Recall the triple angle identity for cosine:
cos(3x) = cos³(x) - 3 sin²(x) cos(x)
… = cos³(x) - 3 (1 - cos²(x)) cos(x)
… = 4 cos³(x) - 3 cos(x)
and the definition of secant,
sec(x) = 1/cos(x)
So we have
sec(x) cos(3x) = 0
(4 cos³(x) - 3 cos(x))/cos(x) = 0
cos(x) (4 cos²(x) - 3)/cos(x) = 0
If cos(x) ≠ 0 (this happens at the endpoints of the interval [-π/2, π/2]), we can simplify this to
4 cos²(x) - 3 = 0
cos²(x) = 3/4
cos(x) = ±√3/2
But since -π/2 < x < π/2, we know cos(x) > 0, so we ignore the negative case:
cos(x) = √3/2
==> x = π/6 and x = -π/6
The solution of the given trigonometric equation by using trigonometric identities is [tex]\frac{\pi }{6} \ and \frac{-\pi }{6}[/tex].
What are trigonometry identities?Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.
Some trigonometric identities are[tex]cos(3x) = 4cos^{3}(x) -3cos(x)[/tex]
According to the given question.
We have an equation
[tex]sec(x)cos(3x) = 0[/tex]
Since, the above equation can be written as by using trigonometric identities
[tex]sec(x)cos(3x) = 0\\\implies \frac{1}{cos(x)} (4cos^{3} x-3cos(x))=0[/tex]
Solve the above equation for x.
[tex]\implies 4cos^{2} x -3= 0[/tex]
[tex]\implies 4cos^{2}x = 3\\ \implies cos^{2} x = \frac{3}{4} \\\implies cos x = \sqrt{\frac{3}{4} } \\\implies cos x = \pm\frac{\sqrt{3} }{2}[/tex]
In the given domain [tex][\frac{-\pi }{2}, \frac{\pi }{2} ][/tex] we know that cosx > 0. Therefore, we take only positive part
[tex]\implies cosx = \frac{\sqrt{3} }{2} \\\implies x = cos^{-1} \frac{\sqrt{3} }{2} \\\implies x = \frac{\pi }{6}, and \frac{-\pi }{6}[/tex]
Hence, the solution of the given trigonometric equation by using trigonometric identities is [tex]\frac{\pi }{6} \ and \frac{-\pi }{6}[/tex].
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can u guys help me with this
Answer:
O
Step-by-step explanation:
We can answer this by using the process of elimination
We want to find the position of 0.5
So let's begin
Looking at the number line...
D and E are both in between negative numbers
0.5 is not a negative number so immediately we can eliminate D and E
So it is between O and M
O is between 0 and 1
.5 is between 0 and 1 so O might be the answer
However let's look at M just to be safe.
M is located between 3 and 4.
Both 3 and 4 are greater than .5 meaning that M cannot represent .5
So we can conclude that O is the best answer
what are the slope and the y-intercept of the linear function that is represented by the table?
The length of a rectangle field is represented by the expression 14 X minus 3X squared +2 Y. The width of the field is represented by the expression 5X minus 7X squared plus 7Y. How much greater is the length of the field than the width?
Answer:
[tex]9x+4x^2-5y[/tex]
Step-by-step explanation:
Hi there!
Length of the field: [tex]14x-3x^2+2y[/tex] units
Width of the field: [tex]5x-7x^2+7y[/tex] units
To find how much greater the length of the field is than the width, subtract the width from the length:
[tex]14x-3x^2+2y-(5x-7x^2+7y)[/tex]
Open up the parentheses
[tex]= 14x-3x^2+2y-5x+7x^2-7y[/tex]
Combine like terms
[tex]= 14x-5x-3x^2+7x^2+2y-7y\\= 9x+4x^2-5y[/tex]
Therefore, the length is [tex]9x+4x^2-5y[/tex] units greater than the width.
I hope this helps!
In Exercise 6, suppose that there is a probability of .01 that a digit is incorrectly sent over a communication channel (i.e., that a digit sent as a 1 is received as a 0, or a digit sent as a 0 is received as a 1). Consider a message that consists of exactly 60% 1s. a. What is the proportion of 1s received at the end of the channel
Answer:
The proportion of 1s received at the end of the channel is of 0.598.
