D) The ratio of Sunflowers to all flowers in a bouquet with 70 total flowers is 0.43:1 or 3:7.
What is the ratio?The ratio is the relative size of one value compared to another.
The ratio is determined as the quotient or average of two relative values.
The number of Dahlias = 8
The number of Sunflowers = 6
Ratio of Dahlias to Sunflowers = 1.33:1 (8:6)
Ratio of Sunflowers to Dahlias = 0.75: 1 (6:8)
The sum of ratios = 14 (8 + 6)
B) When there are 33 sunflowers, there will be 44 Dahlias (33/0.75).
C) The ratio of Dahlias to all flowers:
The total flowers = 21
Dahlias = 12 (21 x 8/14)
The Ratio of Dahlias to all flowers = 12:21 = 0.57:1
The ratio of dahlias to Sunflowers = 12:9
D) The ratio of Sunflowers to all flowers:
The total flowers = 70
Sunflowers = 30 (70 x 6/14)
The ratio of Sunflowers to all flowers = 30:70 = 0.43:1
Ratio of sunflowers to dahlias = 30:40 = 3:4
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What is an important first step when building a theory?
\
Overall, building a theory is a complex process that requires careful planning, attention to detail, and a willingness to remain open to alternative explanations and interpretations.
An important first step when building a theory is to formulate a clear and testable research question or hypothesis. A research question or hypothesis is a statement that suggests a relationship between two or more variables, and it guides the development of the theory. The research question or hypothesis should be specific, measurable, and falsifiable, meaning that it can be tested and potentially proven false through empirical research. Once a clear research question or hypothesis has been formulated, the researcher can then begin to gather and analyze data to support or refute the theory.
Another important first step when building a theory is to conduct a comprehensive review of existing literature and research in the field. This review helps the researcher to identify gaps or inconsistencies in current knowledge and to develop a clear understanding of the state of the field. This step can also help the researcher to refine their research question or hypothesis, as well as to identify potential variables and relationships that may be relevant to the theory.
Another important step when building a theory is to select an appropriate research design and methodology that will allow for the systematic collection and analysis of data. The research design should be aligned with the research question or hypothesis, and should enable the researcher to collect high-quality data that can be analyzed using appropriate statistical or qualitative methods.
In addition, it is important to be mindful of potential biases that can impact the development of a theory, such as confirmation bias or researcher bias. It is important for the researcher to be objective and to consider all possible explanations for their data, rather than solely seeking out evidence that supports their hypothesis or preconceived ideas.
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Priovde the missing reasons for the proof
The missing reason is ∠ ABC ≅ ∠ DBE by vertically opposite angles postulate and Δ ABC ~ Δ DBE by AA rule.
What are vertically opposite angles?Vertically opposite angles are angles that are opposite one another at a specific vertex and are created by two straight intersecting lines. Vertically opposite angles are equal to each other. These are sometimes called vertical angles.
Given that, ∠ C ≅ ∠ E, and we need to find a reason to prove Δ ABC ~ Δ DBE,
As we know that, to prove two triangles similar we can show two corresponding angles congruent in them, then they will be similar by AA rule,
Here, we already have, ∠ C ≅ ∠ E, we need one more,
If we see the figure, we get ∠ ABC and ∠ DBE are vertically opposite angles
We know that, Vertically opposite angles are equal to each other.
Therefore, ∠ ABC ≅ ∠ DBE
Hence, the missing reason is ∠ ABC ≅ ∠ DBE by vertically opposite angles postulate and Δ ABC ~ Δ DBE by AA rule.
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Is the number 1 prime? Explain why or why not.
Please explain why.
1 can only be divided by one other integer except1, hence it is not a prime number.
The reason why 1 is not a prime number.Any natural number higher than 1 that is not the sum of two smaller natural numbers is referred to be a prime number. A composite number is a natural number greater than one that is not prime.
Given this definition, 1 is not a prime number because it can only be divided by one other integer, which is 1 itself.
Based on the explanation above, we can then conclude that 1 is not a prime number.
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.please help!!!!!!!!
Answer:
226.19 cubic units
Step-by-step explanation:
radius = r = 6 units
height = h = 6 units.
To find the volume of resulting solid, subtract the volume of cone from the volume of hemisphere.
