Find the midpoint of the segment below
Find an exact value. sine of negative eleven pi divided by twelve.
The sine of negative eleven pi divided by twelve is -0.5.
Explanation:To find the sine of negative eleven pi divided by twelve, we need to determine the exact value of this trigonometric function. The sine function represents the ratio between the length of the side opposite the angle and the hypotenuse in a right triangle. In this case, the angle is negative, which means it lies on the negative y-axis. Remember that the unit circle can help us determine the values of trigonometric functions for any angle. By mapping the angle on the unit circle, we can see that -11π/12 is equivalent to -330 degrees. Since the sine function has a period of 360 degrees, we can find the reference angle by subtracting 360 degrees. So, the reference angle is 30 degrees (360 - 330).The sine of 30 degrees is 0.5. However, the angle is negative, so the sine function is negative. Therefore, the exact value of sine of -11π/12 is -0.5.
Final Answer:
The exact value of [tex]\(\sin\left(-\frac{11\pi}{12}\right)\)[/tex] is:
[tex]\[\frac{\sqrt{2} + \sqrt{6}}{4}\][/tex]
Explanation:
To find the sine of an angle that is not one of the commonly known angles, such as [tex]\(-\frac{11\pi}{12}\)[/tex], we can use sine sum and difference identities. In this case, we know the sine of [tex]\(-\frac{\pi}{12}\)[/tex] is not a standard angle, but we can express it as a sum or difference of angles like [tex]\(-\frac{\pi}{4}\)[/tex] and [tex]\(-\frac{\pi}{3}\)[/tex] whose sines we do know.
The sum identity for sine is:
[tex]\[\sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b)\][/tex]
We can express the angle [tex]\(-\frac{11\pi}{12}\)[/tex] as the sum of two angles for which we do know the sine and cosine values, for example,[tex]\(-\frac{\pi}{4}\)[/tex] and [tex]\(-\frac{2\pi}{3}\)[/tex] because:
[tex]\[-\frac{11\pi}{12} = -\frac{3\pi}{4} - \frac{2\pi}{3}\][/tex]
Using the sum identity for sine, we can find the sine value as follows:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \sin\left(-\frac{3\pi}{4}\right)\cos\left(-\frac{2\pi}{3}\right) + \cos\left(-\frac{3\pi}{4}\right)\sin\left(-\frac{2\pi}{3}\right)\][/tex]
Now, let's find the values of sine and cosine for these angles.
[tex]\(\sin\left(-\frac{3\pi}{4}\right) = -\sin\left(\frac{3\pi}{4}\right) = -\sin\left(\frac{\pi}{2} + \frac{\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex]since sine is an odd function.
[tex]\(\cos\left(-\frac{2\pi}{3}\right) = \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2}\)[/tex] since cosine is an even function.
[tex]\(\cos\left(-\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2}\)[/tex] since cosine is an even function.
[tex]\(\sin\left(-\frac{2\pi}{3}\right) = -\sin\left(\frac{2\pi}{3}\right) = -\frac{\sqrt{3}}{2}\)[/tex] since sine is an odd function.
Substitute these trigonometric values into the sum identity:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \left(-\frac{\sqrt{2}}{2}\right)\left(-\frac{1}{2}\right) + \left(-\frac{\sqrt{2}}{2}\right)\left(-\frac{\sqrt{3}}{2}\right)\][/tex]
Simplify the expression:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}\][/tex]
Combine the terms with a common denominator:
[tex]\[\sin\left(-\frac{11\pi}{12}\right) = \frac{\sqrt{2} + \sqrt{6}}{4}\][/tex]
Therefore, the exact value of [tex]\(\sin\left(-\frac{11\pi}{12}\right)\)[/tex] is:
[tex]\[\frac{\sqrt{2} + \sqrt{6}}{4}\][/tex]
What is the solution set of the following equation? 3x2 - 6x = 0
The perimeter of a triangle is 510 ft and the sides are in the ratio of 11:16:24. Find the area of the triangle. Need help, is there a specific formula for this?