Step-by-step explanation:
In this question:
60% of the bits sent are 1, 100 - 60 = 40% of the bits sent are 0.
For each bit receive, 0.01 probability of an error, so 0.99 probability it is correct.
a. What is the proportion of 1s received at the end of the channel?
99% of 60%(sent 1s, received 1).
1% of 40%(sent 0s, received 1). So
[tex]p = 0.99*0.6 + 0.01*0.4 = 0.598[/tex]
The proportion of 1s received at the end of the channel is of 0.598.
identify each of the following as rational or irrational.
Answer:
Step-by-step explanation:
A rational number can be written as a fraction of two integers.
√23 is irrational because we do not know what the square root of 23 is in terms of integers in a fraction.
104.42 is rational because it can be expressed as 10442/100 . We can express it as this because
104.42 * 1 = 104.42 * 100/100 = 10442/100, and multiplying something by 1 keeps it the same
√64 is rational because it is equal to 8, and 8/1 is equal to 8
49.396 with the 6 repeating is rational because we can express 49.39 as
49.39 * 1 = 49.39 * 100/100 = 4939/100, and we are then left with
49.396- 49.39 = 0.006 (with the 6 repeating). A repeating decimal can be expressed as x/9, with the x representing all values before the repetition begins multiplied by 10.
For example, 0.6 with the 6 repeating can be represented as 6/9. This is because 0.6 * 10 = 6, and we divide that by 9. Similarly, 0.006 with the 6 repeating can be represented by 0.06/9
We add the two fractions together to get
4939/100 + 0.06/9
multiply both fractions by the other's denominator to even them out and make the numerator of the second fraction an integer
(4939*9)/(100*9) + (0.06*100)/(9*100)
= 44451/900 + 6/900
= 44460/900
Assuming that the last number has only the digits given, that is rational because, similarly to 104.42, it can be multiplied by a power of 10 to result in (integer)/(integer), making it rational. If the dots represent infinite digits, then it is not rational because there are infinite digits that are not repeating, and there is no way to know the last digit, so it is impossible to write a fraction from it
Find the value of x. Round the length to the nearest tenth.
I donno is there anything else
The first side of a triangle measures 5 in. less than the second side, the third side is 3 in. more than the first side, and the perimeter is 17 in. Set up an equation that relates the sides of the triangles in terms of the perimeter of the triangle.
3s - 10 = 17
3s - 7 = 17
2s -7 = 17
2s - 5 = 17
Answer:
3s-10=17
s=9
3×9-10=17
27-10=17
Part C
Now try this one. Write a description of the partitioned function using known function types, including transformations.
Answer:
Please find the complete question in the attached file.
Step-by-step explanation:
All three functions are defined as follows:
For function 1:
The module on the absolute value function instance example also is known as this function:
[tex]f(x)=|x| \ \left \{ {{x, \ \ x>0} \atop {-x \ \ <0}} \right.[/tex]
This is [tex]f(x)= x, \ \ x<0[/tex] in the following diagram.
For function 2:
It is an algebraic function of in this function
It also is a quadratic polynomial functional with a [tex]y=x^2, \ \ x>0[/tex] value.
For function 3:
The cubic polynomial equation is evaluated for in this function
The value of the graph is:
[tex]y=-x^3 \ \ \ \ and\\\\\to y= f(x)\\\\\to y= -f(x)\\\\[/tex]
a.) What is the measurement of the missing side (c)?
b.) What is the angle of (x)?
Work out the mean for the data set below:
403, 404
Give your answer as a decimal.
Answer:
605.5
Step-by-step explanation:
When you do a frequency table for this question,the zigma F becomes 2 and the zigma fx becomes 1211.Because you will multiply the marks by the frequency to get the zigma fx so after getting them you add it.Which becomes 403+808=1211.the formula for finding the mean too is zigma fx/zigma f: 1211/2=605.5
A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\heartsuit$ and $\diamondsuit$) are red. The deck is randomly arranged. What is the probability that the top card is a 3 and the second card is an eight
Answer:
4 / 663
Step-by-step explanation:
Given that :
Number of cards in a standard deck = 52
Number of 3's in s standard deck = 4 (each suit has one card each)
Number of 8's in a standard deck = 4 (each suit has one card each)
Probability, P = required outcome / Total possible outcomes
Choosing without replacement :
P(top card is a 3) = 4 / 52
P(second draw is 8) = 4 / 51
P(top card is a 3 and second is 8) :.
