Answers
The solutions of the trigonometric equation are π / 3, π / 2, 3π / 2 and 5π / 3.
How to solve a trigonometric equation
In this problem we need to find the solutions of a trigonometric equation, which can be done by means of algebra properties and trigonometric formulas. Now we proceed to present the entire procedure:
Step 1 - Introduce the trigonometric equation:
cos 2x - cos x = - 1
Step 2 - Use the trigonometric formula cos 2x = cos² x - sin ² x:
cos² x - sin ² x - cos x = - 1
Step 3 - Use fundamental trigonometric formula:
cos² x - 1 + cos² x - cos x = - 1
2 · cos² x - cos x = 0
Step 4 - Use algebra properties to factor the expression:
cos x · (2 · cos x - 1) = 0
Step 5 - Find the solutions of the expression:
cos x = 0 → x = {π / 2, 3π / 2}
2 · cos x - 1 = 0
2 · cos x = 1
cos x = 1 / 2 → x = {π / 3, 5π / 3}
The solutions of the trigonometric equation are π / 3, π / 2, 3π / 2 and 5π / 3.
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Related Questions
Solve for x
X+59+80+50=180
Answers
Answer: x= -9
Step-by-step explanation:
Add like terms excluding x in: x+59+80+50=180
= x+189=180
[Transposing 189 to RHS] x=180-189
So x= -9
Fully factorise this expression
Answers
Answer:
3k(7k + 4)
Step-by-step explanation:
1. Find the GCF of all terms in the polynomial
2. Express each term as a product of the GCF and another factor.
3. Use distributive property to factor out GCF
how to solve this and answer
Answers
After simplifying the equation, the chemist used 18 liters of 20% solution, 12 liters of 35% solution and 24 liters of 80% solution.
From the given question,
Let that chemist used x, y, z liters of 20%, 35% and 80% solution respectively.
Then according to the given information,
x+y+z=54..........(1)
As we know that in x liters having 20% acid, in y liters having 35% acid, in z liters having 80% acid and in total mixture having 50% acid.
So the equation should be
x ×20%+y×35%+z×80%=54×50%
On simplifying we get
20x+35y+80z=2700
4x+7y+16z=540...............(2)
As given that the number of 80% solution used in 2 times the number of liters of 35% solution used.
So z=2y
Now putting the value of z in equation (1)
x+y+2y=54
x+3y=54
Subtract 3y on both side
x+3y−3y=54−3y
x=54−3y
Now putting the value of x and z in equation (2)
4(54−3y)+7y+16×2y=540
Simplifying
216−12y+7y+32y=540
216+27y=540
Subtract 216 on both side
216+27y−216=540−216
27y=324
Divide by 27 on both side
27/27 y=324/27
y=12
Now putting the value of y in x=54−3y to find the value of x
x=54−3×12
x=54−36
x=18
Now putting the value of y in z=2y to find the value of z
z=2×12
z=24
Hence, the chemist used 18 liters of 20% solution, 12 liters of 35% solution and 24 liters of 80% solution.
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Find the slope and the y-intercept of the line.
y=4x-7
Answers
Answer:
slope: 4
Y intercept: -7
Step-by-step explanation:
slope intercept form is y=Mx+b
m- slope
b- y intercept
Looking at this equation where m is 4 is so the slope is 4
And where b is -7 is so the y intercept is -7
Hopes this helps please mark brainliest
Find the equation of the linear function represented by the table below in slope-intercept form.
Answers
Answer:
y = x + 2
Step-by-step explanation:
Use any two points.
For example, let's use (-3, -1) and (1, 3).
First, we find the slope.
slope = m = (-1 - 3)/(-3 - 1) = -4/(-4) = 1
We have slope = m = 1.
Now we use the equation:
y = mx + b
We plug in the slope we found above.
y = 1x + b
y = x + b
Now we use one of the given points to find b.
We'll use point (1, 3).
3 = 1 + b
b = 2
Now that we know m and b, we write the equation:
y = x + 2
Identify the variation as a direct, inverse, joint, or combined.
xy/z=c
Answers
The given equation is a combined variation.
We have been given an equation is
[tex]\frac{xy}{z}[/tex]
This is an equation in which we have three variables and one constant.