Volume of the resulting solid = Volume of hemisphere - volume of cone
[tex]\sf \bf = \dfrac{2}{3}\pi r^3 - \dfrac{1}{3}\pi r^2h\\\\\\=\dfrac{2}{3}*3.14*6*6*6 - \dfrac{1}{3}*3.14*6*6*6\\\\\\=(3.14*6*6*6)(\dfrac{2}{3}-\dfrac{1}{3})\\\\=3.14*6*6*6**\dfrac{1}{3}\\\\=3.14*6*6*2\\\\=226.19 \ cubic \ units[/tex]
How many times is 7 a factor in 2023!
Answer:
The Prime Factors and Pair Factors of 2023 are 7 × 17 and (1, 2023), (7, 289), (17, 119) respectively.
Answer:
To determine how many times 7 is a factor in 2023:
1. Divide 2023 by 7 to obtain the quotient: 2023/7 = 288.571.
2. If the result of the division is a whole number, 7 is a factor in 2023.
3. Round down 288.571 to obtain the integer 288.
4. Since the result of the division is a whole number, 7 is a factor in 2023 and occurs 1 time.
How much is 10 mL in teaspoons or tablespoons?
10 mL is equal to 2 teaspoons or 0.67 tablespoons. One teaspoon is equal to 5 mL, so to convert 10 mL to teaspoons, divide 10 by 5. One tablespoon is equal to 15 mL, so to convert 10 mL to tablespoons, divide 10 by 15.
10 mL is equal to 2 teaspoons (tsp) or 0.67 tablespoons (tbsp).
One teaspoon is equal to 5 mL, so to convert 10 mL to teaspoons, you can divide 10 by 5, which gives:
10 mL / 5 mL/tsp = 2 tsp
Therefore, 10 mL is equal to 2 teaspoons.
One tablespoon is equal to 15 mL, so to convert 10 mL to tablespoons, you can divide 10 by 15, which gives:
10 mL / 15 mL/tbsp = 0.67 tbsp
Therefore, 10 mL is equal to approximately 0.67 tablespoons.
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types as many equations
basically draw 1 linear line with domain constraints for the first two
y = 1/2x + 6 {7<x < 13}
y = 1/2x + 3 {x<7}
the values might have to be tweaked a little as the stars are not exactly on point
how do i do this please
If FG is parallel to KL , then the ratio of FJ to KJ must be equal to GJ to JL. Therefore FG is not parallel to KL
What are similar triangles?Two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling, possibly with additional translation, rotation and reflection.
The ratio of corresponding sides of Similar triangles are equal.
This means that 32/72 must be equal to 48/64
32/72 = 8/18 = 4/9
48/64 = 12/16 = 3/4
Since the ratio of the corresponding sides are not
equal ∆FGJ is not similar to JKL and thus FG is not parallel to KL
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What’s the value of x?
Answer:
The value of [tex]x[/tex] is [tex]55\textdegree[/tex].
Step-by-step explanation:
A straight line is equal to 180 degrees.
So any angles that forms on the same side of a straight line always add to 180 degrees.
We are given an angle of [tex]130\textdegree[/tex] on the exterior of the triangle.
Lets call the angle next to that [tex]\angle B[/tex].
We can conclude that
[tex]\angle B+130=180[/tex]
[tex]\angle B=180-130[/tex]
[tex]\angle B=50[/tex]
The sum of the interior angles in a triangle is 180 degrees. This is because a triangle is a polygon with three sides and three angles, and the sum of the angles in any polygon can be found by using the formula:
sum of interior angles = (n - 2) x 180 degrees
Note: n is the number of sides.
A triangle has 3 sides. So substituting 3 in for n gives us
(3 - 2) x 180 degrees = 180 degrees
So the sum of the interior angles in a triangle is always 180 degrees.
[tex]x\textdegree=180\textdegree-75\textdegree-50\textdegree\\x\textdegree=55\textdegree[/tex]
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Find & particular solution Yp of the following equation using the Method of Undetermined Coefficients. Primes denote the derivatives with respect to x
y" -y' - 2y = 8x + 5 A particular solution is Yp(x) = ?
(a) The particular solution, y_p is 7
(b) y_p is -4x
(c) y_p is -4x + 7
(d) y_p is 8x + (7/2)
To find a particular solution to a differential equation by inspection - is to assume a trial function that looks like the non-homogeneous part of the differential equation.
(a) Given y'' + 2y = 14.
Because the no non-homogeneous part of the differential equation, 14 is a constant, our trial function will be a constant too.
Let A be our trial function:
We need our trial differential equation y''_p + 2y_p = 14
Now, we differentiate y_p = A twice, to obtain y'_p and y''_p that will be substituted into the differential equation.
y'_p = 0
y''_p = 0
Substitution into the trial differential equation, we have.
