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
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
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%
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.
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.
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
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
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
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:
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!!!
Pls help
bobik: yes, or no
Answers
Answer:
yes
Step-by-step explanation:
Answer:
?? Yes??
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
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:
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
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
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.
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
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
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
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 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:
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
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
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
Need help with question 1 and 2
Answers
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 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
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.
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.
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:
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