The exact value of cos 15° is (√6 - √2)/4.
To find the exact value of cos 15°, we can use the trigonometric identities and special angles to determine its value.
One of the key identities that can be used is the cosine addition formula: cos (A + B) = cos A x cos B - sin A x sin B.
We can rewrite 15° as the sum of two special angles: 15° = 45° - 30°.
Using the cosine addition formula, we have:
cos 15° = cos (45° - 30°) = cos 45° x cos 30° - sin 45° x sin 30°.
Now, we know the exact values of cos 45° (which is (√2)/2), cos 30° (which is (√3)/2), sin 45° (which is (√2)/2), and sin 30° (which is 1/2).
Substituting these values into the equation, we get:
cos 15° = (√2)/2 x (√3)/2 - (√2)/2 x 1/2.
Simplifying further, we have:
cos 15° = (√6 - √2)/4.
This means that cos 15° can be expressed exactly as (√6 - √2)/4 without any decimal approximation.
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Answer:
the answer is check2+check6/4
Step-by-step explanation:
i took an exam and got it right
(i dont know what those check mark things are called or how to put the actual thing on a computer so hopefully you understand the check part)
What speed must you toss a ball straight up so that it takes 4 s to return to you? Show your work.
Given that point U is the circumcenter of triangle XVZ, which segments are congruent?
Answer: [tex]\overline{WX}\cong \overline{WV},\ \overline{VA}\cong\overline{AZ}[/tex]
[tex]\overline{XY}\cong\overline{YZ}[/tex]
[tex]\overline{UV}\cong \overline{UZ}\cong \overline{UX}[/tex]
Step-by-step explanation:
In the given figure we have a triangle , in which U is the circumcenter of triangle XVZ.
We know that the circumcenter is equidistant from each vertex of the triangle.
Since , the line segments which are representing the distance from the vertex and the circumcenter are [tex]\overline{UV},\ \overline{UZ},\ \overline{UX}[/tex]
Also, The circumcenter is at the intersection of the perpendicular bisectors of the triangle's sides.
Then , [tex]\overline{WX}\cong \overline{WV},\ \overline{VA}\cong\overline{AZ}[/tex] and [tex]\overline{XY}\cong\overline{YZ}[/tex]
Hence, the segments which are congruent are [tex]\overline{UV}\cong \overline{UZ}\cong \overline{UX}[/tex]
[tex]\overline{WX}\cong \overline{WV},\ \overline{VA}\cong\overline{AZ}[/tex]
[tex]\overline{XY}\cong\overline{YZ}[/tex]
What is the lcm of 9,45,81?
3x+6y=18. 3y=-3/2x+9 solve as a substitution problem
Stella graphs the equation y=13x – 2y=13x – 2 .
Select all statements about Stella's graph that are true.
The graph is a straight line.
The line passes through the origin.
The line passes through the point (0, –2)(0, –2) .
The slope of the line is 3.
The y-intercept of the line is 2.
The height of a coconut falling from a tree can be represented by the function h(t)=-16t^2 + 24, where h(t) is the height of the coconut, in feet, and t is time, in seconds.
What is the initial height, in feet, of the coconut?
Answer:
The answer is C "The values of h(t) when t = 4 and 5 should be 0."
Use the remainder theorem to determine whether x - 2 is a factor of
f(x) = x^3 + 3x^2 - x - 18
A) Yes, x - 2 is a factor of f(x) because f(2) = 0
B) No, x - 2 is not a factor of f(x) because f(2) = 0
C) Yes, x - 2 is a factor of f(x) because f(-2) = -12
D) No, x - 2 is not a factor of f(x) because f(-2) = -12
Gasoline
Costs $3.37 per gallon.mary's father put 9 gallon of gasoline in the tank of his car.how much will the gasoline cost
The slope of the line tangent to the curve y^2 + (xy+1)^3 = 0 at (2, -1) is ...?
The slope of the line tangent to the curve y^2 + (xy+1)^3 = 0 at (2, -1) is -3/4.
Explanation:The slope of the line tangent to the curve y^2 + (xy+1)^3 = 0 at (2, -1) can be found using the concept of implicit differentiation. To find the slope, we need to differentiate the equation with respect to x and then substitute the coordinates (2, -1) into the resulting equation. Let's solve it step by step.
First, we differentiate the equation implicitly with respect to x:
2y * dy/dx + 3(xy + 1)^2 * (y + x * dy/dx) = 0
Next, we substitute the values x = 2 and y = -1 into the equation:
2(-1) * dy/dx + 3(2(-1) + 1)^2 * (-1 + 2 * dy/dx) = 0
Simplifying the equation:
-2dy/dx - 3 * 1 * (-1 + 2dy/dx) = 0
-2dy/dx + 3 + 6dy/dx = 0
Combining like terms:
4dy/dx = -3
Finally, solving for dy/dx, we get:
dy/dx = -3/4
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The slope of the tangent line to the curve at the point \((2, -1)\) is:
[tex]\[\boxed{\frac{3}{4}}\][/tex]
To find the slope of the tangent line to the curve given by the equation [tex]\( y^2 + (xy + 1)^3 = 0 \)[/tex] at the point [tex]\( (2, -1) \)[/tex], we need to use implicit differentiation.
