Solve the equation cos 2 θ = -5 cos θ - 3 on the interval 0 ≤ θ < 2π
Answers must be exact.
Answers
Answer:
[tex]\theta =\dfrac{2}{3} \pi ,\;\; \theta =\dfrac{4}{3} \pi[/tex]
Step-by-step explanation:
Given:
[tex]\cos 2\theta=-5 \cos \theta-3, \quad \quad 0 \leq \theta < 2 \pi[/tex]
[tex]\boxed{\begin{minipage}{6.5 cm}\underline{Cos Double Angle Identity}\\\\$\cos (A \pm B)=\cos A \cos B \mp \sin A \sin B$\\\end{minipage}}[/tex]
Use the cos double angle identity to create a quadratic:
[tex]\begin{aligned}\cos 2\theta &=-5 \cos \theta-3\\\cos^2\theta - \sin^2 \theta &=-5 \cos \theta-3\\\cos^2\theta - (1-\cos^2 \theta) &=-5 \cos \theta-3\\2\cos^2\theta - 1 &=-5 \cos \theta-3\\2\cos^2\theta+5 \cos \theta +2 &=0\end{aligned}[/tex]
Factor the quadratic:
[tex]\begin{aligned}2\cos^2\theta+5 \cos \theta +2 &=0\\2\cos^2\theta+4 \cos \theta +\cos \theta +2 &=0\\2\cos \theta(\cos \theta +2)+1(\cos \theta +2) &=0\\(2\cos \theta+1)(\cos \theta +2)&=0\end{aligned}[/tex]
Apply the zero-product property:
[tex]2 \cos \theta + 1 = 0 \implies \cos \theta =-\dfrac{1}{2}[/tex]
[tex]\cos \theta + 2= 0 \implies \cos \theta =-2[/tex]
As -1 ≤ cos θ ≤ 1, cos θ = -2 is undefined.
Therefore, the only valid solution is cos θ = -¹/₂
Therefore:
[tex]\implies \cos \theta =-\dfrac{1}{2}[/tex]
[tex]\implies \theta =\cos^{-1}\left(-\dfrac{1}{2}\right)[/tex]
[tex]\implies \theta =\dfrac{2}{3} \pi +2\pi n,\;\; \dfrac{4}{3} \pi+2\pi n[/tex]
Solutions in the given interval 0 ≤ θ < 2π :
[tex]\theta =\dfrac{2}{3} \pi ,\;\; \theta =\dfrac{4}{3} \pi[/tex]
Related Questions
You are organizing party invitations. The stamps come in packs of 20. The invitations
come in packs of 15. What is the least number of stamps and invitations you can buy so
you have an even number of stamps and invitations?
Answers
Answer:
255
Step-by-step explanation:
You need the least common multiple (LCD) of 15 and 17.
The prime factorizations of 15 and 17 are
3 × 5
17
Since there are no common factors, the LCM is their product.
15 × 17 = 255
Answer: 255
The following coordinate plane shows three of the vertices of a rectangle. On an 8 by 8 coordinate grid, T is plotted 1 unit right and 4 units up from the origin. P is plotted 6 units right and 2 units up from the origin. Z is plotted 3 units right and 5 units up from the origin. What are the coordinates of Point Z? ( Choose... ) What are the coordinates of Point P? ( Choose... ) Which point has coordinates ( 1 , 4 ) ? Choose... In order to create a rectangle using these points as three of the vertices, what coordinates should be used for the fourth vertex? ( Choose... )
Answers
The coordinates of the points of the rectangle are given as follows:
Point Z: (3,5).Point P: (6,2).Point T is the point that has coordinates given by (1,4).The coordinates of the fourth vertex are given as follows:
(4,1).
What are the coordinates?The notation of an ordered pair is given as follows:
(x-coordinate,y-coordinate).
Point T is plotted 1 unit right and 4 units up from the origin, hence the coordinate are:
(1,4).
Point P is is plotted 6 units right and 2 units up from the origin, hence the coordinates are:
(6,2).
Point Z is plotted 3 units right and 5 units up from the origin, hence the coordinates are:
(3,5).
The equivalent sides on a rectangle are congruent, hence the equivalent sides are:
TZ and PM.
Hence the missing coordinate M is obtained as follows:
1 unit down from vertex P, hence y = 2 - 1 = 1.2 units left from vertex P, 6 - 2 = 4.These distances are because these vertical and horizontal distances also are the vertical and horizontal distances from the equivalent side TZ.