This can not be the case of direct or inverse variation because in direct or inverse variation we have only two variables. Whereas, here we have three variables.
So, Direct and inverse options are discarded.
So, either It can be joint or combined
In joint variation both the variables are directly proportional
Hence, given equation is not joint because if we rewrite the equation we will get [tex]y = \frac{cz}{x}[/tex] both the variables are not directly proportional.
In combined, one variable is directly proportional and the other one is inversely proportional.
From [tex]y = \frac{cz}{x}[/tex] we can see that z is directly proportional and x is inversely proportional.
Hence the answer is, the given equation is combined variation.
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Rewrite each expression without using absolute value bars.
|3r-15| if r<5
Answers
Rewriting the expression |3r-15| without using absolute value bars gives
3r < 15 for r < 5
How to write the expression without using the absolute value barsAbsolute value do not give negative values it gives only the positive value
The given expression is |3r-15|
|3r - 15| < 0
|3r| < |-15|
3r < 15
If r < 5 then say 4
|3r - 15|
|3 * 4 - 15|
|12 - 15|
|-3| = 3
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Two Tofurkys will serve six people. How many people will fourteen Tofurkys serve?
Answers
Graph the polynomial
1- g(x) = (x+5)^4
2- f(x) = 7- (x) ^4.
3- h(x) = 1/4 ( x-3 )^4
Don't forget to find x-int. and y-int.
Answers
Answer:
The x-intercept is the point where the graph crosses the x-axis; it can be found by setting y = 0 and solving for x.
The y-intercept is the point where the graph crosses the y-axis; it can be found by setting x = 0 and solving for y.
1) g(x) = (x + 5)⁴
y-intercept: g(x) = (0 + 5)⁴ → (0, 625)x-intercept: 0 = (x + 5)⁴ → (-5, 0)2) f(x) = 7 - x⁴
y-intercept: f(x) = 7 - 0⁴ → (0, 7)x-intercept: 0 = 7 - x⁴ → (±[tex]\sqrt[4]{7}[/tex], 0)3) h(x) = ¼(x - 3)⁴
y-intercept: h(x) = ¼(0 - 3)⁴ → (0, [tex]\frac{81}{4}[/tex])x-intercept: 0 = ¼(x - 3)⁴ → (3, 0)The graphs are labeled below.
help please and thankyou6
Answers
In triangle ABC, the value of angle A is 60°.
What is a triangle?A triangle is a geometric figure with three edges, three angles and three vertices. It is a basic figure in geometry.
The sum of the angles of a triangle is always 180°
In the triangle ABC,
b=5cm, c=10cm
And by using Pythagoreans theorem,
a²=c²-b²
a²=10²-5²
a²=75
Use cosine rule to determine the angle A,
cosA=b²+c²-a²/2bc
cosA=25+100-75/2×5×10
cosA=50/100
cosA=1/2
cosA=cos60
⇒A=60°
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help please
.............
Answers
Answer:
x ≈ 41.7 ° (a)
Step-by-step explanation:
b² = a² + c² - 2ac × cosβ
~~~~~~~~~~~
25² = 32² + 37² - 2(32)(37)cosx
625 = 2,393 - 2,368 cosx
cosx = [tex]\frac{-2393+625}{-2368}[/tex] ≈ 0.74662162
x ≈ 41.7 ° (a)
A new pair of Jordans are going to drop in a couple weeks and you know you just have to have them. It looks like the/il
be expensive, so you decide to start babysitting your cousin to make the money. You already have $42 saved up and
you will make $6 for every hour of babysitting. The shoes will cost at least $90.
a) What’s the initial cost
Answers
Answer:
incomplete question..please complete the question to get the answer. there are some missing details
If h(x) = 3x² – 5x – 2, find h(2)
Answers
Answer:
h(2) = 0
Step-by-step explanation:
1) 2 * 2 = 4
2) 4 * 3 = 12
3) 5 * 2 = 10
4) 12 - 10 - 2 = 0
Find the prime factorization of
the number 68.
Order the factors from least to greatest.
Remember, 1 is not a prime number.
[?] × [×]
Enter the number that belongs in the green box.
Answers
The prime factorization of the number 68 is 2×2×17 and least prime number is 2 and greatest prime number is 17.