4/52 * 4/51 = 16 / 2652 = 4 / 663
Question 2.
If 0 is an angle in the first quadrant, and cos(θ) = 0.669, determine the value of
cos(π - θ), cos(π + θ), and cos(4π - θ) using Symmetry.
Answer:
1) -0.669
2) -0.669
3) 0.669
Step-by-step explanation:
Since we are subtracting or adding multipled of pi, we will either obtain 0.669 or -0.669 as our answer for each of the three different questions.
Cosine is the x-coordinate in our orderes pairs. If our point ends up on the right side of the y-axis, the cosine will be positive. If our point ends up on left side, it will be negative.
Choose a thetha (I'm going to choose it in degrees) in the first quadrant to help with a visual.
If theta=70:
1) then 180-70=110 which is in second quadrant, so our cosines will be opposite in value.
2) then 180+70=250 which is in third quadrant, so our cosines will be opposite in value.
3) then 4×180-70=720-70=650 =1(360)+290 which ends up in the 4th quadrant which means the consines will have the same value.
I don't think that there is enough info for this problem, but could someone try to help?
Answer:
12
Step-by-step explanation:
Rectangle : Let length = b & width = a
PQ = QR = 6
LR = QR - PL = 6 - a
ΔPQR ~ ΔMLR
In similar triangles, corresponding sides are in same ratio.
[tex]\frac{PQ}{QR}=\frac{ML}{LR}\\\\\frac{6}{6}=\frac{a}{6-b}[/tex]
[tex]1 =\frac{a}{6-b}[/tex]
6 - b = a
6 = a + b
a +b = 6
Perimeter of rectangle = 2*(length +width}
= 2*(a+ b)
= 2 * 6
= 12
Carlos compro cierta cantidad de panes puso 1/4 de esa cantidad sorbre su bandeja y dejo el resto de panes en la bolsa ¿Cuantos panes dejo Carlos en la bolsa?
Answer:
The quantity of loaves left in the bag is 3/4.
Step-by-step explanation:
Carlos bought a certain amount of loaves, put 1/4 of that amount on his tray and left the rest of the loaves in the bag. How many loaves did Carlos leave in the bag?
Quantity of loaves in tray is 1/4
So, the quantity of loaves in the bag is
[tex]1-\frac{1}{4}=\frac{3}{4}[/tex]
What is the value of y?
B
1
5
5
60°
A
60°
C
5
Answer:
the value of y in the equation is 60°
What is the value of x and the length of segment DE?
StartFraction 5 Over 9 EndFraction = StartFraction 9 Over 2 x + 3 EndFraction
10x + 15 = 9(9)
x = _____
Length of = ___units
Answer:
Step-by-step explanation:
This is a geometric means problem involving the height of 9. 9 is the geometric mean because it is the height of both ΔCFD and ΔEFD. Set up the problem accordingly:
[tex]\frac{5}{9}=\frac{9}{2x+3}[/tex] and cross multiply:
81 = 5(2x + 3) and
81 = 10x + 15 and
66 = 10x so
x = 6.6
Now we will plug that value in for x to find the length of DE:
2(6.6) + 3 = 16.2 units
6. Show how to simplify the expression 8x + 3 + 4x - 2 without using algebra tiles. Describe each step of the process.
Answer:
12x+1
Step-by-step explanation:
8x+3+4x-2
8x+4x+3-2
Arranging Like terms
12x+1
Adding Like terms (8x+4x) and Substracting numbers ! ++- = - So we substrate it and in addition we don't change symbols so 3 is greatest number so we write + there !
On a coordinate plane, a curved line with a maximum value of (negative 1, 2) crosses the x-axis at (negative 3, 0) and (1, 0), and crosses the y-axis at (0, 1.5).
