Use trigonometric identities to solve each equation within the given domain
sec(x) cos(3x) = 0 from [- x /2 , x /2]
Answers
Answer:
Either [tex]x = (-\pi / 6)[/tex] or [tex]x = (\pi / 6)[/tex].
Step-by-step explanation:
Angle sum identity for cosine:
[tex]\cos(a + b) = \cos(a)\, \cos(b) - \sin(a) \, \sin(b)[/tex].
Double-angle identity for cosine and sine:
[tex]\begin{aligned}& \cos(2\, a)\\ &= \cos^{2}(a) - \sin^{2} (a) \\ &= (\cos^{2}(a) + \sin^{2}(a) - 2\, \sin^{2}(a) \\ &= 1 - 2\, \sin^{2}(a)\end{aligned}[/tex].
[tex]\sin(2\, a) = 2\, \cos(a) \, \sin(a)[/tex].
The secant of an angle is the inverse reciprocal of the cosine of that angle:
[tex]\displaystyle \text{$\sec(x) = \frac{1}{\cos(x)}$ if $\cos(x) \ne 0$}[/tex].
Apply these identities to rewrite the left-hand side of the equation.
[tex]\begin{aligned} & \sec(x) \, \cos(3\, x) \\ =\; & \frac{1}{\cos(x)} \cdot \cos(3\, x) \\ =\; & \frac{1}{\cos(x)} \cdot (\cos(x)\, \cos(2\, x) - \sin(x)\, \sin(2\, x)) \\ =\; &\cos(2\, x) \\ & - \frac{1}{\cos(x)} \cdot (\sin(x)\, \sin(2\, x)) \\ =\; & \cos(2\, x)\\ & - \frac{1}{\cos(x)} \cdot (\sin(x) \cdot (2\, \sin(x)\, \cos(x))) \\ =\; & \cos(2\, x) - 2\, \sin^{2}(x) \\ =\; & (1 - 2\, \sin^{2}(x)) - 2\, \sin^{2}(x) \\ =\; &1 - 4\, \sin^{2}(x) \end{aligned}[/tex].
Solve the rewritten equation:
[tex]\displaystyle \sin^{2}(x) = \frac{1}{4}[/tex].
Either [tex]\sin(x) = (1/2)[/tex] or [tex]\sin(x) = (-1/2)[/tex].
The question required that [tex](-\pi / 2) \le x \le (\pi / 2)[/tex]. The sine function is one-to-one over this range. In other words, for each [tex]y[/tex] between [tex](-1)[/tex] and [tex]1[/tex], there is exactly one [tex]x[/tex] between [tex](-\pi / 2)[/tex] and [tex](\pi / 2)[/tex] such that [tex]\sin(x) = y[/tex].
If [tex]\sin(x) = (-1/2)[/tex] and [tex](-\pi / 2) \le x \le (\pi / 2)[/tex], then [tex]x = (-\pi / 6)[/tex].
Otherwise, if [tex]\sin(x) = (1/2)[/tex] and [tex](-\pi / 2) \le x \le (\pi / 2)[/tex], then [tex]x = (\pi / 6)[/tex],
Verify that [tex]\cos(x) \ne 0[/tex] for either of the two [tex]x[/tex] values, such that the [tex]\sec(x)[/tex] in the original equation takes a real value.
Related Questions
What is -(x + 4) - 3x = x in Distributive property?
Answers
Answer:
x = -4/5
Step-by-step explanation:
Original equation: -(x+4)-3x=x
Remember distributive property allows us to get rid of the parenthesis by bringing a multiplication from outside to inside a parenthesis.
-(x+4) = -x-4
= -x-4-3x=x
Now we combine like terms in this case we have (-x) and (-3x)
= -4x-4=x
Subtract x from both sides
= -4x-4=x
-x -x
----------------
-5x-4=0
Add 4 to both sides
-5x-4=0
+4 +4
---------------
-5x=4
Our final step is to divide both sides by -5
-5x=4
---- ----
-5 -5
x = -4/5
Hope this helps!!!
- How are the functions y = x and y = x + 4 related? How are their graphs related?
Answers
Answer:
Either horizontal translation 4 units to the left or vertical translation UP 4 units
Step-by-step explanation:
I need help ASAP!!! (please look at the image and explain how you got the answer, please and thanks)
Answers
Answer:
114
Step-by-step explanation:
so lets call shelf 1 x
shelf 2 is twice size + 18 of shelf 1 so it is 2(x)+18
shelf 3 is 12 less than shelf 1 so x-12
shelf 4 is four more so x+4
Add them all together and it should be 5x+10= 2.5 m or 250 cm
Solve for x using cm
then plug into shelf 2 equation
From the graph, write the linear equation in slope-intercept form
Answers
Answer:
y= [tex]\frac{2}{5}[/tex](x)+2
Step-by-step explanation:
First take two points from the line (x-int (-5,0) and y-int (0,2)). Once you get that you want to find the slope using the formula [tex]\frac{Y_{2} -Y_{1} }{X_{2}- x_{1} }[/tex]. Substitute the values and find the slope. Once you find the slope (m) to be [tex]\frac{2}{5}[/tex], take any of the two points and apply it to the slope intercept formula (y=mx+b) to find b. I took the point (0,2) and applied it to the formula. It should look like 2= [tex]\frac{2}{5}[/tex](0)+b. Solve for b by multiplying 2/5 by zero and your left with 2=b. Plugin the missing varibles for the equation and it should come out to be y= [tex]\frac{2}{5}[/tex](x)+2.