Final answer:
To find the area of a triangle with sides in a given ratio, first find the value of x by setting up and solving an equation using the perimeter. Then, multiply each side by x to find the base and height of the triangle. Finally, substitute the base and height into the formula for the area of a triangle.
Explanation:
To find the area of a triangle, we can use the formula: Area = 1/2 × base × height. In this case, we don't know the base and height directly, but we do know that the sides of the triangle are in the ratio of 11:16:24. Let's assume the sides are 11x, 16x, and 24x. The perimeter of the triangle is 510 ft, which means 11x + 16x + 24x = 510. Simplifying this equation, we get 51x = 510. Therefore, x = 10. Now we can find the base and height of the triangle by multiplying the corresponding sides by 10. The base is 11x * 10 = 110 ft and the height is 16x * 10 = 160 ft.
Now we can substitute these values into the formula to calculate the area of the triangle. Area = 1/2 × base × height = 1/2 × 110 ft × 160 ft = 8800 square feet.
what is a transformation that proportionally reduces or enlarges a figure.
Answer: Dilation is a transformation that proportionally reduces or enlarges a figure.
Step-by-step explanation:
A dilation a transformation that changes the size of the shape by using scale factor in particular ways .It stretches or shrinks the actual figure. It produces similar figures.
Since the corresponding sides of similar figures are in proportion.
⇒ It proportionally reduces or enlarges a figure.
Hence, A dilation is a transformation that proportionally reduces or enlarges a figure.
Final answer:
A scale transformation or dilation is a linear transformation that proportionally enlarges or reduces a figure, maintaining the proportional size relationships within the figure.
Explanation:
A transformation that proportionally reduces or enlarges a figure is known as a scale transformation or dilation. In such a transformation, lines are transformed into lines, and parallel lines remain parallel, consistent with the requirement for a transformation to be linear. This is important because it maintains the proportionality of the figure, meaning the size relationship of the parts of a figure to each other and to the whole figure remains constant, even though the overall size changes. Dilation can be characterized by a scale factor, which dictates how much larger or smaller the figure will become. If the scale factor is greater than 1, the figure enlarges; if it is between 0 and 1, the figure reduces in size.
A chocolate bar measures 40 mm wide, 80 mm long, and 5 and 1 over 2 mm high. Will’s Wrapping Company is making a wrapper to cover the chocolate bar. Use the correct net and determine how much paper will be needed to make the wrapper
Answer:
Will’s Wrapping Company needs 7,720 mm of paper to wrap the chocolate bar.
Carissa is budgeting for her next semester at college. She spent $50 on gas this month. Economists are predicting gas prices to rise 8% per month for the next 6 months. How much should she budget for gas if classes start next month and are 4 months long?
The answer is D.243.33
(I looked up the question and couldnt find it so i thought i'd post it for anyone who needed it)
On a road trip you and your family stop a truck stop to take a break and to let your puppy Fido stretch. You notice that they have a triangular dog park that is fenced in on two sides. The third side of the field is formed by a creek. If the fences measure 150 feet and 98 feet, and the side along the creek is 172 feet, what are the measures of the angles made by the dog park?
Answer:
Angles are 34.59°, 85.08° and 60.33°
Step-by-step explanation:
Let ABC is a triangle, ( that show the dog park)
In which,
AB = 150 feet
BC = 98 feet
CA = 172 feet,
By the cosine law,
[tex]BC^2=AB^2+AC^2-2(AB)(AC)cos A[/tex]
[tex]2(AB)(AC)cos A=AB^2+AC^2-BC^2[/tex]
[tex]\implies cos A=\frac{AB^2+AC^2-BC^2}{2(AB)(AC)}-----(1)[/tex]
Similarly,
[tex]\implies cos B=\frac{AB^2+BC^2-AC^2}{2(AB)(BC)}-----(2)[/tex]
[tex]\implies cos C=\frac{BC^2+AC^2-AB^2}{2(BC)(AC)}-----(3)[/tex]
By substituting the values in equation (1),
[tex]cos A=\frac{150^2+172^2-98^2}{2\times 150\times 172}[/tex]
[tex]=\frac{22500+29584-9604}{51600}[/tex]
[tex]=\frac{42480}{51600}[/tex]
[tex]\approx 0.8233[/tex]
[tex]\implies m\angle A\approx 34.59^{\circ}[/tex]
Similarly,
From equation (2) and (3),
m∠B ≈ 85.0°, m∠C ≈ 60.33°
Zeus Industries bought a computer for $2011. It is expected to depreciate at a rate of 24% per year. What will the value of the computer be in 5 years?