What are the x-intercepts of the graphed function?
(–3, 0) and (0, 1.5)
(–3, 0) and (1,0)
(–1, 2) and (1, 0)
(0, 1.5) and (1, 0)
9514 1404 393
Answer:
(–3, 0) and (1,0)
Step-by-step explanation:
The x-intercepts are where the curve crosses the x-axis. Your problem statement tells you ...
"crosses the x-axis at (negative 3, 0) and (1, 0)"
This means the x-intercepts are (-3, 0) and (1, 0).
Answer:
The answer would be B= (–3, 0) and (1,0)
Can someone help me with this math homework please!
Answer:
1.33 & n - 6 = 0.75(t - 8)
Explanation:
Speed can be calculated as miles ÷ minutes
4 ÷ 3 = 1.33 and 8 ÷ 6 = 1.33 as well
Since we know the n and t, we can substitute one into the equation to find the other (substitute t = 8 into all four equations and see do we get n = 6)
You can do this with t = 4 and n = 3 as well
Answer:
you are a great woman removing the sum o saare and pa and pa to the lord and your life to you for drawing the sum o the streets
write y=4x+3 in mapping notation
Answer:
y=7x
Step-by-step explanation:
because y= 4x+3=7x
what is the value of 5 in the number 68.513?
Answer:
00000000.1
Step-by-step explanation:
1+1=2-1%000.1
calculate the surface area of a hockey puck with a height of 1 in. and a diameter of 3 in
Answer:
23.56
Step-by-step explanation:
2([tex]\pi r^{2}[/tex]) + 2[tex]\pi[/tex]rh
2([tex]\pi[/tex] [tex]1.5^{2}[/tex])+ 2[tex]\pi[/tex](1.5)(1)
23.56
23.56 in² is the surface area
What is surface area?A solid object's surface area is a measurement of the overall space that the object's surface takes up.
Given
2(πr²) + 2πrh
2π( 1.5)²+ 2π(1.5)(1)
= 23.56 in²
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Solve the inequality -10 is greater or equal to -2x
Answer:
-10 and - 2x
The x is unknown number so, we don't know which one if -2x or -10 is greater.
Calculate the maximum absolute uncertainty for R if:
R = 9A / B
A = 32 +/- 2 seconds
B = 11 +/- 3 seconds
1 second
0.33 seconds
9 seconds
2 seconds
6 seconds
Answer:
1.1 small inches of the day
Step-by-step explanation:
Find the area of a sector of a circle whose radius is 16 cm and whose central angle is
36 degrees.
Answer:
1πcm
Step-by-step explanation:
do you have another question to ask pls if you have ask
Year 10 girls = 70, year 10 boys = 140, what fraction of year 10 are boys?
Answer:
2/3
Step-by-step explanation:
140/(140+70)
=140/210
simplified
=2/3
Sage is 7 years older than Jonathan. If Jonathan is x years old, how old was Sage 10 years ago?
Answer:
(x-3) years
Step-by-step explanation:
We are given that
Age of Jonathan= x years
Sage is 7 years older than Jonathan
It means
Age of Sage=(x+7) years
We have to find the age of Sage 10 years ago.
10 Years ago,
Age of Jonathan=(x-10) years
Age of Sage=(x+7-10) years
Age of Sage=(x-3) years
Hence, 10 years ago, age of Sage =(x-3) years
There are 3 pizzas. Each child will get 1/2 of the pizza. How many 1/2's will there be in 3 pizzas?
Answer:
There are 6 halves of the pizza
Step-by-step explanation:
3/0.5=6
4. Three friends are collecting football cards. They have 240 cards between them. Zukile has 60 more cards than Adam, and Siya has twice as many cards as Adam. If Adam has x number of cards, write and solve an equation to find how many cards they each have.
Answer:
Zukile has 105, Adam has 45, Siya has 90
Step-by-step explanation:
Not sure how right this is, but I did: 240=2x+x+(60+x)
Simplify to: 240=60+4x
Simplify to: 180=4x
Simplify to: 45=x
HELP!! pie/3 is the reference angle for:
Answer:
A, C and D.
Step-by-step explanation:
Answer:
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