The price of a notebook has risen to $3.45 today. Yesterdays price was $3.20. find the percentage increase, round your answer to the nearest tenth of a percent.
Answers
Answer:
7.8%
Step-by-step explanation:
Yesterday’s price =$3.20
Today’s price =$3.45
Increase =today’s price - yesterday’s price
That’s $3.45 - $3.20
= $0.25
Percentage increase = increment /yesterday’s price x 100%
That’s $0.25/$3.20 x 100%
0.0781 x 100%
7.81
7.8%
When using empirical data (data and information collected by scientists in an experiment), sometimes, that data will not form a perfect line on a graph. When this happens, the line that is created using a linear regression is known as the “Line of Best Fit.” The following speed data was collected from a car moving with a constant acceleration.
Time (s) Velocity (m/s)
2 13.1
4 19.8
6 28.1
8 34.6
What is the Line of Best Fit for this data? Perform a linear Regression on this table to find the equation.
Based on your equation, what was the car’s approximate initial velocity (at time=0)?
If the car continued at the same acceleration (meaning the slope of the line does not change), how fast would it be going in 15 seconds?
Answers
Answer:
98.25
Step-by-step explanation:
The functions j(x)=x−1 and k(x)=3x+1 intersect at:
A. Infinitely many points
B. 2 points
C. 1 point
D. There are no intersection points
Answers
Answer:
C
Step-by-step explanation: for j(x) = k(x)
My English is not good, please forgive me
what is green and blue? HELP
Answers
Answer:
green horizontal asymptote: y = -2
blue horizontal asymptote: y = -4
Step-by-step explanation: Exponential functions have a horizontal asymptote
please help me do this
Answers
Answer:
125
Step-by-step explanation:
add 21 and 34. with that you get 55. then subtract 180 from 55 then get 125
If 40 meters of a cloth costs Rs 1940 how many meters can be bought for Rs 727.5
Plz explain
Answers
Answer:
15
Step-by-step explanation:
suppose x is the no. of cloth that can be bought in rs 727.5
then cost of 1meter cloth is rs 727.5×.....(i)
and cost of 40mtr cloth is rs 1940
then 1mtr cloth is 1940/40....(ii)
so from i and ii
we get
727.5/x = 1940/40
60% of a mussems fossils are from the jurassic period if the number of fossils from the jurassic period is 15 what is the total of number of fossils ate the musuem ?
Answers
The total number of fossils at the museum are, 25
3 ft
10 ft
Perimeter:
Area:
Answers
Step-by-step explanation:
The area = 3 × 10 = 30
The perimeter = 2(3 + 10) = 26
please give me a brainliest answer
i need help. ill mark brain to the first to answer.
Answers
Answer:
a and f
Step-by-step explanation:
1500 is double 750, which is the combined calories of lunch and breakfast, which is only 1 third.
Answer:
1st and 2nd options
Step-by-step explanation:
Daily total = (400+350)*3 = 2250
Dinner = 2250 * 2/3 = 1500
equation: 2/3*(x+400+350) = 2/3 * daily total = Dinner = x
helping my sister with her homework
Answers
Answer:
5 : 10
6 : 12
7 : 14
8 : 16
9 : 18
Step-by-step explanation:
10/5=2
there is only one number in between 7 and 5
so now you have to go up 2 each time in the right collum and go up once in the left collum.
HELP
In order to isolate the variable term in the equation 6 -2x = -3 using the subtraction property of equality, which number
should be subtracted from both sides of the equation?
-3
3
6
-2
Answers
Plzzz help!! just atleast look at my question!! plzz I really need help!
Answers
no
Step-by-step explanation:
-2 - -3 doesn't equal -5
Answer:
no
Step-by-step explanation:
-2 - -3 = 1 not 5
please help me its due in 3 mins
Answers
Answer:
p = 35°
Step-by-step explanation:
Angles on a straight line add up to 180° so:
2p + 5 + 3p = 180° (simplify)
5p + 5 = 180° (subtract 5 from both sides)
5p = 175° (divide both sides by 5)
p = 35°
Hope this helps!
Answer:
p=35
Step-by-step explanation:
You have to add the two together and set them equal to 180, so 2p+5+3p=180 so 5p=175, so p=35.
What are the values of v and w ?
