Answers
Answer:
[tex]\cos(\alpha +\beta)=\frac{2}{3}(1-\frac{\sqrt{5}}{5})[/tex]
Step-by-step explanation:
Let the hypotenuse of the smaller triangle be h units.
Then; from the Pythagoras Theorem.
[tex]h^2=4^2+2^2[/tex]
[tex]h^2=16+4[/tex]
[tex]h^2=20[/tex]
[tex]h=\sqrt{20}[/tex]
[tex]h=2\sqrt{5}[/tex]
From the smaller triangle;
[tex]\cos (\alpha)=\frac{4}{2\sqrt{5} }=\frac{2}{\sqrt{5} }[/tex] and [tex]\sin(\alpha)=\frac{2}{2\sqrt{5} }=\frac{1}{\sqrt{5} }[/tex]
From the second triangle, let the other other shorter leg of the second triangle be s units.
Then;
[tex]s^2+4^2=6^2[/tex]
[tex]s^2+16=36[/tex]
[tex]s^2=36-16[/tex]
[tex]s^2=20[/tex]
[tex]s=\sqrt{20}[/tex]
[tex]s=2\sqrt{5}[/tex]
[tex]\cos(\beta)=\frac{2\sqrt{5} }{6}=\frac{\sqrt{5} }{3}[/tex]
and
[tex]\sin(\beta)=\frac{4}{6}=\frac{2}{3}[/tex]
We now use the double angle property;
[tex]\cos(\alpha +\beta)=\cos(\alpha)\cos(\beta) -\sin(\alpha)\sin(\beta)[/tex]
we plug in the values to obtain;
[tex]\cos(\alpha +\beta)=\frac{2}{\sqrt{5} }\times \frac{\sqrt{5} }{3}-\frac{1}{\sqrt{5} }\times \frac{2}{3}[/tex]
[tex]\cos(\alpha +\beta)=\frac{2}{3}(1-\frac{\sqrt{5}}{5})[/tex]
Answer:
[tex]cos(\alpha + \beta)=cos(68.4\°) \approx 0.37[/tex]
Step-by-step explanation:
Little triangle.We know both legs, we can use the tangent trigonometric reason to find the angle.
[tex]tan\alpha =\frac{2}{4}\\ tan \alpha=\frac{1}{2}\\ \alpha=tan^{-1}(\frac{1}{2} )\\ \alpha \approx 26.6\°[/tex]
Larger triangle.We know the hypothenuse and the opposite leg. We can use the sin trigonometric reason to find the angle
[tex]sin\beta =\frac{4}{6}\\ sin\beta=\frac{2}{3}\\ \beta=sin^{-1} (\frac{2}{3} )\\\beta= 41.8\°[/tex]
So, the sum of them is
[tex]\alpha + \beta = 26.6+41.8= 68.4\°[/tex]
Then,
[tex]cos(\alpha + \beta)=cos(68.4\°) \approx 0.37[/tex]
Therefore,
[tex]cos(\alpha + \beta)=cos(68.4\°) \approx 0.37[/tex]
Related Questions
The following equation describes a circle.
2x^2 - 20x + 2y^2 + 8y =40
What are the center and radius of the circle?
Answers
Based on the given equation of the circle, the center of the circle is (5, -2), and the radius is 7.
How to determine the center and radius of the circle
To determine the center and radius of the circle described by the equation rewrite the equation in the standard form of a circle equation, which is
[tex](x - h)^2 + (y - k)^2 = r^2.[/tex]
First, complete the square for both the x and y terms:
[tex]2x^2 - 20x + 2y^2 + 8y = 40[/tex]
To complete the square for the x-terms, factor out a 2 from the [tex]x^2[/tex] and x terms:
[tex]2(x^2 - 10x) + 2y^2 + 8y = 40[/tex]
Now, add and subtract the square of half the coefficient of x, which is [tex](-10/2)^2[/tex] = 25:
[tex]2(x^2 - 10x + 25 - 25) + 2y^2 + 8y = 40[/tex]
Simplifying inside the parentheses:
[tex]2((x - 5)^2 - 25) + 2y^2 + 8y = 40[/tex]
Similarly, for the y-terms, add and subtract the square of half the coefficient of y, which is
[tex](8/2)^2[/tex] = 16:
[tex]2((x - 5)^2 - 25) + 2(y^2 + 4y + 4 - 4) = 40[/tex]
Simplifying inside the parentheses:
Now, let's distribute the 2 on both terms:
[tex]2(x - 5)^2 - 50 + 2(y + 2)^2 - 8 = 40\\2(x - 5)^2 + 2(y + 2)^2 - 58 = 40[/tex]
Rearrange the equation:
[tex]2(x - 5)^2 + 2(y + 2)^2 = 98[/tex]
Divide both sides by 2 to simplify:
[tex](x - 5)^2 + (y + 2)^2 = 49[/tex]
Now the equation is in the standard form of a circle equation.