0 + 2A = 14
A = 6/2 = 7
Therefore, the particular solution, y_p = A is 7
(b) y'' + 2y = −8x
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = -8x
2Ax + 2B = -8x
By inspection,
2B = 0 => B = 0
2A = -8 => A = -8/2 = -4
The particular solution y_p = Ax + B
is -4x
(c) y'' + 2y = −8x + 14
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = -8x + 14
2Ax + 2B = -8x + 14
By inspection,
2B = 14 => B = 14/2 = 7
2A = -8 => A = -8/2 = -4
The particular solution y_p = Ax + B
is -4x + 7
(d) Find a particular solution of y'' + 2y = 16x + 7
Let y_p = Ax + B
y'_p = A
y''_p = 0
0 + 2(Ax + B) = 16x + 7
2Ax + 2B = 16x + 7
By inspection,
2B = 7 => B = 7/2
2A = 16 => A = 16/2 = 8
The particular solution y_p = Ax + B
is 8x + (7/2)
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The cost of tickets to the amusement park is $19.50 for 1 ticket and $78 for 4 tickets
What is the slope
Is it positive negative zero or undefined
The slope is 1/19.5 and it is positive slope.
What is slope of a line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
The formula to find the slope of a line is slope = (y₂-y₁)/(x₂-x₁).
Given that, the cost of tickets to the amusement park is $19.50 for 1 ticket and $78 for 4 tickets.
Here, the coordinates are (19.50, 1) and (78, 4)
Now, the slope = (4-1)/(78-19.50)
= 3/58.5
= 1/19.5
Therefore, the slope is 1/19.5 and it is positive slope.
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If one object has twice as much mass as another object it also has twice as much.
If one object has twice as much mass as another object it also has twice as much Inertia.
Let the mass of object A= m₁
mass of object B = m₂
Given that m₁ = 2m₂
Speed, velocity and acceleration are subject to motion of the object. Given the insufficiency of information in the question, nothing can be said about these.
This is basically Newton's first law of motion, this law states that, every object continues in a state of rest of uniform motion in a straight line, unless acted upon by an external force
On the other hand, inertia is directly dependent on the mass. Hence, inertia of object A would be twice the inertia of object B.
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Gregory is six years younger than Janet. 7 years ago, Janet was thrice as old as Gregory. How old are they know?
The ages of Gregory and Janet are 10 and 16 years respectively.
How to find there ages?Gregory is six years younger than Janet. 7 years ago, Janet was thrice as old as Gregory. Therefore, the ages of Gregory and Janet are as follows:
Let
x = age of Janet
Gregory age = x - 6
x - 7 = 3(x - 6 - 7)
Hence,
x - 7 = 3(x - 13)
x - 7 = 3x - 39
x - 3x = -39 + 7
-2x = -32
divide both sides by -2
x = -32 / -2
x = 16 years
Therefore,
Janet age = 16 years
Gregory's age = 16 - 6 = 10 years
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Janet's current age is 16 and Gregory's current age is 10
How to determine how old they are know?Let's start by using algebra to represent the information in the problem.
Let's assume that Janet's current age is "J" and Gregory's current age is "G".
From the first sentence, we know that:
G = J - 6
From the second sentence, we know that 7 years ago, Janet was thrice as old as Gregory.
J - 7 = 3(G - 7)
So, we have
J - 7 = 3(J - 6 - 7)
This gives
J - 7 = 3J - 39
So, we have
2J = 32
This gives
J = 16
This also means that
G = 16 - 6
G = 10
Hence, Gregory's age is 10
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please help hurry i need help hurry calculating slope
The slopes for the given pairs of points are:
015/14-2/5-25/21014How to find the slopesTo find the slope of a line passing through two points (x1, y1) and (x2, y2), we use the formula:
slope = (y2 - y1) / (x2 - x1)
Using this formula for each pair of points, we get:
(19, 3), (20, 3)
slope = (3 - 3) / (20 - 19) = 0/1 = 0
(3,0), (-11, -15)
slope = (-15 - 0) / (-11 - 3) = -15 / -14 = 15/14
(19,-2), (-11, 10)
slope = (10 - (-2)) / (-11 - 19) = 12 / (-30) = -2/5
(6, -10), (-15, 15)
slope = (15 - (-10)) / (-15 - 6) = 25 / (-21) = -25/21
(12,-18), (-15, -18)
slope = (-18 - (-18)) / (-15 - 12) = 0 / (-27) = 0
(3,-20), (5, 8)
slope = (8 - (-20)) / (5 - 3) = 28 / 2 = 14
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You have cubes with edge lengths of 1/4 inch, 1/3 inch, and
1/2 inch. If you only use ONE size of cube, which can you use to fill the container? Explain your reasoning.
