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
on a shopping trip, Gina buys two books priced $4.99 and $12.99. She also buys 2 notebooks, 2 packs of pencils, and 3 composition books. The books are on sale for 10% off the list price and school supplies are 15% off. The store charges 6.5% sales tax. Item Cost ($) Notebooks 2.88 Composition Books 0.79 Pencil Packs 1.89
Answers
The total cost of items Gina bought on her shopping trip given the percentage discount is $26.31
Given:
cost of two books = $4.99 + $12.99
= $17.98
Amount paid for books = $17.98 - 10% of $17.98
= 17.98 - 0.1 × 17.98
= 17.98 - 1.798
= $16.182
Cost of notebooks = quantity × price
= 2 × $2.88
= $5.76
cost of composition books = quantity × price
= 3 × $0.79
= $2.37
cost of pencil packs = quantity × price
= 2 × $1.89
= $3.78
Total cost = $5.76 + $2.37 + $3.78
= $11.91
Amount paid for school supplies = $11.91 - 15% of $11.91
= 11.91 - 0.15 × 11.91
= 11.91 - 1.7865
= $10.1235
Therefore,
total cost of all items Gina bought = $16.182 + $10.1235
= $26.3055
Approximately, $26.31
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Help please!!!!!!!!!!!! I need help
Answers
Slope of second graph = 4
Slope of Function A is less than that of Function B
Answer:
The slope of Function A is less than the slope of Function B.
Step-by-step explanation:
Using the slope formula:
m = (y2 - y1)/(x2 - x1)
and the following ordered pairs from Function A:
(x1, y1) = (-5, -3)
(x2, y2) = (-10, -8)
m = (y2 - y1)/(x2 - x1)
[tex]m = \frac{-8 - (-3)}{-10 - (-5)} = \frac{-8 +3}{-10+5} = \frac{-5}{-5} = 1[/tex]
Since Function A's slope = 1, and Function B's slope = 4, then it means that the slope of Function A is lesser than the slope of Function B.
1/3x7 ...............
Answers
Answer:
7
Step-by-step explanation:
Given the proportion a/b=5/19, what ratio completes the equivalent proportion a/5=?
Answers
Answer:
[tex]\frac{b}{19}[/tex]
Step-by-step explanation:
If you cross-multiply in the first proportion you get 19a = 5b
To keep this, if we have a/5, we need b/19 to cross-multiply and still get 19a = 5b
Increase a price of £30 by 10%
Answers
Answer:
£33
Step-by-step explanation:
A 10% increase is the same as multiplying a value by 1.1.
30 * 1.1 = 33
Answer:
10 x 30 /100 = 3 so 30 +3 = 33
Please answer this qusetion
Answers
Answer:
A
Step-by-step explanation:
With way one, we can easily count the tally marks, but with way 2 it becomes much harder for us to count the tallies when we are done.
Answer:
a
Step-by-step explanation:
its easy to count way 1 but way 2 is harder to count
Help
Ill mark u as brainlest if ur answer is clear
Thank you
Answers
Answer:
[tex]x=55[/tex]
Step-by-step explanation:
To solve this question you need to know some properties of a parallelogram which is the shape given.
Firstly, remember that in a parallelogram opposite angles are equal.
Secondly, remember that the diagonals of a parallelogram bisect each other.
Lastly, remember that the sum of the interior angles of a quadrilateral or a parallelogram is 360°.
So, by using the first property we can see that because [tex]<[/tex]PQR=70° then [tex]<[/tex]PSR must be equal to 70° as well.
Now, we can use the last property to find [tex]<[/tex]SPQ then, use the second property to find [tex]x[/tex].