Comparing it to [tex](x - h)^2 + (y - k)^2 = r^2[/tex], we can see that the center of the circle is (h, k) = (5, -2), and the radius squared is r^2 = 49.
Take the square root of both sides
r = 7.
Therefore, the center of the circle is (5, -2), and the radius is 7.
Plzzzzzzzzzzz Help me
Answers
-3 < 0 Because -3 is less than 0.
Hope this helps ;P
Answer:
-3 > 3
Step-by-step explanation:
If you look at the number line, a negative number is just the positive number upside down. But a negative number is always greater than the positive number for your number line.
Point P (1, 7) is located on circle C. The center of circle C is (-2, 3). What is the radius of the circle?
Answers
Answer:
Radius = 5 units
Step-by-step explanation:
We can use the formula of getting magnitude of a line joining two points;
(x1, y1) and (x2, y2)
Length = √((x2-x1)² + (y2-y1)²)
Therefore;
In this case the radius will be; PC
Length of PC = √((3-7)² + (-2-1)²)
= √((-4)² + (-3)²)
= √(16 +9)
= √ 25
= 5 units
Thus the radius is 5 units
Proof by mathematical induction
Answers
Answer:
See below.
Step-by-step explanation:
I will assume that 3n is the last term.
First let n = k, then:
Sum ( k terms) = 7k^2 + 3k
Now, the sum of k+1 terms = 7k^2 + 3k + (k+1) th term
= 7k^2 + 3k + 14(k + 1) - 4
= 7k^2 + 17k + 10
Now 7(k + 1)^2 = 7k^2 +14 k + 7 so
7k^2 + 17k + 10
= 7(k + 1)^2 + 3k + 3
= 7(k + 1)^2 + 3(k + 1)
Which is the formula for the Sum of k terms with the k replaced by k + 1.
Therefore we can say if the sum formula is true for k terms then it is also true for (k + 1) terms.
But the formula is true for 1 term because 7(1)^2 + 3(1) = 10 .
So it must also be true for all subsequent( 2,3 etc) terms.
This completes the proof.
Inductive reasoning can include making generalizations based on specific instances, such as predicting that water freezes at 32°F based on past observations. The binomial theorem, a series expansion, is an example that can be proven using mathematical induction.
Explanation:Understanding Inductive Reasoning
Inductive reasoning is a method of reasoning in which the premises seek to supply strong evidence for the truth of the conclusion.
While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.
This form of reasoning allows us to create generalizations based on observations or patterns we have seen.
For instance, when we see that water freezes at 32° Fahrenheit, and we observe a temperature drop to 30°F at night, we can induce that there will be ice in the morning.
Mathematical Induction
Mathematical induction is a specific type of inductive reasoning that is used for proving that a statement is true for all natural numbers. It is a two-step process where we first prove that a base case is true, typically for n=1.
Then, we show that if the statement is true for an arbitrary positive integer n, this implies that it's also true for n+1.
This creates a domino effect, proving the initial statement for all natural numbers.
The binomial theorem is a classic example of series expansion that can be proven using mathematical induction.
The theorem expands powers of binomials in a form that involves summation of terms, and each term is a product of binomial coefficients and the variables raised to appropriate powers.
Series Expansions and Inductive Inferences
Series expansions in mathematics, such as the binomial theorem, are formulated based on patterns observed in specific instances.
