How to show (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1 ?
Answers
Following shows (sin 7x)/sin x = 64(cos x)^6 - 80(cos x)^4 +24(cos x)^2 - 1:
12sin^3 x - 8sin^
To show that (sin 7x)/sin x is equal to 64(cos x)^6 - 80(cos x)^4 + 24(cos x)^2 - 1, we can use trigonometric identities and algebraic manipulation.
Let's start with the left-hand side of the equation:
(sin 7x)/sin x
Using the trigonometric identity for sin(A + B):
sin(A + B) = sin A cos B + cos A sin B
We can rewrite sin 7x as sin (6x + x):
sin (6x + x) = sin 6x cos x + cos 6x sin x
Now we can substitute sin 7x with sin 6x cos x + cos 6x sin x:
(sin 6x cos x + cos 6x sin x)/sin x
Next, we can simplify this expression by dividing both terms by sin x:
(sin 6x cos x)/sin x + (cos 6x sin x)/sin x
The sin x term cancels out, leaving us with:
sin 6x cos x + cos 6x
Now, we can use the double-angle identity for sin 2A:
sin 2A = 2sin A cos A
To rewrite sin 6x cos x, we can treat it as sin 2A with A = 3x:
sin 6x cos x = 2sin 3x cos 3x
Next, we can use the triple-angle identity for sin 3A:
sin 3A = 3sin A - 4sin^3 A
To rewrite sin 3x, we can treat it as sin A with A = x:
sin 3x = 3sin x - 4sin^3 x
Substituting this into our expression:
2sin 3x cos 3x = 2(3sin x - 4sin^3 x) cos 3x
Expanding further:
= 6sin x cos 3x - 8sin^3 x cos 3x
Now, we can use the double-angle identity for cos 2A:
cos 2A = cos^2 A - sin^2 A
To rewrite cos 3x, we can treat it as cos A with A = x:
cos 3x = cos^2 x - sin^2 x
Substituting this into our expression:
6sin x cos 3x - 8sin^3 x cos 3x = 6sin x (cos^2 x - sin^2 x) - 8sin^3 x (cos^2 x - sin^2 x)
Expanding further:
= 6sin x cos^2 x - 6sin x sin^2 x - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
Now, we can use the Pythagorean identity for sin^2 x + cos^2 x:
sin^2 x + cos^2 x = 1
To rewrite sin^2 x, we can subtract cos^2 x from both sides:
sin^2 x = 1 - cos^2 x
Substituting this back into our expression:
= 6sin x (cos^2 x - (1 - sin^2 x)) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
= 6sin x (2sin^2 x) - 8sin^3 x cos^2 x + 8sin^3 x sin^2 x
= 12sin^3 x - 8sin^
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Related Questions
Develop a single formula you can use to find the profit earned by both the Alcone Company & Besame Beauty for their lip color. Explain why your formula will work for both companies
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Answer:
I have no idea!!!!!!!!!!!!!
why did the girl wear glasses during math class
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Answer:
Because she found it improves division.
Step-by-step explanation:
answer for brainlist + extra points
Which operation will not change the value of any nonzero number?
A. Multiplying by One
B. Dividing by Zero
C. Multiplying by Zero
D. Adding One
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Answer:
A. Multiplying by One
Step-by-step explanation:
semoga membantu
What is the radius of F?
Answers
The radius of circle F is equal to: C. 12.
What is Pythagorean theorem?In Mathematics and Geometry, Pythagorean's theorem is modeled or represented by the following mathematical equation (formula):
a² + b² = c²
Where:
a, b, and c represents the length of sides or side lengths of any right-angled triangle.
By substituting the given side lengths into the formula for Pythagorean's theorem, we have the following;
a² + b² = c²
a² + 9² = 15²
a² = 225 - 81
a = √144
a = 12 units.
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PLEASE HELP ME ASAP!!! ILL MARK BRIANLIEST AND GIVE EXTRA POINTS1. When writing a linear equation for a word problem, what are some things to keep in mind or some keywords to watch out for? What do those keywords mean/ why should you watch out for those things?
Answers
Answer:
Some keywords include "starting amount," you should look out for this because it will be the y-intercept for the equation. Another keyword may be "daily" or "monthly" or "yearly," which would be the coefficient of x, or the amount that y increases with an increase of 1 in x.
pls mark brainliest :)
helppp asap Given:
Prove: ΔKVM ~ ΔBVG
Answers
Triangle KVM is similar to triangle BVG because angle M = angle G = 90° and angle V is common to both triangles.
What are similar triangles?Two triangles are similar if the angles are the same size or the corresponding sides are in the same ratio.