Given the curve:
[tex]\[y^2 + (xy + 1)^3 = 0\][/tex]
We differentiate both sides with respect to x . Using the chain rule and implicit differentiation, we get:
[tex]\[\frac{d}{dx} [y^2] + \frac{d}{dx} [(xy + 1)^3] = 0\][/tex]
First, differentiate [tex]\( y^2 \):[/tex]
[tex]\[\frac{d}{dx} [y^2] = 2y \frac{dy}{dx}\][/tex]
Next, differentiate [tex]\( (xy + 1)^3 \)[/tex] using the chain rule:
[tex]\[\frac{d}{dx} [(xy + 1)^3] = 3(xy + 1)^2 \cdot \frac{d}{dx} [xy + 1]\]\[= 3(xy + 1)^2 \cdot (y + x \frac{dy}{dx})\][/tex]
Putting it all together, we get:
[tex]\[2y \frac{dy}{dx} + 3(xy + 1)^2 (y + x \frac{dy}{dx}) = 0\][/tex]
Now, substitute [tex]\( x = 2 \) and \( y = -1 \)[/tex] into the equation:
[tex]\[2(-1) \frac{dy}{dx} + 3((2)(-1) + 1)^2 \left( -1 + 2 \frac{dy}{dx} \right) = 0\]\[-2 \frac{dy}{dx} + 3(-2 + 1)^2 \left( -1 + 2 \frac{dy}{dx} \right) = 0\][/tex]
[tex]\[-2 \frac{dy}{dx} + 3(-1)^2 \left( -1 + 2 \frac{dy}{dx} \right) = 0\]\[-2 \frac{dy}{dx} + 3(1) \left( -1 + 2 \frac{dy}{dx} \right) = 0\][/tex]
[tex]\[-2 \frac{dy}{dx} + 3(-1 + 2 \frac{dy}{dx}) = 0\]\[-2 \frac{dy}{dx} + 3(-1 + 2 \frac{dy}{dx}) = 0\][/tex]
[tex]\[-2 \frac{dy}{dx} + 3(-1) + 6 \frac{dy}{dx} = 0\][/tex]
[tex]\[-2 \frac{dy}{dx} - 3 + 6 \frac{dy}{dx} = 0\][/tex]
[tex]\[4 \frac{dy}{dx} - 3 = 0\][/tex]
[tex]\[4 \frac{dy}{dx} = 3\][/tex]
[tex]\[\frac{dy}{dx} = \frac{3}{4}\][/tex]
So, the slope of the tangent line to the curve at the point \((2, -1)\) is:
[tex]\[\boxed{\frac{3}{4}}\][/tex]
What is the length of the altitude of the equilateral triangle below
Method 1
Applying the Pythagorean Theorem
we know that
[tex]10^{2}= 5^{2} +a^{2}[/tex]
Solve for a
[tex]100= 25 +a^{2}[/tex]
[tex]a^{2}=100-25[/tex]
[tex]a^{2}=75[/tex]
[tex]a=\sqrt{75}=5 \sqrt{3}\ units[/tex]
therefore
the answer is
the length of the altitude is [tex]5 \sqrt{3}\ units[/tex]
Method 2
we know that
[tex]sin(60\°)=\frac{\sqrt{3}}{2}[/tex] -------> equation A
and
in this problem
[tex]sin(60\°)=\frac{a}{10}[/tex] --------> equation B
equate equation A and equation B
[tex]\frac{\sqrt{3}}{2}=\frac{a}{10}\\\\a=\frac{10\sqrt{3}}{2}\\\\a=5 \sqrt{3}\ units[/tex]
therefore
the answer is
the length of the altitude is [tex]5 \sqrt{3}\ units[/tex]
please help with math
1. Provide a counterexample for the statement in parentheses. (1 point)“If a figure has four sides, then it is a square”
2. Identify the hypothesis of the statement in parentheses. (1 point)“If two angles form a linear pair, then they are adjacent and supplementary.”
3. Identify the conclusion of the statement in parentheses. (1 point)“If two angles form a linear pair, then they are adjacent and supplementary.
4. Write the statement in parentheses as a biconditional. (1 point)“If two angles form a linear pair, then they are adjacent and supplementary.”
5. Is the statement in parentheses a good definition? Explain. (2 points)“Obtuse angles are fairly large.”
Forty-six and seven thousandths in decimal form
Shape 1 and shape 2 are plotted on a coordinate plane. Which statement about the shapes is true?
Shape 1 is congruent to shape 2, which can be shown using a sequence of dilations and translations.
Shape 1 is not congruent to shape 2 because the shapes do not have the same absolute coordinates.
Shape 1 is congruent to shape 2, which can be shown using a translation.