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twelve more than the product of 16 and 3
Answers
Answer:
60
Step-by-step explanation:
16(3)+12
=48+12
=60
Answer:
60
Step-by-step explanation:
I need to find x and y
Answers
Using the same-side interior angles theorem,
[tex]8x+30+6x+24=180 \\ \\ 14x+54=180 \\ \\ 14x=126 \\ \\ \boxed{x=9}[/tex]
Using vertical angles,
[tex]6(9)+24=10y+2 \\ \\ 78=10y+2 \\ \\ 10y=76 \\ \\ \boxed{y=7.6}[/tex]
Find the values of X and Y. State which theorem(s) you used.
Answers
Using alternating and corresponding angle theorem, the value of x and y are (8, 146), (86, 94) and (21, 29.5) respectively
Alternate and Corresponding Angle TheoremAlternate angle theorem states that when two parallel lines are cut by a transversal, then the resulting alternate interior angles or alternate exterior angles are congruent. Corresponding and Alternate angles are the most fundamental angles formed when two parallel lines are intersected by a transversal line.
5)
3x = 5x - 64
Reason: Alternate angles are equal
5x + 3x = 64
8x = 64
x = 8
y + 5x - 6 = 180
Reason: Angles on a straight line are equal to 180
y + 5(8) - 6 = 180
y + 40 - 6 = 180
y + 34 = 180
y = 146
6) x = 86
Reason: Alternating angles are equal to one another
y + 86 = 180
Reason = 94
7) 2y = 59
Reason: Vertical angles are equal
y = 59 / 2
y = 29.5
2y = 3x - 4
Reason: Alternate angles are equal
2(29.5) = 3x - 4
59 = 3x - 4
3x = 59 + 4
3x = 63
x = 21
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24)Jessica wants of make a garland out of 8 roses. 12 daisies and 48 marigolds. How many
arrangements can she make with those flowers so that, there is no flower left & what does they look
like.
Answers
Answer:
4
Step-by-step explanation:
Given: Number of roses = 8, Number of daisies =12 and number of marigolds 48.
Since we have given,
Number of roses = 8 , Number of daisies = 12 and number of marigolds = 48.
The factors of 8 = 2 * 2 * 2
The factors of 12 = 2 * 2 * 3
The factors of 48 = 2 * 2 * 2 * 2 * 3
Therefore, greatest common factor of 8, 12 and 48 is 4.
Therefore, the total number of arrangements = 4
Answer: 4
Write and solve the equation that represents the sentence below.
The product of 18 and a number is 109.8 .
Which statements are correct? Select three options.
Answers
The linear equation we want to write is:
18*x = 109
And the solution is x = 6.06
How to write and solve the equation?Remember that the product between two numbers is just a multiplication A*B.
Here we know that the product of 18 and a number, is 109.
If we define the variable x as the number, then we can write the linear equation:
18*x = 109
Where the product between 18 and x is equal to the number 109.
Now, we want to find the value of x, to do that, we need to divide both sides of the equation by 18, then we will get:
18*x/18 = 109/18
x = 109/18 = 6.06
The solution of the linear equation is 6.06
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Francesca is way behind on email. Over winter break, she resolves to deal with 5 emails per
day.
Write an equation that shows the relationship between the number of days Francesca has
been on winter break, x, and the total number of emails she has dealt with, y.
y =
Submit
Answers
The equation for the total number of emails Francesca has dealt with is; y = 5x.
Explain the term linear equation?A linear equation is just an algebraic expression of the form y=mx+b, where m represents the slope while b is the y-intercept, while only a constant and the first-order (linear) term are included. The variables in the preceding equation are y and x, and it is occasionally referred to as a "linear equations of two variables."For the stated question-
Five emails are handled by Francesca each day.Assume that Francesca has been away on winter break for 'x' days.Let y represent the overall amount of emails she has handled.Then, the linear equation forms is-
y = 5x
Thus, the equation for the total number of emails Francesca has dealt with is; y = 5x.
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Use the information to complete the task.
A family is driving at a constant speed of 60 miles per hour on the second day of a trip. The expressions 60h + 300 and 60(h+ 5) are equivalent.
They represent the total number of miles the family has driven on the trip, where h is the number of hours they have driven on the second day.
Choose "True" or "False" for each statement.
Statement
The value 5 represents the speed the family drove on the first day.
The value 60 represents the miles the family drove on the first day.
The value 300 represents the miles the family drove on the first day.