What is Number system?A number system is defined as a system of writing to express numbers.
We need to find the prime factorization of the number 68
Prime factorization is a process of writing all numbers as a product of primes.
The prime factorization of 68 is 2×2×17
The prime factors of 68 are 2 and 17.
The least prime number is 2 and greatest prime number is 17.
Hence the prime factorization of the number 68 is 2×2×17 and least prime number is 2 and greatest prime number is 17.
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(w+1)°
91 ° algebraic
Answers
The value of w in Angle (w + 1)° is 88
Given,
Angle (w + 1)°
Angle 91°
We have to find the value of w.
Here,
Linear pair of angles;
When two lines cross at one point, a linear pair of angles is created. If the angles follow the intersection of the two lines in a straight line, they are said to be linear. A linear pair's total angles are always equal to 180°.
Then,
Here the given angles are linear pair.
So,
(w + 1)° + 91 = 180
w + 1 + 91 = 180
w + 92 = 180
w = 180 - 92
w = 88
Then,
Angle (w + 1)° = 88 + 1 = 89°
Therefore,
The value of w in Angle (w + 1)° is 88
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4. The first three terms of an infinite geometric sequence are m- 1,6, m + 8.
(a)write down two expressions for r.
(b) the find two possible values of m.
(c) Hence, find two possible values of (r)
Answers
For the given infinite geometric sequence,
(a) the two expressions for r are [tex]r=\frac{6}{m-1}[/tex] and [tex]r= \frac{m+8}{6}[/tex]
(b) The two possible values of m are -8 and 5.
(c) The two possible values of r are [tex]-\frac{2}{3}[/tex] and [tex]\frac{3}{2}[/tex]
What are Geometric Series?
A geometric sequence's finite or infinite terms are added together to form a geometric series. The geometric series that corresponds to the geometric sequence a, [tex]a,ar, ar^{2} ,..., ar^{n-1} ,...[/tex] is [tex]a+ar+ar^{2}+...+ar^{n-1} +...[/tex] Clearly, "series" means "sum." The phrase "geometric series" refers specifically to the total of words with a common ratio between every adjacent pair of them. Finite and infinite geometric series are both possible.What is the Common Ratio of an Infinite Geometric Sequence?
The common ratio (r) of a geometric sequence or series is defined as the ratio between the two consecutive terms of it.The common ratio of an infinite geometric sequence, [tex]a_{0} , a_{1} ,a_{2} ,a_{3} ,...[/tex] is given as [tex]r=\frac{a_{1} }{a_{0} } =\frac{a_{2}}{1}[/tex]In the question, the first three terms of an infinite geometric sequence are given as, m-1, 6, m+8
[tex]\implies a_{0} =m-1, a_{1}=6,[/tex] and [tex]a_{2}=m+8[/tex]
So, the common ratio is calculated as,
[tex]r=\frac{6}{m-1}=\frac{m+8}{6}[/tex]
Here, there are two expressions,
[tex]r=\frac{6}{m-1}[/tex] ------(1), and [tex]r= \frac{m+8}{6}[/tex] ------(2)
Simplifying (1), we get
[tex]r(m-1)=6[/tex] -----(3)
Simplifying (2), we get
[tex]6r=m+8\\\implies r= \frac{m}{6}+\frac{8}{6}[/tex] ------(4)
Now, substituting the values of (4) in (3), we get
[tex](\frac{m}{6}+\frac{8}{6})(m-1)=6\\\implies \frac{m^{2} }{6}-\frac{m}{6} +\frac{2m}{3}-\frac{2}{3} =6\\\implies m^{2}+3m-4=36\\[/tex]
Solving further, we get
[tex]\implies m^{2}+3m-40=0\\\implies m=5, m=-8[/tex]
So, when [tex]m=5, r=\frac{6}{5-1}=\frac{3}{2}[/tex]
Also, when [tex]m=-8, r=\frac{6}{-8-1}=-\frac{2}{3}[/tex]
Therefore, (a) the two expressions for r are [tex]r=\frac{6}{m-1}[/tex] and [tex]r= \frac{m+8}{6}[/tex]
(b) The two possible values of m are -8 and 5.
(c) The two possible values of r are [tex]-\frac{2}{3}[/tex] and [tex]\frac{3}{2}[/tex]
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I NEED A ANSWER RN PLS!!