how many integers from 42 to 92, inclusive, have a remainder of 4 when divided by 6
If they have a remainder of 4 when divided by six the sequence of terms are: 6n+4
42≤6n+4≤92
subtract 4 from both ends
38≤6n≤88
divide both ends by 6
6.3≤n≤14.6
since n must be an integer...
7≤n≤14 so the number of integers is:
14-7+1=8
they are 6n+4 for n=[7,14]...46,52,58,64,70,76,82,88
There are 8 integers between 42 and 92, inclusive, that have a remainder of 4 when divided by 6. These integers can be expressed in the form of 6n + 4 and can be found by incrementally adding 6, starting from the smallest integer greater than 42 that meets this condition.
To find how many integers from 42 to 92, inclusive, have a remainder of 4 when divided by 6, we need to identify the numbers that fit the condition. These numbers are of the form 6n + 4, where n is an integer. Starting with the smallest number greater than 42 that fits the condition, which is 46 (as 6*7 + 4 = 46), we can find the next by adding 6 to get 52, then 58, and so on, up until we reach the largest number less than or equal to 92 that fits the condition, which is 88 (as 6*14 + 4 = 88).
To find the total count, we calculate (88 - 46) / 6 + 1 = 42 / 6 + 1 = 7 + 1 = 8. Therefore, there are 8 numbers between 42 and 92 that have a remainder of 4 when divided by 6.
10-2v=-5-50
+5 +5
10-3v=-50
+3 +3
10=47
An object is dropped from a height of 1,600 feet. The amount of time, in seconds, the object takes to hit the ground can be found by solving the equation −16t2+1,600=0. How many seconds will it take to hit the ground?
To find the time it takes for the object to hit the ground, we can solve the quadratic equation -16t^2 + 1600 = 0 using the quadratic formula. The positive solution to this equation gives us the time in seconds. Therefore, the object will take √102400/32 seconds to hit the ground.
Explanation:To find the time it takes for the object to hit the ground, we need to solve the equation -16t^2 + 1600 = 0. This is a quadratic equation, so we can use the quadratic formula. Plugging in the values, we get t = (-b ± √(b^2 - 4ac))/(2a). In this case, a = -16, b = 0, and c = 1600. Plugging in these values, we get t = (± √(0 - 4(-16)(1600)))/(2(-16)). Simplifying the equation further, we get t = (± √(0 + 102400))/(32). This gives us two possible values for t: t = √102400/32 and t = -√102400/32. Since time cannot be negative in this context, we can discard the negative solution. Therefore, the object will take √102400/32 seconds to hit the ground.
To make lemonade, you used one gallon of water, four quarts of lemon juice, and 32 ounces of honey. How much lemonade did you make?
0.43 is 10 times as mush as what
The number that is 10 times less than 0.43 is 0.043, found by dividing 0.43 by 10.
To find the number that 0.43 is 10 times as much as, we need to divide 0.43 by 10. This is the same as multiplying by 10 raised to the power of -1, which is a division by 10.The operation looks like this:
0.43*10^{-1} = 0.043.
Therefore, 0.43 is 10 times as much as 0.043
Which statistical test would be most appropriate for examining the relationship between temperature and the number of ice cream cones sold?
The most appropriate statistical test for examining the relationship between temperature and the number of ice cream cones sold would be a correlation analysis, specifically a Pearson correlation coefficient.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
Correlation coefficient test measures the strength and direction of the linear relationship between two continuous variables, which is suitable for examining the relationship between temperature and the number of ice cream cones sold.