These patterns are then generalized to hold for all numbers within the constraints of the theorem. In contrast, weak inductive inferences are those which cannot firmly establish a conclusion due to insufficient or irrelevant evidence.
Two hundred people attended the history museum exhibit on Monday. Twenty-five of the people who attended the exhibit were surveyed at random. Twelve of the surveyed people answered "yes" to the question.
How many of the total number of people who attended the history museum would you expect to answer "yes"?
12
24
48
96
Answers
Answer:
96
Step-by-step explanation:
Total number of people attending the history museum is 200
Surveyed people are 25
Out of the 25, 12 people said yes
Therefore; the fraction which said yes = 12/25
To find the total number of people expected to say yes out of those that attended the history museum;
Fraction of the sampled that said yes × total number of people that attended.
= 12/25 × 200
= 96
200/25=8
8x12=96
Convert 7.8 liters to milliliters.
0.0078 mL
78 mL
780 mL
7,800 mL
Answers
Answer:
7,800mL
Step-by-step explanation:
Answer:
D
Step-by-step explanation:
1 Liter = 1000 mL
7.8 L = x
Cross multiply
7.8 * 1000 = 1*x
x = 7800 mL
D
Por Favor, help!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Using the distributive property, the sum of 44 + 12 can be expressed as 4(11 + 3).
Which math word identifies the number 4 in the rewritten expression?
A) Addend
B) Common Difference
C) Least Common Multiple
D) Greatest Common Factor
How could you use the GCF and the distributive property to express the sum of 63 + 81?
A) 3(21 + 27)
B) 7(9 + 3)
C) 7(9 + 13)
D) 9(7 + 9)
Answers
Answer:
it is a for addend
Step-by-step explanation:
Answer:
D) GCD
D) 9(7 + 9)
Step-by-step explanation:
Given in the question an expression
44 + 12
4(11 + 3)
2 | 44, 12
2 | 22 , 6
| 11 , 3
D) GCF = 2x2 = 4
C) LCM = 2x2x11x 3
= 132
B) The difference between each number in an arithmetic sequence is called common difference.
A) A number which is added to another is called addend.
63 + 81
3 | 63 , 81
3 | 21 , 27
| 7 , 9
GCF = 3x3
= 9
Distributive property = 9(7 , 9)
A spinner has 6 equal sectors labeled A, B, C, D, E and F on it. The spinner is spun twice.
What is the probability of spinning B and then E, given that the spinner shows B on the first spin?
A. 1/2
B. 1/3
C. 1/6
D. 1/36
Answers
Answer: C. [tex]\dfrac{1}{6}[/tex]
Step-by-step explanation:
Given: A spinner has 6 equal sectors labeled A, B, C, D, E and F on it. The spinner is spun twice.
If it is given that the spinner shows B on the first spin, then the total outcomes will be{ (B,A), (B,B), (B,C), (B,D), (B,E), (B,F)}=6
The number of favorable outcome = 1
Now, the probability of spinning B and then E, given that the spinner shows B on the first spin=[tex]\dfrac{1}{6}[/tex]
Answer: C. 1/6
Step-by-step explanation:
If f(x) = x2 – 2x and g(x) = 6x + 4, for which value of x does (f + g)(x) = 0?
Answers
Answer:
x = - 2
Step-by-step explanation:
f(x) + g(x) = x² - 2x + 6x + 4 = x² + 4x + 4, hence
x² + 4x + 4 = 0 ← This is a perfect square in the form
(x + a)² = x² + 2ax + a²
Hence 2a = 4 ⇒ a = 2, thus
(x + 2)² = 0
x+ 2 = 0 ⇒ x = - 2 ( 2 equal roots )
Value of x = - 2
Answer:
x = - 2
Step-by-step explanation:
the length of a rectangle is 9 units more than the width of the rectangle, w. if the width of the rectangle is more than 20 units, what equation could be used to find the possible values of the width?
Answers
Answer:
[tex]W>20[/tex]
Step-by-step explanation:
Let
L-----> the length of a rectangle
W---> the width of a rectangle
we know that
[tex]L=W+9[/tex] -----> equation A
[tex]W>20[/tex] -----> inequality B
The inequality B represent the equation that could be used to find the possible values of the width
The solution for the width of the rectangle is all real numbers greater than 20 units
which of the following are right triangle congruence theorems
Answers
Answer:
The correct answer options are A, B, C and D.