A cylindrical jar has a radius of 6 inches and a height of 10inches. The jar is filled with marbles that have a volume of 20 in3. Use 3.14 for pi. Show work. Complete sentences.
a. What is the volume of the jar?
b. How many whole marbles can fit in the jar? *Each marble has the volume of 20 cubic inches
Answers
The volume of the jar is 1130.4 cubic inches and the number of marbles that can fit is 56.
What is a cylinder?In geometry, it is defined as a three-dimensional shape having two circular shapes at a distance called the height of the cylinder.
We know the volume of the cylinder is given by:
[tex]\rm V = \pi r^2 h[/tex]
It is given that:
A cylindrical jar has a radius of 6 inches and a height of 10inches.
The volume:
V = π(6)²(10)
V = (3.14)(6)²(10)
V = 1130.4 cubic inches
Number of marbles = 1130.4/20 = 56.52 = 56 marbles
Thus, the volume of the jar is 1130.4 cubic inches and the number of marbles that can fit is 56.
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this was due the 7th i didn't even relize i had this due!!! pls help
Answers
The function for motorboat's distance from the shore is y = -4x + 50.
A function is a formula, rule, or law that specifies how one variable (the independent variable) and another variable are related. The linear equation is of the form Ax+By+C=0.
Where A and B are the coefficient and C is the constant.
After 9 minutes, boat is 14 km from the shore, so finding the rate of change,
Rate of Change [tex]=\frac{14-50}{9}\\[/tex]
[tex]=\frac{-36}{9}\\ =-4[/tex]
So, the rate of change of distance is 4 km / minutes.
Now, finding the function,
The function for the distance of motorboat from the shore will be,
y = -4x + 50
Where y will be the final distance and x is the minutes.
Therefore, the function for motorboat's distance from the shore is y = -4x + 50.
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PLEASE HELP!!!
for points
Complete each row of the table.
Answers
The Hiking club will be 0.6.
The completion for the row of the table will be:
Grade 7
Dance club: 0.25
Hiking club: 0.75
Total : 1
Grade 8:
Dance club: 0.40
Hiking club: 0.60
Total : 1
How to calculate the values?
The dance club for grade 7 will be calculated as:
= Number of dancers / Total number
= 5/20
= 0.25
The hiking club will be:
= 15/20
= 0.75
The dance club for grade 8 will be calculated as:
= Number of dancers / Total number
= 20/50
= 0.4
The hiking club will be:
= 30/50
= 0.6
Hence, The Hiking club will be 0.6.
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what is the probability of rolling a yahtzee on the first round? round your answer to 4 decimal places. enter your answer into variable
Answers
The probability of rolling a yahtzee on the first round is calculated to be 6/46656 or 1/7776
Given that,
A dice is rolled for one time .
To find : The probability that a yahtzee will be rolled on the first round.
Your first die will be one of the 6 possible results 6/6 times
That will match on the second die 1/6th of the time.
The third, fourth, and fifth dice all share this characteristic.
Multiplying the all probabilities that the first die will be a yahtzee which is
P( yahtzee will be rolled on the first round) = Product of all possible outcomes of each time a dice is rolled.
6/6*1/6*1/6*1/6*1/6*1/6=6/46656 or 1/7776
Therefore, probability that a yahtzee will be rolled on the first round is 1/7776.
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Given h(x)=-x+5h(x)=−x+5, find h(-3)h(−3).
Answers
The expression we get from the given functions h for all real numbers x.
h(-3)= 8
Given that,
All real numbers x are defined by the functions h.
h(x)= -x+5
We have to find the expression
h(-3)
The core concept of mathematics' calculus is functions. The unique varieties of relations are the functions. In mathematics, a function is represented as a rule that produces a distinct result for each input x. In mathematics, a function is indicated by a mapping or transformation.
Take the functions,
h(x)= -x+5
Now, Take
h(-3) =-(-3)+5 = 3+5= 8
Therefore, The expression we get from the given functions h defined for all real numbers x.
h(-3)=8
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Determine the equation of the polynomial, f(x), of minimum degree whose graph is shown above. Write your answer in factored form.
f(x)=____
Answers
Answer:
[tex]f(x)=\frac{5}{6}(x+2)^2(x-1)(x-3)[/tex]
Step-by-step explanation:
The root -2 has a multiplicity of 2, and corresponds to a factor of [tex](x+2)^2[/tex].
