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
-2(x-4)=10x+2
show ur work
Answers
Answer:
Step-by-step explanation:
-2(x-4) =10x+2
-2x--8= 10x+2
-2x+8= 10x+2
-2x+10x= 8+2
8x= 10
x= 0.8
Answer:
Step-by-step explanation:
To solve for x you first isolate the variable you want on one side. We will do this by using PEMDAS and expanding our terms fully first before we move our terms around.
[tex]-2(x-4)=10x+2\\-2x-8=10x+2[/tex]
We then isolate our term, remember to add numbers and variables to both sides so you aren't changing the value of the equation, and solve for x.[tex]-2x-8=10x+2\\-2x-8-2+2x=10x+2-2+2x\\-10=12x\\\frac{-10}{12}=x\\\frac{-6}{5}=x\\-1.2=x\\[/tex]
Compare solving equations using addition with solving inequalities using addition
Answers
Answer:
give picture
Step-by-step explanation:
Solving inequalities with addition is similar to solving equations with addition in the initial steps, but it differs in the final answer due to the inequality sign. In both cases, you use addition to isolate the variable. However, with inequalities, you must consider the direction of the sign throughout the process.
When solving inequalities with addition, like equations, the goal is to isolate the variable on one side. You use addition to simplify the expression. Begin by treating the inequality like an equation, adding or subtracting the same value to both sides to eliminate constants and isolate the variable.
For example, if you have [tex]\(2x + 5 > 10\)[/tex], subtract 5 from both sides to get [tex]\(2x > 5\)[/tex].
Here's where it diverges: since inequalities involve the relationship between two quantities, you need to be mindful of the inequality sign. When you multiply or divide both sides by a negative number, the inequality direction flips.
For example, if you multiply [tex]\(x < 3\)[/tex] by -2, the sign changes to [tex]\(x > -6\)[/tex].
Remember, you're looking for a range of values that satisfy the inequality. So, when representing your final answer, use the appropriate inequality sign. For instance, if [tex]\(3x + 2 \leq 8\)[/tex], after simplifying, you'll have [tex]\(x \leq 2\)[/tex], indicating that any value of [tex]\(x\)[/tex] less than or equal to 2 satisfies the inequality.
In summary, while solving inequalities with addition is akin to solving equations with addition in the initial steps, the crucial distinction lies in managing the inequality sign throughout the process and representing the solution as a range of values that satisfy the inequality.
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Complete Question:
How is solving inequalities with addition is similar to or different from solving equations with addition ?
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Answer:
look at the photo..................
Solve the math problem
Answers
x+50+70+4x=180
combine like terms
50+70=120
subtract 120 by both sides
x+4x=60
combine like terms
5x=60
divide by 5
x=12
4x12=48
B=48
12+50=62
48+62=110
110+70=180
x=12
A=62
B=48
What is 5 1/4 x 4 1/5
Answers
Answer:
The answer to your question is 22 1/20
Answer: [tex]22\frac{1}{20}[/tex] or 22.05
Step-by-step explanation:
[tex]\frac{21}{4}[/tex] × [tex]\frac{21}{5}[/tex]
[tex]=\frac{441}{20}[/tex]
[tex]22\frac{1}{20}[/tex] or 22.05
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Find the slope of each line.
А
B
с
A)
A
B)
C)
Answers
Answer:
A=b B=C C=A
Step-by-step explanation:
which of these points is closest to the point (7,1)?
A. (4,1)
B. (7,-1)
C. (7,4)
D. (11,1)
Answers
Answer:
7,-1 should be the answer but I am not fix ok wait for others comment
the width of a rectangular room is 3m shorter than its length . if its perimeter must not exceed 42m , calculate the greatest length the room can have
Answers
The width of a rectangular room is 3 m shorter than its length. If its perimeter must not exceed 42 m, calculate the greatest length that the room can have.
An insurance agent earns a base salary of 32,000 and commission of 3% of his sale he also receives an additional fee of 16.00
Answers
Answer:
EHEHEHEHEHEHEHEGRGEHEHE
Step-by-step explanation:
22,324
A lightbulb consumes 1440 watt-hours Per day how many watt-hours does it consume in three days and 12 hours?
Answers
Answer:
it consumes 5040 watt hour in 3 days and 12 hours. Happy to Help :)
Step-by-step explanation:
what is the slope of this line?
a) -1/3
b)1/3
c) -2/3
d) 2/3
Answers
Answer:
look at the picture i have sent
Which expression is equivalent to 6m + (m – 2)(m + 7) + 3?
Answers
Answer:
m^2 + 11m - 11
Step-by-step explanation:
Step 1: Distribute.
[tex]6m + (m)(m) + (m)(7) + (-2)(m) + (-2)(7) + 3[/tex] [tex]6m + m^2 + 7m -2m -14 + 3[/tex]Step 2: Combine like terms.
[tex](m^2)+(6m+7m-2m)+(-14+3)[/tex] [tex]m^2 + 11m - 11[/tex]Therefore, the answer is [tex]m^2 + 11m - 11[/tex].
Sophie has 9 cards.
There is a number on each card.
a
5
4 42
31|15|9|6
Sophie takes at random two of the cards.
She works out the product of the numbers on the two cards.
Work out the probability that the product is an even number.
a marked
Answers
Answer:4
Step-by-step explanation:
Help me please!Please look at the pic then tell me if im wrong on something.Thank you! :)
Answers
Answer: for every 3 triangles there are 5 squares. not 4. so the ratio is 3:5.