True or False?
True False
True False
True
False
A
Answers
Answer:
False
False
True
Step-by-step explanation:
Knowing that they are traveling 60 miles per hour, 60h is equivalent to how many miles they are traveling. When adding 300 you know that would be the a1 term going along the rules of sequences.
Therefore 60h is the rate of increase.
300 is the miles traveled on the first day.
With this information in mind we can now answer the True and false statements.
1. 5 represents the hours they drove on the first day, therefore 1 is a false statement.
2. 60 represents the mph they are traveling on the second day, therefore 2 is a false statement.
3. 300 represents the miles they traveled on the first day, therefore 3 is a true statement.
Answer:
false
false
true
the person above is correct :)
NEED HELP WITH GEOMETRY ASAP!!!
Answers
Answer:
LK = 5
LM ≈ 5.83
KM ≈ 7.28
NOT isosceles
Step-by-step explanation:
L(-7,4)
K(-4,8)
[tex]LK=\sqrt{(-7--4)^{2}+(4-8)^{2}}=\sqrt{(-3)^{2}+(-4)^{2}}=\sqrt{9+16}=\sqrt{25}[/tex]
M(-2,1)
[tex]LM=\sqrt{(-7--2)^{2}+(4-1)^{2}}=\sqrt{(-5)^{2}+(3)^{2}}=\sqrt{25+9}=\sqrt{34}[/tex]
[tex]KM=\sqrt{(-4--2)^{2}+(8-1)^{2}}=\sqrt{(-2)^{2}+(7)^{2}}=\sqrt{4+49}=\sqrt{53}[/tex]
If I get $147.00 every two weeks for child support? In a year how much would that equal up to?
Answers
Answer:
Step-by-step explanation:
52 weeks in a year
Divide 52 by 2 = 26 weeks
26 weeks x 147 dollars = 3822 dollars
Thus the answer is 3822 dollars yearly
Answer:$3,528
Step-by-step explanation:
since there are 4 weeks in a month that would mean you would have to multiply 147 by two and that would = to $294.00/month so then you would have to multiply that by 12 so then that would = to $3,528.00 a year.
A card is chosen at random from a standard
deck. What is the probability that the card is
a spade or a king, given that it is black?
Answers
the probability of getting a king or a spade is 30.7%.
What is probability?
Simply put, probability measures how probable something is to occur. The probability formula can be used to calculate the likelihood of an event by simply dividing the favorable number of possibilities by the total number of options. The likelihood of an occurrence might vary from 0 to 1 because the favourable number of outcomes can never be greater than all possible outcomes.
The formula for calculating the probability of an event is
Probability of an event = Favorable outcomes/Total number of outcomes.
Favorable outcomes = 16 {13 spade cards which include one king and the remaining 3 king cards.}
The total number of outcomes = 52.
Probability of choosing a king or a spade= 16/52 = 8/26 = 4/13=0.307
Hence, the probability of getting a king or a spade is 30.7%.
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Con Beads for Everyone Mrs. Harris wants her nineteen students to each have twenty-four beads to make a bracelet. Mrs. Harris finds beads in bags at the store. There are twelve beads in each bag. How many bags of beads does Mrs. Harris need to buy so she has exactly enough beads for her nineteen students? Each bag of beads costs $0.59. How much will Mrs. Harris spend on the beads? Show all your mathematical thinking.
Answers
Answer:
i.She needs 96 bags of beads
ii.She will spend $56.64 on the beads
Step-by-step explanation:
i.19×24=456
12÷456=96
96 bags
ii.96×0.59
=56.64
$56.64
*1/2
*4 to 4
*2:1
*8/4
Answers
Answer:
1/2
Step-by-step explanation:
shaded sections = 4
total sections = 8
4/8
(4:4)/(8:4)
1/2
Answer:
The answer is 4:4.
Step-by-step explanation:
what is -57.17 as a mixed number
Answers
The value of 57.17 as a mixed number is - 57 17/100.
In simple terms, what is a mixed number?A mixed number is said to be in its simplest form if its fractional part's highest common factor, or HCF, is 1. When mixed numbers are simplified, the fraction value remains constant. We can say that the simplified mixed number and the actual mixed number are fractions that are equivalent.
A mixed number is made up of three components: a whole number, a numerator, and a denominator. The numerator and denominator are both components of the proper fraction that results in the mixed number.
In this case, -57.17 will be:
= -57 + (17/100)
= -57 17/100.