Let's name [tex]<[/tex]SPQ, [tex]y[/tex].
and That gives:
[tex]2y+140=360[/tex]
Subtract 140 from both sides:
[tex]2y=360-140[/tex]
[tex]2y=220[/tex]
[tex]y=\frac{220}{2}[/tex]
[tex]y=110[/tex]
So, [tex]<[/tex]SPQ=110 and
Now using the second property we can find that:
[tex]<[/tex]SPQ=[tex]2x[/tex]
And [tex]2x=110[/tex]
[tex]x=\frac{110}{2}[/tex]
[tex]x=55[/tex]
Hope this helped.
Dana went to the local salon to get a haircut. The cost was $38. Dana tipped the hair stylist 20%. What was the total cost of the haircut including the tip? Round to the nearest cent.
Answers
Answer:
Dana paid $45.60 in total.
Step-by-step explanation:
20% of 38 is 7.60. Add this to the total price of $38. (38+7.60=$45.60)
Brita has a monthly budget of x dollars. Every month, $1,425 is spent on her mortgage. One-third of her remaining budget is spent on recreational activities. Which equation below can be used to determine the amount, in dollars, spent on recreational activities?
y=x−1,4253
y=1,425−x3
y=1,425+x3
y=x+1,4253
Answers
Answer:
y=1,425-×3
Step-by-step explanation:
the x has to be divided by the 1/3
please help due asap
Answers
Answer:
I think it's angle 10
Step-by-step explanation:
I'm not 100% sure, but I think this is it because angles 10 and 2 have the same relative position at the intersection.
I hope this helps
help please this is due rn
Answers
i need help with get more math
Answers
Answer:
I'm not sure I understand but if you want more math to do look up Quizlet and do some of those tests
Before the sale, the store had 100 pairs of flip-flops in stock. After selling some they noticed that 3/5 of the flip flops they have left are blue. If the store has 39 pairs of blue flip flops, how many pairs of flip flops (any color) have they sold?
Answers
Answer:
35
Step-by-step explanation:
The following graph shows how many gallons of water leaves a bucket every hour.
- right the slope intercept equation. Find the constant rate of change and the vertical intercept of the line. Explain how they are related to this situation
Answers
Answer:
thanks for your points and the time
58ml : 1120ml : 104ml need answer asap!
Answers
Answer:
i dont know the answer sorry
Step-by-step explanation:
Here is a list of numbers:
1.9, 0.1, 7.7, 6.8, 6, 7, 7.8, 2.8, 6.9, 3.5
State the median.
Give your answer as a decimal.
Answers
Answer:
So I added, 1.9, 0.1, 7.7, 6.8, 6, 7, 7.8, 2.8, 6.9, 3.5, which equals 50.5.
Then, I divided 50.5 by 9 since that is how many number sections there are.
Therefore, I get 5.61.
If Zayed creates the greatest number of identical packages possible, how many pencils will be in each package?
Answers
Answer:
Step-by-step explanation:
Sam started cooking at 7;43 AM and finished at 8;32 AM. how long did it take him?
Answers
Answer:
49 minutes
Step-by-step explanation:
7: 43
8: 32
17 minutes 'till 8:00
Another 32 'till 8:32
so add 17+32
Help please ill mark brainliest!
Answers
The polygon is 5 units in length and not 4
so using the given formula P = 2l + 2w
(2 x 5) + (2 x 3)
10 + 6
= 16 units
if the price of an item before tax is £720 and the tax percentage is 25%, what is the selling price?
Answers
Original x percent needed (125%) = total price
please help me this hard
Answers
Answer:
P= 6
Step-by-step explanation:
6P=-6+7P
6P-7P=-6
-P=-6 -------divide both side by negative
P=6 you get
MARKING BRAINLEIST IF YOUR RIGHT !!
Answers
Step-by-step explanation:
slope =3/2
that is the answer
Jayla reads 35% of her book over 7 days. If Jayla reads the same number of pages each night, how many more days will it take her to finish her book?
Answers
Answer:
13 days
Step-by-step explanation:
Amount of book left = 100% - 35% = 65%
If 35% requires 7 days,
65% requires x days.