The root 1 has a multiplicity of 1, and corresponds to a factor of [tex](x-1)[/tex].
The root 3 has a multiplicity of 1, and corresponds to a factor of [tex](x-3)[/tex].
So, [tex]f(x)=a(x+2)^2(x-1)(x-3)[/tex].
Since [tex]f(0)=10[/tex],
[tex]10=a(0+2)^2(0-1)(0-3) \\ \\ 10=12a \\ \\ a=\frac{5}{6} \\ \\ \therefore f(x)=\frac{5}{6}(x+2)^2(x-1)(x-3)[/tex]
1. George plays basketball in a week-long camp. On day 3, he scored 10 points. On day 6, he scored 16 points. Explain how to measure his average rate of change.
PLEASE HELP WILL GIVE BRAINLIEST
Answers
Answer:
The average rate of change is 2
Step-by-step explanation:
First set divide your data into 2 sets of separate data:
Independent Data= Day 3
Dependent Data = 10 points
Independent Data= Day 6
Dependent Data = 16 points
Then subtract your two independent data: 6-3=3
Then subtract your two dependent data: 16-10=6
Next divide: difference in dependent/ difference in independent
6/3=2
So, your rate of change is 2.
Consider function f.
f(x) = 2² - 2 + 6
Which statement is true about the parabola modeled by function ?
O A.
OB.
O C.
O D.
The parabola has a minimum value of 5.75.
The parabola has a maximum value of 5.75.
The parabola has a maximum value of 0.5.
The parabola has a minimum value of 0.5.
Answers
The parabola has a minimum value of 5.75.
From the question, we have
f(x)=x^2-x+6
a=1, b=-1, c=6
Δ=b^2-4ac
=1-24
=-23<0
So f(x)=0 has no solution.
x=-b/2a=-(-1)/2=1/2
f(1/2)=1/4-1/2+6
=23/4 is the minimum
=5.75
The parabola has a minimum value of 5.75.
Multiplication:
Mathematicians use multiplication to calculate the product of two or more numbers. It is a fundamental operation in mathematics that is frequently utilized in everyday life. When we need to combine groups of similar sizes, we utilize multiplication. The fundamental concept of repeatedly adding the same number is represented by the process of multiplication. The results of multiplying two or more numbers are known as the product of those numbers, and the factors that are multiplied are referred to as the factors. Repeated addition of the same number is made easier by multiplying the numbers.
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PLEASE HELP ME HERE ITS DUE TOMMOROW! I'LL GIVE U 30 POINTS FOR THIS!!!
Answers
Answer:
1.9
2.3
3.6
4.5
5.20
Step-by-step explanation:
hope it helps
A spherical balloon has a 8-in. diameter when it is fully inflated. Half of the air is let out of the balloon. Assume that the balloon remains a sphere.
a. Find the volume of the fully-inflated balloon.
b. Find the volume of the half-inflated balloon.
c. What is the radius of the half-inflated balloon?
Answers
The volume of the fully inflated balloon is 267.95 cubic inches, the volume of half-inflated balloon is 133.97 cubic inches and it's radius is 3.17in
Volume of a SphereThe volume of sphere is the measure of space that can be occupied by a sphere. If we draw a circle on a sheet of paper, take a circular disc, paste a string along its diameter and rotate it along the string. This gives us the shape of a sphere.
The formula of a sphere is given as
v = 4/3πr³
But diameter = 2 * radius
radius = diameter / 2
radius = 8/2
radius = 4in
The volume of the fully inflated balloon will be
v = 4/3πr³
v = 4/3 * 3.14 * 4³
v = 267.95in³
The volume of the fully inflated balloon is 267.95in³
b) The volume of the half-inflated balloon will half the size of the current volume
v = 267.95in³ / 2 = 133.97in³
c) v = 4/3πr³
133.97 = 4/3 * 3.14 * r³
r = 3.17in
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Mrs.Galicia bakes and then sells cookies and brownies from her bakery. Her revenue for each baked good (in
dollars) is modeled by the equations, where x is the number of days since the bake goods started to sell.