Step-by-step explanation:
A. Leg-angle (LA):
According to this theorem, if the leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent.
B. Hypotenuse-leg (HL):
If any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.
C. Hypotenuse-angle (HA):
If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of another right triangle, then the two triangles are congruent.
D. Leg-leg (LL):
If any two right triangles that have a congruent hypotenuse and a corresponding, congruent leg are congruent triangles.
Rebecca can make 180 cupcakes at her bakery each day. Currently, she has orders for more than 540 cupcakes.
1)
Which inequality represents how many days it will take Rebecca to complete her orders?
A) x > 3
B) x < 3
C) x ≥ 3
D) x ≤ 3
2)
Which graph shows the solution set for the inequality that represents how many days it will take Rebecca to complete her orders?
A) A
B) B
C) C
D) D
* Hover over answer image to enlarge
Answers
Answer:
(1) It is a 180x3=540. 0x3=0 3x8=24 send the tens on top of the 1 and add 1+2 tens and you get 540! I hope this is helpful
Step-by-step explanation:
Prolly A
1) The correct inequality which represent number of days it will take Rebecca to complete her order is,
⇒ x > 3
2) The correct graph is shown in image.
What is Inequality?A relation by which we can compare two or more mathematical expression is called an inequality.
Given that;
Rebecca can make 180 cupcakes at her bakery each day. Currently, she has orders for more than 540 cupcakes.
Let number of days is, x
Now, We can formulate;
⇒ 180x > 540
⇒ x > 3
Hence, 1) The correct inequality which represent number of days it will take Rebecca to complete her order is,
⇒ x > 3
And, The correct graph of x > 3 is shown in image.
Learn more about the inequality visit:
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A bag contains 10 counters. 6 of them are white. a counter is taken at random and not replaced. A second counter is taken out of the bag at random. Calculate the probability that only one of the two counters are white.
PLEASE HELP!!
Answers
Step-by-step explanation:
Given:
There are 10 counters.
6 of the counters are white.
Two counters are drawn one after another without replacement.
To Find:
The probability that only one of the two counters is white
Explanation:
P(Only one of the two counters is white)
= P(white) x P( not white) + P(not white) x p(white)
Solution:
P(Only one of the two counters is white) = 6/10 x 4/9 + 4/10 x 6/9
P(Only one of the two counters is white) = 4/15 + 4/15
P(Only one of the two counters is white) = 8/15
Answer: The probability is 8/15
The probability that only one of the two counters is white will be [tex]8\div15[/tex] and this can be determined by using the probability formula.
Given :
A bag contains 10 counters, 6 of them are white. The counter is taken at random and not replaced.The probability that only one of the counters is white will be:
[tex]\rm =(probability\; of \;white\; counter) \times (probability\; of\; not \;a\; white \;counter) + (probability \;of\; not\; white\;p counter )\times (probability\; of\; white \;counter)[/tex]
[tex]= \left(\dfrac{6}{10}\times \dfrac{4}{9}\right)+\left (\dfrac{4}{10}\times \dfrac{6}{9}\right )[/tex]
[tex]= \left(\dfrac{12}{45}\right)+\left (\dfrac{4}{15}\right )[/tex]
[tex]=\dfrac{8}{15}[/tex]
Therefore, the probability that only one of the two counters is white will be[tex]8\div15[/tex].
For more information, refer to the link given below:
https://brainly.com/question/23017717
i need help with this question
Answers
Answer:
2
Step-by-step explanation:
I don't think you are asked to prove that these triangles are similar. The diagram leads you to believe that they are. The easiest way to solve this is to take 2 corresponding lines and compare them.
So how do you get from FE to BA
BA / FE = 6/3
BA = 6
FE = 3
BA / FE = 6/3
BA / FE = 2
The scale factor is 2
combine like terms to create an equivalent expression 6(1/2w-3/4)
Answers
The answer and work is the attached sheet!
Hope I helped~! :)
Answer:
3w - 4 1/2
Step-by-step explanation:
Expand the brackets.