For two triangles to be similar, the corresponding angles must be congruent i.e equal.. Also the ratio of the corresponding sides of similar triangles are equal.
angle M and G are both 90° , this means they are equal.
angle KVM = BVG
therefore angle K = angle B
Since all the corresponding angles are equal, we can say triangle KVM is similar to triangle BVG
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A circle whose center is at (1, -3) passes through the point (7, 5). whats the radius?
Answers
The distance from the center to any point on the circle exists comprehended as the radius. The radius is 10cm.
What defines the radius of a circle?The radius of a circle is the separation between any two points on the circle and its center. The radius of a circle is the separation between any two points on its circumference. R or r is typically used to indicate it.
The distance along a line connecting a circle's center to a point on its perimeter is known as a circle's radius. Therefore, the radius is equal to half the diameter's length. Therefore, the radius is equal to half the diameter's length.
The radius of a circle is the separation between any two points on the circle and its center.
In this instance (using the Pythagorean Theorem)
r = \(\sqrt{(7-1)^{2}+(5-(-3))^{2} }\)
= √(36 + 64)
r = 10
If a circle whose center is at (1, -3) passes through the point (7, 5) then the radius of the circle exists 10.
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Write an expression (Simplified) for the perimeter of the given figure.
PLS ANSWER ASAP
Answers
Answer:
p = 6x + 10
Step-by-step explanation:
p = 2(l + w)
p = 2(6 + (3x - 1))
Simplify terms inside parenthesis
p = 2(3x + 5)
Distribute
p = 6x + 10
y<-3x+3
y
a. (1,-5)
b.(1,5)
c.(5,1)
d.(-1,5)
Answers
Answer: a
Step-by-step explanation:
substitute the coordinate values for x and y
y<3x+3, (1,-5)
-5<3(1)+3
-5<3+3
-5<6
the coordinate (1,-5) works, so a is the answer
Click and drag like terms onto each other to simplify fully.
2-5+7y-5x+7x-2y
Answers
Answer: 5y + 2x - 3
(In simplest form)
consider an invertible symmetric n×n matrix a. when does there exist a nonzero vector v in rn such that av is orthogonal to v? give your answer in terms of the signs of the eigenvalues of a.
Answers
If an invertible symmetric n×n matrix A has a nonzero vector v in R^n such that Av is orthogonal to v, then it means that the dot product of Av and v is equal to zero. In other words, Av . v = 0. Using th peroperties of the dot product, we can rewrite this equation as:
v . A^T v = 0
Since A is symmetric, A^T = A, and we can rewrite the equation as:
v . A v = 0
This equation can be further simplified using the properties of the dot product:
v . A v = (A v) . v = 0
From this equation, we can see that the eigenvalues of A must be either positive or negative. If the eigenvalues of A are all positive, then v . A v > 0, which contradicts the equation v . A v = 0. Similarly, if the eigenvalues of A are all negative, then v . A v < 0, which also contradicts the equation v . A v = 0.
Therefore, the only way for there to exist a nonzero vector v in R^n such that Av is orthogonal to v is if the eigenvalues of A are both positive and negative. In other words, A must have both positive and negative
eigenvalues.
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HELP ME PLZ I WILL GIVE 30 POINTS!!!!!
What is the length of the hypotenuse of the triangle?
Triangle A B C. Side A C is 7 feet and side C B is 4 feet. Hypotenuse A B is unknown.
a.StartRoot 22 EndRoot ft
b.StartRoot 33 EndRoot ft
c.StartRoot 57 EndRoot ft
d.StartRoot 65 EndRoot ft
Answers
Answer:
d
Step-by-step explanation:
Step-by-step explanation:
Using Pythagoras' identity on the right triangle.
The square on the hypotenuse is equal to the sum of the squares on the other 2 sides, that is
AB² = 7² + 4² = 49 + 16 = 65 ( take the square root of both sides )
AB = \sqrt{65} → D
Instructions: Find the missing side lengths. Leave your answers as radicals in simplest form.
Answers
Answer:
x = 16
y = 8√3
Step-by-step explanation:
This is a special right triangle with angle measures of 30° 60° 90° and side lengths x, x√3, and 2x
The side length that sees angle measure of 30° is equal to x
and the side length that sees 60° is equal to x√3 and the side length that sees 90° is equal to 2x
The image of the point B(8,-3) under the translation T6,0?
Answers
Answer:
(14, - 3 )
Step-by-step explanation:
Under the translation < 6, 0 >
Add 6 to the x- coordinate and 0 to the y- coordinate, that is
B(8, - 3 ) → B'(8 + 6, - 3 + 0 ) → B'(14, - 3 )
Which of the following numbers are greater than 6 and less than 8?
Answers
Answer:
\(\sqrt{60}\)Explanation:
We are required to determine which of the given numbers are greater than 6 and less than 8.