Using the 'cross the heart' or 'cross multiplication' method:
35x = 7 * 65
x = (7 * 65)/35
= 455/35
= 13
Hence, it would take 13 more days to complete the book.
Hope this helps and be sure to have a great time ahead at Brainly! :D
Find the measure of each angle indicated 12,13,14
Answers
Answer:
Step-by-step explanation:
12. 15 (3x5x1)
13. 8 (4x2x1)
14. 18 (6x3x1)
Jason deposits $1700 in an account that earns 5% interest compounded weekly. How much will Jason have after 8 years?
Answers
Answer:
$2511.674254, rounded to $2512
Step-by-step explanation:
1700x1.05^8
write an equation of the line that passes through the point (1,3) and has a slope of -2
Answers
Answer:
y= -2x+1
Step-by-step explanation:
y=mx+b
y= -2x +b
3= -2(1) +b
3= -2 +b
b=1
y= -2x+1
14% of the towns population is over the age 65. If there are 320 residents over the age of 65, approximately what is the towns population?
Answers
Answer:
about 315
Step-by-step explanation:
2,250×0.14
hope this helps!
If (x+iy)³ = y+vi then show that (y/x + v/y) = 4(x²-y²)
Answers
Answer:
Step-by-step explanation:
[tex] \large\underline{\sf{Solution-}}[/tex]
Given that,
[tex]\rm \longmapsto\: {(x + iy)}^{3} = y + iv[/tex]
[tex]\rm \longmapsto\: {x}^{3} + 3 {x}^{2}(iy) + 3x {(iy)}^{2} + {(iy)}^{3} = y + iv[/tex]
[tex]\rm \longmapsto\: {x}^{3} + 3 {x}^{2}yi + 3x {i}^{2} {y}^{2} + {i}^{3} {y}^{3} = y + iv[/tex]
We know,
[tex] \purple{\rm \longmapsto\:\boxed{\tt{ {i}^{2} = - 1 \: \: \: and \: \: \: {i}^{3} = - i }}} \\ [/tex]
So, using this, we get
[tex]\rm :\longmapsto\: {x}^{3} + 3 {x}^{2}yi - 3x{y}^{2} - i{y}^{3} = y + iv[/tex]
[tex]\rm \longmapsto\: ({x}^{3} - 3x {y}^{2}) + i(3{x}^{2}y- {y}^{3})= y + iv[/tex]
On comparing real and Imaginary parts, we get
[tex]\rm \longmapsto\:y = {x}^{3} - 3 {xy}^{2} [/tex]
and
[tex]\rm \longmapsto\:v = {3x}^{2}y - {y}^{3} [/tex]
Now, Consider
[tex]\rm \longmapsto\:\dfrac{y}{x} + \dfrac{v}{y} [/tex]
[tex]\rm \: = \: \dfrac{ {x}^{3} - {3xy}^{2}}{x} + \dfrac{ {3yx}^{2} - {y}^{3} }{y} [/tex]
[tex]\rm \: = \: \dfrac{ x({x}^{2} - {3y}^{2})}{x} + \dfrac{y({3x}^{2} - {y}^{2})}{y} [/tex]
[tex]\rm \: = \: {x}^{2} - {3y}^{2} + {3x}^{2} - {y}^{2} [/tex]
[tex]\rm \: = \: 4{x}^{2} - {4y}^{2}[/tex]
[tex]\rm \: = \: 4({x}^{2} - {y}^{2})[/tex]
Hence, Proved
what fraction of an hour is 5 minutes in its simplest form
Answers
Answer:
1 / 12
Step-by-step explanation:
one hour = 60 minutes
so
5 / 60
= 1 / 12
Answer:
[tex]\frac{1}{12}[/tex]
Step-by-step explanation:
There are 60 minutes in 1 hour , then 5 minutes as a fraction of an hour is
[tex]\frac{5}{60}[/tex] = [tex]\frac{1}{12}[/tex] ← in simplest form
What's the equation 3c - 4y =28 on a graph?