6*1/2w - 6*3/4
6/2w - 18/4
3w - 18/4
3w - 4 1/2
Can somebody help me answer this plz?
Answers
It’s D because clock wise is right then you enlarge it to fit the new triangle
Answer:
Its the last choice.
Step-by-step explanation:
That would be a 90 degree clockwise rotation about the origin, followed by an enlargement of factor 3.
Can I get help on this please
Answers
Answer:
B
Step-by-step explanation:
Angles g and h are supplementary, which means they add up to 180 degrees.
So the equation would go g + h = 180.
Substitute 117 for g to get
117 + h = 180
Subtract 117 from each side.
h = 63.
The answer is B
Don’t trust me on it because I sorta use google
Solve x 11, 13, 21,13
Answers
Answer:
12
Step-by-step explanation:
n the proportion 1 over z equals fraction numerator begin display style 4 over 5 end style over denominator 8 end fraction which number is equal to z in the proportion?
Answers
Answer:
[tex]z=10[/tex]
Step-by-step explanation:
we have
[tex]\frac{1}{z}=\frac{(4/5)}{8}[/tex]
solve for z
[tex]z(\frac{4}{5})=8[/tex]
[tex]z=8*5/4[/tex]
[tex]z=10[/tex]
Help please and thank you
Answers
Answer:
C
Step-by-step explanation:
how i figured this out was not by using the table, but you can use the table
i just asked myself what two numbers multiplied together has a product of -12, one number must me negative
the two numbers must also add up to -4, meaning the bigger number had to be negative
the only answer that worked was 2 and -6
A 128 ounce container of orange juice cost $5.12 at the local supermarket. What price should the supermarket sell 96 ounce container of orange juice so that the unit rate of the container is the same?
Answers
Answer:
Step-by-step explanation:
To figure out the unit rate, divide 5.12 by 128. You should get 0.04, which is the unit rate or cost per ounce. To figure out how much the 96 oz can should be, multiply $0.04 times 96.
Your answer should be $3.84.
Final answer:
To maintain the same unit rate as the 128 ounce container of orange juice that costs $5.12, the supermarket should price the 96 ounce container at $3.84.
Explanation:
The student is asking how to calculate the price for a 96 ounce container of orange juice, given that a 128 ounce container costs $5.12, with the condition that the unit rate remains the same. First, we need to determine the unit rate for the orange juice in the 128 ounce container. To do that, divide the total cost by the number of ounces to find the cost per ounce:
$5.12 \/ 128 ext{ounces} = $0.04 ext{per ounce}
Now that we have the unit rate ($0.04 per ounce), we can calculate the cost for a 96 ounce container by multiplying the unit rate by the number of ounces:
96 ext{ounces} \times $0.04 ext{per ounce} = $3.84
Therefore, the supermarket should sell the 96 ounce container of orange juice for $3.84 to keep the unit rate the same as the 128 ounce container.
HURRY!!!!!!!!!!!!!Which equation represents g(x)?
Answers
Answer: I think its C but im not positive!
Step-by-step explanation:
Answer:
Option D. is the correct option
Step-by-step explanation:
The graph of g(x) is the translation of y = ∛x, which passes through origin (0, 0).
The graph in the figure is showing a shift of 1.5 units downwards.
Therefore, the new translated function g(x) will be represented by
g(x) = ∛x - 1.5
This graph can create a confusion of being shifted on x-axis but when we plug in the values of x in the options A and B, parent graph gets changed.
Therefore, option D. is the answer.
in a family of 4 children, what is the probability of having all girls, given that the oldest child is a girl
Answers
1 in 16
The probability of having all boys is 1 in 16
The probability of three boys and 1 girl is 4 in 16
The probability of two boys and two girls is 6 in 16
The probability of one boy and three girls is 4 in 16
The probability of four girls is 1 in 16
PLZ HELP 20 POINTS
Point P (−3, −4) is reflected across the x axis.
What are the coordinates of its image after this transformation?