\(6^2=36\text{ and }8^2=64\)\(\begin{gathered} \sqrt{7}<\sqrt{36}=6 \\ \text{Therefore, }\sqrt{7}\text{ is less than 6.} \end{gathered}\)Secondly:
\(\begin{gathered} 6=\sqrt{36}<\sqrt{60}<\sqrt{64}=8 \\ 6<\sqrt{60}<8 \\ \text{Therefore, }\sqrt{60}\text{ is greater than 6 and less than 8.} \end{gathered}\)Finally:
\(\begin{gathered} 6=\sqrt{36}<\sqrt{80}>\sqrt{64}=8 \\ \sqrt{80}>8>6 \\ \text{Therefore, }\sqrt{80}\text{ is greater than 6 and 8.} \end{gathered}\)Therefore, the number which is greater than 6 and less than 8 is the square root of 60.
find an equation for the curve that passes through the point (1,0) and whose slope at (x,y) is xe^y.
Answers
An equation for the curve that passes through the point (1,0) and whose slope at (x,y) is xe^y is y = -1/x -1.
Define slope.The value of the steepness or the direction of a line in a coordinate plane is referred to as the slope of a line, also known as the gradient. Given the equation of a line or the coordinates of points situated on the straight line, slope can be determined using a variety of approaches. The slope is nothing more than the tangent to the angle formed with the x-axis measured. As a result, it is simply an angle's measurement.
Given,
The curve that passes through the point (1,0) and whose slope at (x,y) is xe^y.
Since x-2 is equal to the slope of the function f(x), we can express that as
dy/dx = x^-2
dx multiplication and integration of both sides:
y = -1/x + C
Connecting the provided point:
0 = -1/1 + C
C = -1
Consequently, the curve's equation is
y = -1/x -1
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Let x and y be some integers. Consider the following statement: If xis odd and y is odd, then xy is odd. Proof. Assume that x is odd and y is odd. Then there exists some k integers such that x=2k+1 and m=2k+1. Then xy=(2k+1)(2k+1)=4k2+4k+1=2(2k2+2k)+1. Since 2k2+2k is an integer because k is an integer then xy is odd. The proof is correct but the statement is incorrect The statement is correct, but the proof is incorrect. The statement and proof are incorrect. The statement and the proof are correct.
Answers
hence proved If number is 2t + 1 where t belongs to integer, then it is odd integer.
Odd number integers = 2k + 1, where k is integer
Even number integer = 2k
Odd integer + even integer
= 2k + 1 + 2k
= 4k + 1
= 2(2k) + 1
Let 2k = t, where t is integer
= 2t + 1
= Odd integer by definition
If number is 2t + 1 where t belongs to integer, then it is odd integer.
Hence proved.
The question is incomplete. The complete question is :
Each statement below involves odd and even integers. An odd integer is an integer that can be expressed as 2k+1, where k is an integer. An even integer is an integer that can be expressed as 2k, where k is an integer. Prove each of the following statements using a direct proof. (a) The sum of an odd and an even integer is odd
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write down five numbers, so that the mean is 6, the median is 5 and the mode is 4.
Answers
Answer:
10, 4, 5, 7, 4
Ecological approach to algal bloom control. Algal blooms can have negative effects on an ecosystem by dominating its phytoplankton communities. Gonyostomum semen is a nuisance alga infesting many parts of northern Europe. Could the overall biomass of G. semen be controlled by grazing zooplankton species? A research team examined the relationship between the net growth rate of G. semen and the number of Daphnia magna grazers introduced in test tubes. A net growth rate was computed by comparing the initial and final abundances of G. semen in the experiment, with a negative value indicating a decrease in abundance. Here are the findings: 13 1 2 3 4 5 6 Number of grazers Net growth rate -1.9 -2.5 -2.2 -3.9 -4.1 -4.3 a. Make a scatterplot of number of grazers and net growth rate. Do you think that D. magna is an effective grazer of the G. semen alga? b. Find the correlation 1. How does it support your interpretation?
Answers
a. D. magna is an effective grazer of the G. semen alga. b. a very strong negative correlation between the number of grazers and the net growth rate of G. semen.
a. The scatterplot of the data shows a clear negative correlation between the number of D. magna grazers and the net growth rate of G. semen. As the number of grazers increases, the net growth rate of G. semen decreases. This suggests that D. magna is an effective grazer of the G. semen alga.
b. The correlation coefficient between the number of grazers and the net growth rate is -0.996. This indicates a strong negative correlation between the two variables, which supports the interpretation that D. magna is an effective grazer of G. semen. The closer the correlation coefficient is to -1, the stronger the negative correlation between the two variables. In this case, the correlation coefficient is very close to -1, which indicates a very strong negative correlation between the number of grazers and the net growth rate of G. semen.
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DESCRIBE PRINCIPAL ROOTS AND TELLS WHETHER THEY ARE RATIONAL OR IRRATIONAL?