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Urgent, please help
I'll mark brainliest
Answers
Answer:
Step-by-step explanation:
The function rule g(x) after the given transformations of the graph of f(x)=4x is g(x) = -1/2x
Transformation techniques
Transformation is the process of changing the position of an object on the xy-plane.
Given the parent function
f(x) = 4x
If the function is not reflected over x-axis, the resulting function will be f(x) = -4x
If the resulting function is compressed by a factors of 1/8, hence the final function rule will be;
g(x) = 1/8(-4x)
g(x) = -1/2x
Hence the function rule g(x) after the given transformations of the graph of f(x)=4x is g(x) = -1/2x
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perimeter of a rectangle is 36 cm . If the length of the rectangle is 12 cm . Find its area.
Answers
Answer: 72 cm
Step-by-step explanation:
So since we know its length is 12 cm, that leaves us with finding the width. You could do that by adding the two length's together to get 24, then subtract that from the perimeter to get 12. The widths being added together must equal 12 and be the same, so we can divide by 2 to get 6 cm as the width. Then multiply 6 times 12 to get 72
Answer: A=72 cm²
Step-by-step explanation:
Let the width of the rectangle be x
Hence, P=36 cm l=12 cm
P=2(l+x)
P=2l+2x
P-2l=2x
Divide both parts of the equation by 2
Thus,
[tex]\displaystyle\\x=\frac{P-2l}{2} \\\\x=\frac{36-2(12)}{2}\\\\x=\frac{36-24}{2} \\\\x=\frac{12}{2} \\\\x=6 cm\\\\[/tex]
Area A=l*x
A=12*6
A=72 cm²
a stick of length 1.75m was measure by a boy as 1.80. find the percentage error in his measurement
Answers
Answer:
2 6\7
Step-by-step explanation:
Solution Error in measurement = 1.8-1.75m =+0.05mPercentage error. = error ÷true value ×100 =+ 0.05÷1.75×100 =+ 5÷1.75 = +20÷7 =+ 2 6\7
In circle M with m ∠ L M N = 106 and LM=4 units, find the length of arc LN. Round to the nearest hundredth.
Answers
The length of arc LN of the circle is 7.40 units rounded to the nearest hundredth.
How to calculate for the length of arc of a circleTo calculate for the length of an arc of a circle, we use 2πr × (θ/360), where r is the radius and θ is the angle formed from the arc.
Thus we evaluate for the length of the arc as follows;
for π = 3.14, r = 4 and θ = 106°
2 × 3.14 × 4 × (106/360)°
25.12 × 0.2944
7.3953
Therefore, the length of the arc 7.3953 and rounded up to 7.40 to the nearest hundredth.
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PLS HELP
During a 90-minute school play, the main character was on stage 80% of the time.
What amount of time in minutes was the main character on stage?
88.9 minutes
112.5 minutes
80 minutes
72 minutes
Answers
Answer:
72 minutes
Step-by-step explanation:
you convert the 80% to decimal form so its now 0.80.
then you multiply how long the school play was (90) times the 'percentage' of how long the main character was on the stage (0.80).
90 times .80 is 72.
hope this helped:)
What is the equation of a line that is parallel to 2x + 3y = 3 and passes through the point (3, -4)?
Enter your answer as a slope intercept equation (y=mx+b) in the box.
Answers
Answer:
y = - [tex]\frac{2}{3}[/tex] x - 2
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + b ( m is the slope and b the y- intercept )
given
2x + 3y = 3 ( subtract 2x from both sides )
3y = - 2x + 3 ( divide through by 3 )
y = - [tex]\frac{2}{3}[/tex] x + 1 ← in slope- intercept form
with slope m = - [tex]\frac{2}{3}[/tex]
• Parallel lines have equal slopes , then
y = - [tex]\frac{2}{3}[/tex] x + b ← is the partial equation
to find b substitute (3, - 4 ) into the partial equation
- 4 = - 2 + b ⇒ b = - 4 + 2 = - 2
y = - [tex]\frac{2}{3}[/tex] x - 2 ← equation of parallel line
Question 2 (1 poing
This table shows the total shipping costs for objects of
different weights.
Shipping Costs
$6.00
$3.00
$1.50
O $1.25
Weight of Object
(in pounds)
4
8
12
16
Total Shipping Cost
(in dollars)
5
11
17
23
Based on the information in the table, by how many dollars
does the total shipping cost increase if the weight of an
object increases by 1 pound?