P' (-3,4)
P' (3,-4)
P' (-4,3)
P' (4,-3)
Answers
Answer:
P' (-3,4)
Step-by-step explanation:
Reflected across the x axis so only y changes (opposite) but x will stay the same
If Point P (−3, −4) is reflected across the x axis then P' will be (-3, 4)
Answer:
P' (-3,4)
Answer:
The answer is (3, -4)
Step-by-step explanation:
Match the description to the best example so that all terms have a matching example.
Answers
Answer:
I'm pretty sure a set a decimal fractions is {1/4, 1/2, 3/4}
I think a set of whole numbers is {3, 4, 5.}
A set of integers might be {-2, -1, 0, +1.}
I'm am sure a set of natural numbers is {1, 2, 3.}
The set if rational numbers is {0.1, 0.2, 0.3} I think.
I am not that sure if my answers are right or not so sorry if this is incorrect.
Answer:
1-e
2-b
3-c
4-q
5-d
Step-by-step explanation:
In this question, we have to match the left hand side to corresponding right hand side. On the right side we are given
a){3,4,5}
b){0,1,2,3}
c){-2,-1,0,1}
d){[tex]\frac{1}{4}[/tex], [tex]\frac{1}{2}[/tex], [tex]\frac{3}{4}[/tex]}
e){0.1, 0.2, 0.3}
1) a set of decimal numbers:
It is clear the the set of decimal numbers is e) {0.1, 0.2, 0.3}
2) a set of whole numbers is b) {0,1,2,3}
3) a set of integers is c) {-2,-1,0,1}
4) a set of natural numbers is a) {3,4,5}
5) a set of rational numbers is d) {[tex]\frac{1}{4}[/tex], [tex]\frac{1}{2}[/tex], [tex]\frac{3}{4}[/tex]}
6= a/4 +2
two step equations
Answers
Answer:
a=16
Step-by-step explanation:
We want to take the 6 on the left side of the equal sign and do the equation in the reverse.
First since we are adding 2, we want to substract 2 away from 6.
6-2 = 4
Then, since we are dividing by 4, we want to multiply by 4.
4*4 = 16
We can check this to make sure it's correct by replacing a with 16:
16/4 = 4
4+2 = 6
The solution of the expression is,
⇒ a = 16
We have to given that;
Expression is,
⇒ 6 = 2 + a/4
Now, WE can simplify as;
⇒ 6 = 2 + a/4
⇒ 6 - 2 = a/4
⇒ 4 = a/4
⇒ a / 4 = 4
⇒ a = 4 × 4
⇒ a = 16
Thus, The solution of the expression is,
⇒ a = 16
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Julie has the three segments shown in the graph. Can she make a triangle with these segments.
A) Yes, the segments meet the triangle inequality theorem.
B) No, the segments do not meet the triangle inequality theorem.
C) No, the segments do not meet the pythagorean theorem.
D) There is not enough information to tell.
Answers
Answer:
D
Step-by-step explanation:
Answer:
ppl the answers A
Step-by-step explanation:
According to the triangle inequality if a, b, and c are lengths of the legs of a triangle then a + b > c. If you measure the segments, their measurements are 5, 5, and
10
and 5 + 5 >
10
. Therefore, Yes, the segments do meet the triangle inequality.
Plz Hellllllllllp meeeeee
Answers
Answer:
The answer would be 8 cubic inches
Step-by-step explanation:
Answer: The answer is 14 Cubic In.
Step-by-step explanation: There is 10 Cubic In. in the bottom + the vertical 4 Cubic In. = 14 Cubic In.
At their annual car wash, the Wfms boys soccer team washes 30 cars in 45 minutes. At this rate, how many cars will they wash in one hour?
Answers
Answer:
Cars in one hour: 40 cars
Step-by-step explanation:
30 cars ---> 45 min
Xc ----------> 60 min
Xc = [(60min)*(30 cars)]/45 min
Xc = 40
Best regards
In the length of each leg of an isosceles triangle is 13 and the base is 24, the length of the altitude to the base is
Answers
Answer:
altitude = 5Step-by-step explanation:
Look at the picture.
Use the Pythagorean theorem:
[tex]a^2+12^2=13^2[/tex]
[tex]a^2+144=169[/tex] subtract 144 from both sides
[tex]a^2=25\to a=\sqrt{25}\\\\a=5[/tex]