TANONG MO SA NANAY MO NANG NASAGUTAN MO BRUH
Answers
Answer:
The principal roots are rational numbers
Step-by-step explanation:
Principal roots :
The principal root is determine if the radicand is a perfect n t h power of a number then the root is rational.
Therefore the principal root is a rational number
Examples of a principal root
a) √4 = 2
b) \(\sqrt{0.81} =0.9\)
c) \(\sqrt{\frac{9}{36} } = \frac{\sqrt{9} }{36} = \frac{3}{6} = \frac{1}{2}\)
i will give the crown thing if answered correctly
Answers
Answer: x=1/4
Step-by-step explanation:
Answer
=0
Step-by-step explanation:
Diego rounded a number to the nearest whole number and got 12.
Which number could be the number Diego rounded to the nearest whole number?
Answers
Answer:
Below
Step-by-step explanation:
N, the number can be anything greater than or equal to 11.5 and less than 12.5
11.5 ≤ n < 12.5
Any rational number can be expressed as the ratio of two integers.
Answers
The given statement about rational numbers being defined as can be expressed as the ratio of two integers is; True
What is a Rational Number?There are different types of numbers in maths such as rational number, irrational numbers, real numbers, integers e.t.c
Now, a rational number can be defined as any number which can be represented in the form of p/q where q ≠ 0. This means as the ratio of two integers.
Thus, we can say that any fraction fits under the category of rational numbers, provided the denominator and numerator are integers and the denominator is not equal to zero.
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How do I solve the inequality -y>-31
Answers
\(-y>-31\)
When you multiply bothsides of an inequality by a negative sign , the inequality sign changes
\(\begin{gathered} -y>-31 \\ multiply\text{ }bothsides\text{ }with-1 \\ y<31 \end{gathered}\)Part B at 10:00 a.m., lin's was 7 meters. Add this point to the graph
at 1:00 p.m., lin was at sea level. add this point to the graph.
Answers
Answer:
add a graph so we can help you
Step-by-step explanation:
ASAP PLZ HELP ME WILL AMRI AS BRAINLIST
Answers
Answer:
x=4
Step-by-step explanation:
Answer:
x=4
Step-by-step explanation:
Find the measure of Angle ADE
Answers
Answer:
30°
Step-by-step explanation:
because the sum of all interior angles in a triangle is 180 degrees
180°= ∡ADE+∡DAE+∡AED
∡ADE = 180°-∡DAE -∡AED
because the diagonals are perpendicular so they form 90° angles, and ∡BRC≅∡AED(vertical angles)
∡AED =90
To find ∡DAE we need to find the value of y
11y+2 =90
11y =90-2
y=88/11
y=8
∡DAE=7y+4=7 ·8 +4= 56+4 =60°
now we know what we need to find ∡ADE
∡ADE = 180°-∡DAE -∡AED
∡ADE = 180°-60° -90°
∡ADE = 30°
Can someone pls help me
Answers
Answer:
10
Step-by-step explanation:
use Pythagoras Theron
a^2+b^2=c^2
let ms know if you have further questions
Using Pythagorean theorem
\(\\ \sf\longmapsto H^2=P^2+B^2\)
\(\\ \sf\longmapsto H^2=(5√3)^2+5^2\)
\(\\ \sf\longmapsto H^2=75+25\)
\(\\ \sf\longmapsto H^2=100\)
\(\\ \sf\longmapsto H=10\)
Are these two triangles similar? If so, because of which postulate?
Answers
SSS, or Side-Side-Side Similarity, is what makes these two triangles alike.
What are a triangle's three sides?A right triangle's hypotenuse is its longest side, its "opposite" side was the one that faces a specific angle, and its "adjacent" side is the one that faces the angle in question. To define the edges of right triangles, we use specific terminology.
What are the kinds of triangles?A triangle represents a closed, three-sided object in two dimensions. The various kinds of circles go by various titles. Three different kinds of triangles—scalene, isosceles, and equilateral—are distinguished by the lengths of their sides.
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the length of human pregnancies (with no complications) from conception to birth caries according to a normal distribution with a mean of 260 days and a standard deviation of 10 days. five percent of pregnancy lengths are shorter than what value?
Answers
By using the concept of z-score, the pregnancy length value below which the shortest 5% of pregnancies fall is approximately 243.55 days (rounded to two decimal places).
The pregnancy length that is shorter than 5% is the value at the 5th percentile. The z-score for this value can be calculated as shown:
\($z=\frac{x-\mu}{\sigma}$$\)
Where x is the unknown value, µ is the mean of the distribution and σ is the standard deviation.
Hence, we have that,
\($-1.64=\frac{x-260}{10}$$\)
\($x=260+(-1.64)\cdot10 = 243.6$$\)
Therefore, by using the z-score formula 5% of pregnancy lengths are shorter than 243.6 days.
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