Answers
Based on the information in the table, the dollars that the total shipping cost will increase if the weight of an object increases by 1 pound is C. $1.50
What is an equation?An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario.
It should be noted that from the information, the table shows the total shipping costs for objects of different weights.
In this case, let the dollar amount be x.
Based on the information, this will be:
4x = 6
Divide
x = 6/4
x = 1.5
In conclusion, the correct option is C.
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HELP ME. I HATE MATH DUDE.
Answers
Given the graph of f(x) above, find the following and write your answers using interval notation (Separate multiple intervals with a comma):
(a) Domain: 7
(b) Range:
(c) Interval(s) on which f(x) is increasing:
(d) Interval(s) on which f(x) is decreasing:
(e) Interval(s) on which f(x) is constant:
(f) Local maxima: 3
(g) Local minima: -5
Answers
Answer:
a) [-9,8)
b) [-5,5]
c) (-4,0), (1,6)
d) [-9,-4), (6,8)
e) [0,1]
f) just the y-value: 5; as a point: (-8,5)
g) just the y-value: -5; as a point: (-4,-5)
Step-by-step explanation:
a) Domain is all of the x-values that are defined in the function. The smallest x-value in the graph is -9, and the largest is 8. And all values in between are defined (have corresponding y-values). But notice that there's an open dot on (8,0).
b) Range is found the same way as Domain, but with the y-values. The smallest y-value of this function is -5, and the largest is 5.
For c-e, notice where the graph changes direction and draw a vertical line from the x-axis through the turning point. These lines are the boundaries between intervals of increasing/decreasing/constant. You should have vertical lines at x=-4, x=0, x=1, and x=6.
c) Interval(s) on which f(x) is increasing: Reading the graph from Left To Right, between which vertical lines is the graph going up?
d) Interval(s) on which f(x) is decreasing: Reading the graph from Left To Right, between which vertical lines is the graph going down?
e) Interval(s) on which f(x) is constant: Reading the graph from Left To Right, between which vertical lines is the graph staying flat?
f) Look for the highest non-infinity point on the graph
g) Look for the lowest non-infinity point on the graph
Which statements are always true regarding the diagram? Select three options.
m∠5 + m∠3 = m∠4
m∠3 + m∠4 + m∠5 = 180°
m∠5 + m∠6 =180°
m∠2 + m∠3 = m∠6
m∠2 + m∠3 + m∠5 = 180°
Answers
The statements that are always true regarding the diagram of interior and exterior angles of a triangle include the following:
C. m∠5 + m∠6 =180°.
D. m∠2 + m∠3 = m∠6.
E. m∠2 + m∠3 + m∠5 = 180°.
What is the Linear Pair Postulate?The Linear Pair Postulate states that the measure of two (2) angles would add up to 180° provided that they both form a linear pair. This ultimately implies that, the measure of the sum of two (2) adjacent angles would be equal to 180º when two (2) parallel lines are cut through by a transversal:
m∠5 + m∠6 =180°.
What is the exterior angle property?The exterior angle property can be defined as a theorem which states that the measure of an exterior angle in a triangle is equal in magnitude to the sum of the measures of the two remote or opposite interior angles of that triangle:
m∠2 + m∠3 = m∠6.
In conclusion, the sum of all the angles formed by a triangle is equal to 180º:
m∠2 + m∠3 + m∠5 = 180°,
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10. Solve 0.1(x+8)-0.5x = 0.2(2x - 6) +5.2. Justify each step.
Answers
x + 4y = 22
4x + y = 13
Solve by elimination
Answers
Using elimination method, the solution of the equation is as follows:
x = 2y = 5How to solve system of equation by elimination method?System of equation can be solved using different method such as elimination method, substitution method and graphical method.
Therefore, let's use elimination method to solve the equation as follows;
x + 4y = 22
4x + y = 13
multiply equation (i) by 4 to eliminate variable x.
Therefore,
4x + 16y = 88
4x + y = 13
subtract equation(ii) from equation(i)
15y = 75
divide both sides by 15
y = 5
x = 22 - 4y
x = 22 - 4(5)
x = 22 - 20
x = 2
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Find the equation of this line.
[?]
y =
x + [ ]
Answers
the slope intercept form is y = mx + b
b is the y-intercept
m is the slope
use rise over run to find the slope
Can somebody help me solve
Answers
Answer:17.5
Step-by-step explanation: trust me
Can someone help me solve these problems?
Answers
Answer:
2. A is false
B is false
C is false
D is true