The size of the cube that can be used to fill the [tex]1\frac{2}{3}[/tex] inches, [tex]\frac{2}{3}[/tex] inch, [tex]2\frac{1}{3}[/tex] inches is the cube with edge length of 1/3 inches
What is a cube?A cube is a three dimensional solid with six congruent square faces.
The expression the height and length dimensions of the rectangular prism as improper fractions, indicates that we get;
Height = (1 2/3) cm = (3 × 1 + 2)/3 cm = 5/3 cm
Length = (2 1/3) cm = (3 × 2 + 1)/3 cm = 7/3 cm
The denominators of the dimensions of the height, length and width of the rectangular prism are equivalent to 3. The edge length of the cube that can be used to fill the container should have a denominator that is a factor of 3.
The size of the cube that can be used to fill the container is therefore the cube with side length of 1/3 inches
The volume of the container = (5/3) in × (2/3) in × (7/3) in = 70/27 in³
The volume of the (1/3) inch cube = (1/3) in ×(1/3) in ×(1/3) in = 1/27 in³
The number of 1/3 inch cubes that fills the container = (70/27)/(1/27) = 70
The possible dimensions of the rectangular prism container in the question obtained from a similar question posted online are;
Height = (1 2/3) centimeters
Width = (2/3) centimeters
Length = (2 1/3) centimeters
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What is the area of this figure?
Please hurry!
Answer: 89 mm^2; Option 1
Step-by-step explanation: Solve the area of the largest rectangle by multiplying 7 and 8. That equals 56. Then do 4 times six to get the area of the medium rectangle and thats 24. Do the same thing with the small square and you get nine. Add them up to get 89.
solve for solutions 18. y = -6x-2
12x + 2y = -6
The solution to the system of equations:
y = -6x - 2
12x + 2y = -6
This is an inconsistent system of equations
What is an Equation ?
An equation is a mathematical term, which indicates that the value of two algebraic expressions are equal. There are various parts of an equation which are, coefficients, variables, constants, terms, operators, expressions, and equal to sign.
For example, 3x+2y=0.
Types of equation
1. Linear Equation
2. Quadratic Equation
3. Cubic Equation
we can substitute the value of y from the first equation into the second equation:
12x + 2(-6x - 2) = -6
12x + -12x - 4 = -6
0x - 4 = -6
-4 = -6
This is an inconsistent system of equations, which means that there are no real solutions that satisfy both equations.
This means that there is no point at which both lines intersect and that the two lines are parallel, never meeting.
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F
G
9x-7
E
17x-2
Solve for x
The solution for x in the circle is x = 12
How to solve for the value of x in the circle?Since the measure of angle EFG is m∠EFG = 9x-7 and the measure of arc GE is 17x-2
We can determine the value of x using the inscribed angle theorem.
Thus, we have:
m∠EFG = 1/2 (arc GE)
9x-7 = 1/2 * (17x-2)
2(9x-7) = 17x - 2
18x - 14 = 17x - 2
18x - 17x = -2 + 14
x = 12
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The image on a movie poster was shrunk to make the DVD cover art for the movie, so that the cover art is a scale image of the poster. The poster is 72 inches wide, and the DVD cover art is 6 inches wide. If the diagonal of the poster is 8 feet, what is the diagonal of the DVD cover art?
A.
8 inches
B.
24 inches
C.
7 inches
D.
0.67 inch
The diagonal of the DVD cover art has a length of 8 inches. (Correct choice: A)
How to use scales to find the length of the diagonal of the DVD cover art
In this question we need to estimate the length of the diagonal of the DVD cover art, this can be used by means of scale factor relative to the width of the movie cover. Scales are an useful resource to represent systems at affordable dimensions. By assuming that both DVD cover art and movie cover have similar geometries, we can estimate the diagonal of the DVD cover art by proportion formula:
6 in / 72 in = x / [(8 ft) · (12 in / ft)]
x = (6 in / 72 in) × 96 in
x = 8 in
The length of the diagonal of the DVD cover art is equal to 8 inches.
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Find the solution to the
system 2x + 3y = 6 and
-2/3x
Answer:
I got 7 as the answer
Step-by-step explanation:
I hope this helps!!
A chef uses a bowl shaped like the one below to prepare her avocado lime sauce. Due to the amount of mixing involved in this recipe she does not want to fill the bowl up entirely to the top. She estimates she can only fill the bowl up to about 2/3 its capacity. What is the volume of contents the chef will be able to fill the bowl with using her estimation?