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What is the coefficient in the expression 8−x7? Question 7 options: −7 7 17 −17
Answers
Answer:
7
Step-by-step explanation:
Which of the following best describes the algebraic expression 10x +2?
A. The sum of ten times a number and two.
B. Ten times the sum of a number and two.
C. The product of ten plus a number and two.
D. Ten times the quantity of a number plus two.
Answers
Step-by-step explanation:
option A describes the algebraic expression 10x + 2.
hope this answer helps you dear!
What is the significance of a correlation coefficient of − 0 . 97 ?
Answers
Answer:
The points all lie very close to the line of fit, and the
line is decreasing.
Step-by-step explanation:
Brainliest maybe
Answer: A.) The points all lie very close to the line of fit, and the line is decreasing.
find the decimal equivalent of ¾ ounce to the thousandths place
Answers
0.75
Explanation:
Every 1/4 is 25. If you want to calculate them, then you're going to need 1/4 to find 25. In some other cases, 25, can also be written as another decimal. Such as, 4/100.
100/4 is 25. For example:
Woah! You have 3/4 of a dollar. You're really saving up money fast!
This example means that the person has 75 cents.
For an ounce, that would mean 0.04 pounds.
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Did George Washington have a middle name
Answers
Answer:
Lol, I would like to answer the question tho...
(George Washington did not have a middle name.)
Step-by-step explanation:
helpppppppppppppppppppppppppp
Answers
Answer:
with what?
Step-by-step explanation:
determine if the givem binomial x-c is a factor of the given polynomial p(x)
[tex]p(x) = {x}^{4} - {x}^{3} + {x}^{2} + x - 1: x - 2[/tex]
Answers
Can I get steps on how to solve this?
Answers
Answer:divided the total buy 9 and if I’m reading correctly the total is two
Step-by-step explanation:
The length of a rectangle is 8 cm more then 4 times the with. If the perimeter of the rectangle is 46 cm, what are the dimensions
Answers
Answer:
15
Step-by-step explanation:
let the breadth be x
let the length be x+8
perimeter of rectangle=2 (length +breadth)
so, 2 (x+x+8)=44cm
=2 (2x+8)=44
=4x + 16=44
=4x= 44-16
=4x= 28
=x = 28÷4
=x= 7
therefore,
breadth is 7 (x)
length is 7+8 (x+8)
=15
Bob's gift shop sold a record number of cards for Mother's Day. One salesman sold 42
cards, which was 4% of the cards sold for Mother's Day. How many cards were sold
for Mother's Day?
Multiply/scale up to solve.
cards
42
100
4
percent
Answers
Answer:
Bob sold 1,050 cards for Mother’s Day
Step-by-step explanation:
what is 5713 – 75 =
show your workings out
Answers
Answer:
5713 - 75
= 5638
Step-by-step explanation:
brainliest plz
Divide Rs 2500 in the proportion of 1:2:3:4 in four parts.
Answers
Answer:
250 : 500 : 750 : 1000
Step-by-step explanation:
2500 ÷ 1+2+3+4
2500 ÷ 10
250
Therefore the answer is,
1 : 2 : 3 : 4
250 : 500 : 750 : 1000
I NEED HELP ASAP WILL GIVE BRAINLIEST FOR ALL 3
Answers
2) i don't know answer for 2, but i know that it isn't right triangle because from Pythagoras' Theory we know that for right triangle: a^2 + b^2 = c^2 but in this triangle 30^2 + 38^2 ≠ 54^2
3) from Pythagoras' theory we know that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides: a^2 + b^2 = c^2
we have to check this in this triangle:
12^2 + 16^2 = 400
√400 = 20
we know that third side of this triangle is 20 so this triangle is right triangle
3) in square all sides are same so If a is the one side of the square then area of this square will be a x a = 242
a^2 = 242
so from Pythagoras' theorem we can found the diagonal:
a^2 + a^2 = c^2 ( c is hypothenuse)
242 + 242 = c^2
484 = c^2
c = 22
hope it's correct and it'll help you :)
Can someone help me, please?
Answers
please help me!!!!!
Answers
Answer:
The equation is 39=3r-6
Step-by-step explanation:
Let x = # of years from the start of the Great Depression to the first Presidential election of Nixon
Let r = Raul's age
The question gives us what x equals so...
x=39
Now we just turn the sentence into an equation.
"The number of years from the start of the Great Depression to the first Presidential election of Nixon [(x)] is six years less than three times Raul's age [(3r-6)]"
Remove the icky words...
x=3r-6
replace x with what it equals (39)
39=3r-6
Now you can go on to olve it
Overal Raul is only 15 :)
Find the Area. Use 3.14 for pi
Answers
Answer: 36.56
Step-by-step explanation:
To solve the equation, first acknowledge that the radius of the quarter circle shown at the top of the figure is 4 cm, so the rectangle's dimensions are 4 by (10-4) or 4 by 6, meaning the area is 24 cm^2.
Then address the circle, with an area of pi * r^2.
pi * r^2
3.14 *(4)^2
3.14 * 16
50.24
Then, because the figure shows 1/4 of a circle, divide 50.24 by 4.
50.24/4
12.56
Then add 24 to 12.56 to get 36.56.
Hope it helps :) and let me know if you want me to elaborate.