Brownies: M (d) = 2x² + 8x - 4
Cookies: R(d) = 2x + 4
(a) Solve the system algebraically. After how many days is the revenue for each item the same? Show
your work and explain your answer.
(b) Graph this system to show both solutions. Label each axis appropriately.
Answers
a) After 1 day, the revenue for each item is the same.
b) Graph is shown in the attachment.
What is an equation?
An equation is a mathematical statement that proves two mathematical expressions are equal in algebra, and this is how it is most commonly used. In the equation 3x + 5 = 14, for instance, the two expressions 3x + 5 and 14 are separated.
M(d) = 2x² + 8x - 4
R(d) = 2x + 4
a) Find x by taking y=M(d)=R(d). ( same revenue=y)
y= 2x² + 8x - 4
y= 2x + 4
This shows that right-side expressions are equal.
2x² + 8x - 4 = 2x + 4
2x² + 8x - 4 - 2x - 4=0
2x² +6x-8=0
Find the factors
2(x-1)(x+4)=0
x-1=0, x+4=0
x= 1, x=-4
consider positive value for days
So, after 1 day, the revenue for each item is the same
b) The equations:
y= 2x² + 8x - 4
y= 2x + 4
By taking some random values for x and finding the corresponding y values, we get the ordered pairs in the form of (x,y).
By plotting and joining those points, we get the graph as shown in the attachment.
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Sam can paint a house in 5 hours. Gary can do it in 3 hours. How long will it take the two working together?
Answers
It takes them approximately 1.8 hours to paint the house together
Given,
Sam can paint a house in 5 hrs.
and, Gary can do it same work in 3 hrs.
To find the how long it will take the two working together?
Sam can paint a house in 5 hours. This means that Sam's unit rate of working is 1/5
Gary can paint the same house in 3 hours.
This means that Gary's unit rate of working is 1/3
If they work together, it would take them lesser time because they are working simultaneously. This means that their unit rates are additive. Combining their unit rates, it becomes
1/5 + 1/3 = 8/15
Assuming that if they work together, it will take them t hours. Therefore, Their unit rate of working will be 1/t.
Therefore,
8/15 = 1/t
8t = 15
t = 15/8 = 1.875
Hence, It takes them approximately 1.8 hours to paint the house together.
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Element X is a radioactive isotope such that every 11 years, its mass decreases by half. Given that the initial mass of a sample of Element X is 70 grams, how long would it be until the mass of the sample reached 14 grams, to the nearest tenth of a year?
Answers
The time taken for 14g of the radioactive isotope to remain is 25.70 years.
What is the time taken?We know that the half life is the time that it would take for us to have only half of the number of the original radioactive isotopes in the sample still remaining in the sample.
We have the following information;
Half life of the sample = 11 years
Amount of the sample initially present = 70 grams
Amount of the sample after time t = 14 grams
Given that;
N/No = (1/2)t/[tex]t_{\frac{1}{2} }[/tex]
N = amount of radioatve isotpe at time t
No = amount of radioactive isotope initially present
t = time taken
[tex]t_{\frac{1}{2} }[/tex] = half life
14/70 = (1/2)^t/11
0.2 = (1/2)^t/11
ln 0.2 = t/11 ln0.5
t/11 = ln 0.2/ ln0.5
t = 11 * ln 0.2/ ln0.5
t = 11 * (-1.61/-0.69)
t = 25.70 years
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Answer: 2.5
Step-by-step explanation:
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R’S’T’ is the image of RST under a dilation through point C. RS=5 and R’S’=8
Answers
The scale factor that was used in the dilation is 1.6.
What is dilation?A change called dilation lets you resize an object. Dilation is a method for enlarging or contracting objects. The result of this alteration is a picture with the exact same shape as the original. The form, however, has a size difference.
Now,
Scale factor = Dimension of the new shape/Dimension of the original shape
We have △R′S′T′ is the image of △RST under a dilation through point C. Also, given that RS=5 and R′S′=8.
So, the scale factor is,
Scale factor = Dimension of the new shape/Dimension of the original shape
Scale factor = R′S′ / RS
= 8/5
= 1.6
Hence, the scale factor that was used in the dilation is 1.6.
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