Answers
Answer:
V = 72 & W = 72
Step-by-step explanation:
All angles of a triangle will be equivalent to 180 degrees.
We start by subtracting first.
180 - 36 = 144
Look back at the triangle. FG and HG are congruent, which means the values of V and W will be the same.
The last step is to divide.
144 / 2 = 72
V = 72 & W = 72
Solve:
4x + 1.21 > 5
HELP ME PLEASE
Answers
Hi there!
»»————- ★ ————-««
I believe your answer is:
[tex]x > 0.9475[/tex]
»»————- ★ ————-««
Here’s why:
⸻⸻⸻⸻
[tex]4x + 1.21 > 5\\\rule{150}{0.5}\\\rightarrow 4x + 1.21 - 1.21 > 5 - 1.21\\\\\rightarrow 4x > 3.79\\\\\rightarrow \frac{4x>3.79}4}\\\\\rightarrow \boxed{x > 0.9475}[/tex]
⸻⸻⸻⸻
»»————- ★ ————-««
Hope this helps you. I apologize if it’s incorrect.
what is the equation for the line that is perpendicular to the x-axis and passes through the point (4, -7)?
Answers
Step-by-step explanation:
perpendicular means to cross at a right angle (90 degrees).
so, any line perpendicular to the x-axis is parallel to the y-axis.
and any line parallel to the y-axis has only a defining x value. all points on that line have the same x value (while any value for y from -infinity to +infinity is valid).
the y-axis itself is
x = 0
the specified point here has an x value of 4.
so, the line we are looking for is
x = 4
to point out : this is NOT a function, because for the same x value we get more than 1 valid y values.
Explain the parts of an angle
Answers
Answer:
line segments are the lines
rays are line segments that have one vertex
a vertex is the endpoint of a line(usually represented by a dot)
Step-by-step explanation:
Simplify (a – b)(a2 – 5ab + 2)
Answers
Answer:
5ab^2-6a^2b+a^3-2b+2a
Step-by-step explanation:
The simplified expression of expression (a – b)(a² – 5ab + 2) is a³ – 6a²b + 5ab² + 2a – 2b
What is Expression?An expression is combination of variables, numbers and operators.
To simplify the expression (a – b)(a² – 5ab + 2), we need to apply the distributive property, which states that:
(a + b)(c + d) = ac + ad + bc + bd
Using this property, we can multiply the two binomials as follows:
(a – b)(a² – 5ab + 2) = a(a² – 5ab + 2) – b(a² – 5ab + 2)
Now we can distribute the terms of each binomial:
= a³ – 5a²b + 2a – a²b + 5ab² – 2b
Simplifying the terms, we can combine like terms:
= a³ – 6a²b + 5ab² + 2a – 2b
Therefore, the simplified expression is a³ – 6a²b + 5ab² + 2a – 2b
To learn more on Expressions click:
https://brainly.com/question/14083225
#SPJ2
Solve for x: 16 = x + 32
Answers
Answer:
-16
Step-by-step explanation:
Use everyone's favourite thing: ALGEBRA!!!
16=x+32
16-32=x
-16=x
Hope that helps
Step-by-step explanation:
16=x+32
16-x=32
-x=32-16
-x=16
X=-16
Need help with question 1 and 2
Answers
A bottle of grape juice costs 6.50$ of you buy 6 bottles you can get 10%discount workout how much you save if you buy 6 bottles
Answers
$39-3.90= $35.10
Your answer is you save $3.90 by buying six bottles
How many solutions does the nonlinear system of equations graphed below have?
A. One
B. Four
C. Two
D. Three
Answers
Answer:
There are three possible types of solutions for a system of nonlinear equations involving a parabola and a line.
Step-by-step explanation:
Pls help
bobik: yes, or no
Answers
Answer:
yes
Step-by-step explanation:
Answer:
?? Yes??
Step-by-step explanation:
Molly has 3 pounds of blueberries. She is freezing the berries into 1 half pound portions. How many smaller portions of blueberries will Molly make
Answers
Answer:
6 smaller portions
Step-by-step explanation:
1 pound (lb) = 16 ounces (oz)
3 lbs = 48 oz
1/2lb = 8oz
48oz (total)/8oz (amount in each portion) = 6 portions
Solve the equation x2-81 = 0.
Answers
Answer:
40.5
Step-by-step explanation:
x2 - 81 = 0 --> + 81 to both sides
2x = 81 --> divide by 2 on both sides
x = 40.5
Answer:
2x-81=0
2x=0+81
2x=81
x=81
2
x=40.5 ANSWER.
MARK ME AS A BRAINLIST PLZ.
ebgjwbjhkjfhdsn wmiuehdbnsmkji
Answers
Answer:
im srry what
Step-by-step explanation:
DYK?
— Pretty face is everywhere, but a woman who cheer you up at your weakest and stay after knowing about your flaws and bad sides are rare, so you better keep her dude!.
#moodchallenge