Additionally, a scatterplot could be used to visually assess the relationship between the two variables before conducting the statistical test.
Hence, statistical test for examining the relationship between temperature and the number of ice cream cones sold would be a correlation analysis
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Find the number of permutations of the first 8 letters of the alphabet taking 4 letters at a time.
For the function h defined by h(x)=2x2−2, find h(−12)
To find h(-12), substitute -12 into the function h(x)=2x²-2. The value of h(-12) is 286.
Explanation:To find h(-12), we substitute -12 into the function h(x)=2x²-2:
h(-12) = 2(-12)² - 2
h(-12) = 2(144) - 2
h(-12) = 288 - 2
h(-12) = 286
Therefore, h(-12) is equal to 286.
Write the equations in graphing form, then state the vertex of the parabola or the center and radius of the circle.
y=2x^2+16
A card is chosen at random from a standard deck of 52 playing cards. What is the probability that the chosen card is a jack or a queen?
Find the value of x in the equation √ x + 5 = √ x + 45.
An ostrich that is 108 inches tall is 20 inches taller than 4 times the height of a kiwi. What is the height of a kiwi in inches?
A polygon with 12 sides is called a/an
In geometry, a dodecagon or 12-gon is any twelve-sided polygon.
hope this helps!
which of the following fractions will convert to repeating decimals?
a. 1/3
b. 1/6
c. 1/4
d. 3/8
e. 1/5
A square prism has a base length of 5 m, and a square pyramid of the same height also has a base length of 5 m. Are the volume the same?
Melissa has scored 83 , 84 , 93 , 77 , and 85 on her previous five tests. what score does she need on her next test so that her average (mean) is 83 ?
83 x 6 = 498
all six of her scores need to equal at least 498 for an 83 average
83 + 84 + 93 + 77 + 85 = 422
498-422 = 76
she needs a 76
Please help me with my geometry!
Special air bags are used to protect scientific equipment when a rover lands on the surface of Mars. On Earth, the function approximates an object's downward speed in feet per second as the object hits the ground after bouncing x ft in height. The corresponding function for Mars is compressed vertically by a factor of about 2/3. Estimate to the nearest tenth how fast a rover will hit Mars' surface after a bounce of 15 ft in height.
A) 20.7 ft/s
B) 25.3 ft/s
C) 7.3 ft/s
D) 46.5 ft/s
The answer is
20.7 ft/s
Without the specific function that calculates the object's downward speed on Earth, it is not possible to estimate the impact speed on Mars even with knowing the vertical compression factor.
Explanation:Estimating Impact Speed on Mars
The question asks us to estimate how fast a rover would hit the surface of Mars after bouncing 15 ft in height, with the Mars function being compressed vertically by a factor of about 2/3 compared to the function used on Earth. Although the specific Earth function isn't provided, we can still proceed with a general understanding. To find this, we would normally use the Earth function for speed and multiply the result by 2/3 to adjust for Mars' weaker gravity. However, since we are only given the option to select an answer, no calculation can be performed without the Earth function. Therefore, it's not possible to provide a confident answer to this question.
Which is a stretch of an exponential decay function?
f(x) =4/5(5/4)^x
f(x) =4/5(4/5 )^x
f(x) =5/4(4/5)^x
f(x) =5/4(5/4)^x
The length of a rectangle is 6 feet more than twice the width. if the length is 24 feet, what is the width?
Find the measure of an angle whose supplement is 9 times the measure of its complement
The equation is 180 - x = 9 (90 - x).Then the measure of the angle will be 78.75 degrees.
What is an angle?The inclination is the separation seen between planes or vectors that meet. Degrees are another way to indicate the slope. For a full rotation, the angle is 360°.
Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.
Complementary angle - Two angles are said to be complementary angles if their sum is 90 degrees.
Let 'x' be the angle. Then the equation is given as,
180 - x = 9 (90 - x)
180 - x = 810 - 9x
8x = 630
x = 78.75 degrees
The measure of the angle will be 78.75 degrees.
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