Answer: DON'T COPY AND PASTE!! a-lot of people say they get a plagiarism charge so try to switch or add more words to this but i hope it helps!!!
Step-by-step explanation:
Claim - The volume of the bowl is 715.17
evidence - The hemisphere has a radius of 8
- so its volume is 2/3πr³ r = radius
= 2/3 × π × (8)³
she can fill = 2/3 × 2/3 × 22/7 × (8)³ cm³
= 715.17cm³
reasoning - Therefore according to my calculations The volume will be 715.17 cm³
mike painted 1/3 of his room using 2/5 of a can of paint. how much paint will it take to paint the whole room
Answer:
He will need 6/5 cans of paint.
The answer could also be written as 1 1/5 cans of paint or 1.2 cans of paint.
Step-by-step explanation:
Since 2/5 filled 1/3 of the room, he will need 3 times that amount.
2/5 * 3/1 = 6/5
In 2010, the population of a city was 88,000. From 2010 to 2015, the population grew by 4%. From 2015 to 2020, it fell by 3%. To the nearest 100 people, what was the population in 2020?
The population in 2020 is 2,240,700
What is population?A population is the set of people concentrated in a particular area.
Given that, in 2010, the population of a city was 88,000, from 2010 to 2015, the population grew by 4%. From 2015 to 2020, it fell by 3%. We need to determine the population in 2020,
For population growing we will multiply the population is 104 and for population reducing we will multiply by 97,
Therefore, the population in 2020 =
88000 × 5(104/100) × 5(97/100)
= 88000 × 5.25 × 4.85
= 2,240,700
Hence, the population in 2020 is 2,240,700
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Raju used a piece of wire to make shape
(p +10)
(p+10) 25 cm
a) what was the original length of the piece of wire?
b) given that p=6 find the perimeter of the figure
Question 3 3.1.1. Express the following as single trigonometry ratio: 3.1.1. Cos2x · Cos3x - Sinax. Sin3x 2 B12. Sinzx Cosx+Cosex Sinx 3.2. Determine the value of the following without using a calcula 3.2.1. Sın 85° Cos25°- Cos 85°. Sin 25° 322 Cos 160°. Cos10° + Sin160°. Sin 10° 44
Using trigonometric identity, the value of the expressions are (Cosx + Cos5x)/2 + Sin3x.Cosx, √3/2, √3/2 and -√3/2
What is the expression of the trigonometric identityTo solve this problem, we need to apply trigonometric identity;
1.
Cos2x · Cos3x - Sinax. Sin3x:
Using the identity Cos(A - B) = CosA.CosB + SinA.SinB, we can write:
Cos2x · Cos3x = (Cos(2x - 3x) + Cos(2x + 3x))/2 = (Cos(-x) + Cos(5x))/2 = (Cosx + Cos5x)/2
So the expression becomes:
(Cosx + Cos5x)/2 - Sinax. Sin3x
Using the identity Sin(A - B) = SinA.CosB - CosA.SinB, we can write:
Sinax. Sin3x = (Sin(ax - 3x) - Sin(ax + 3x))/2 = (Sin(-2x) - Sin(4x))/2 = (-Sin2x - Sin4x)/2
So the expression becomes:
(Cosx + Cos5x)/2 + (Sin2x + Sin4x)/2
Using the identity SinA + SinB = 2.Sin((A+B)/2).Cos((A-B)/2), we get:
(Sin2x + Sin4x)/2 = Sin3x.Cosx
Therefore, the expression simplifies to:
(Cosx + Cos5x)/2 + Sin3x.Cosx
2.
Sinzx Cosx+Cosex Sinx:
Using the identity Sin(A + B) = SinA.CosB + CosA.SinB, we can write:
Sinzx Cosx+Cosex Sinx = Sin(x + z)
So the expression is equivalent to Sin(x + z).
3.
Sın 85° Cos25°- Cos 85°. Sin 25°:
We can use the identity Sin(A - B) = SinA.CosB - CosA.SinB:
Sin 85° Cos25°- Cos 85°. Sin 25° = Sin(85° - 25°) = Sin60° = √3/2
4.
Cos 160°. Cos10° + Sin160°. Sin10°:
We can use the identity Cos(A - B) = CosA.CosB + SinA.SinB:
Cos 160°. Cos10° + Sin160°. Sin10° = Cos(160° - 10°) = Cos150° = -√3/2
Learn more on trigonometric identity here;
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