Shape 1 is not congruent to shape 2 because a sequence of rigid transformations will not map shape 1 onto shape 2.
Answer:
D
Step-by-step explanation:
(Michigan online school.)
Solve the following equations for all solutions of x
2sin^2x+3cos-3=0
what is f (x)=2x^2+5 multiplied by g (x)=2x
A) 7x+2x
B) 4x+10x
C)4x^2+10x
D)4x^3+10x
Which expression is equivalent to 7(xy)
7x+y
7x-y
x(7y)
xy/7
Answer: [tex]x(7y)[/tex]
Step-by-step explanation:
The given expression: [tex]7(xy)[/tex]
i.e. a product of 7 and xy.
The operation used here: Multiplication.
Commutative property of multiplication :-
[tex]a\times b=b\times a[/tex] for any numbers a and b.
Associative property of multiplication :-
[tex]a\times(b\times c)=(a\times b\times c)[/tex] for any numbers a , band c.
Now, [tex]7(xy)=(7x)y[/tex] [Associative property of multiplication]
[tex]=(x7)y[/tex] [Commutative property of multiplication]
[tex]=x(7y)[/tex] [Associative property of multiplication]
How much simple interest would you earn for 5 years at 7% with a beginning principal of $8,000.00
A. $2,800 B. $3,200 C. $3,300 D. $3,500 I couldn't figure it out and I need some help asap.
What is two plus two plus two plus two??????
what is the solution to this system of equations? 5X + 2y =29 X + 4y=13
The solution of the system of equation [tex]5x+2y = 29[/tex] ; [tex]x+4y =13[/tex] will be (3,2) and this can be determined by using the arithmetic operations.
Given :
[tex]5x+2y = 29[/tex] ----- (1)
[tex]x+4y =13[/tex] ----- (2)
Now, solve for x in equation (2).
[tex]x = 13-4y[/tex] --- (3)
Now, put the value of x obtained above in equation (1).
[tex]5(13-4y)+2y=29[/tex]
[tex]65-20y+2y=29[/tex]
[tex]36=18y[/tex]
[tex]y=2[/tex]
Now, put the value of y in equation (3).
[tex]x=13-(4\times 2)[/tex]
[tex]x=5[/tex]
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The range of the function f(x)=x+5 is (7,9). What is the function's domain?
Is it
(2,4)
(-2,-4)
(12,14)
(-12,-14)
(0,5)
Please help.. what is the function rule for the perimeter P of a building with a rectangular base if the width w is two times the length L?
A.) P=2L B.) P=6L C.) P=6w
Answer:
B.) P = 6LStep-by-step explanation:
The perimeter is the sum of all sides of the figure. In this case, the perimeter of the rectangle would be: [tex]P=W+L+W+L[/tex]; where [tex]W[/tex] is width, and [tex]L[/tex] is length.
According to the problem, the width is two times the length:
[tex]W=2L[/tex]
Replacing this relation in the perimeter equation, we have:
[tex]P=2W+2L\\P=2(2L)+2L\\P=4L+2L\\P=6L[/tex]
Therefore, the perimeter is six times the length of the rectangle. The correct answer is B.
cos2x- sqrt 2 sinx=1 Find all solutions
To solve the equation cos2x - sqrt 2 sinx = 1, rewrite cos2x as 2cos^2x - 1. Use the quadratic formula to solve for cosx. Substitute the values back into the equation cos2x - sqrt 2 sinx = 1 and solve for x.
Explanation:To solve the equation cos2x - √2 sinx = 1, we can use trigonometric identities and equations. First, we can rewrite cos2x as 2cos^2x - 1. So, the equation becomes 2cos^2x - √2 sinx - 1 = 0. To solve this quadratic equation, let's set 2cos^2x - √2 sinx - 1 = 0 and solve for cosx.
Next, we can use the quadratic formula to solve for cosx. The quadratic formula states that x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 2, b = -√2 sinx, and c = -1. Plugging in these values, we can solve for cosx.
After solving for cosx, we can substitute the values back into the equation cos2x - √2 sinx = 1 and solve for x. The student's question is about solving the equation cos(2x) - √2 sin(x) = 1 for all solutions. We can use the trigonometric identities cos(2x) = 1 - 2sin2(x) or cos(2x) = 2cos2(x) - 1 to rewrite the equation. Since we have a sine term in the original equation, let's use the former identity:
cos(2x) - √2 sin(x) = 1
(1 - 2sin2(x)) - √2 sin(x) = 1
2sin2(x) + √2 sin(x) - 1 = 0
This is a quadratic equation in sin(x).
We can solve this quadratic equation for sin(x), then find x using inverse trigonometric functions. By factoring the quadratic or using the quadratic formula, we get solutions for sin(x). Then, we solve for x by considering all possible angles in the unit circle that correspond to the found sine values.
The equation cos(2x) - √2 sin(x) = 1 can be solved by using trigonometric identities to simplify and factor the equation, leading to solving for sin(x) using inverse operations.
Explanation:The original equation given is cos(2x) - √2 sin(x) = 1. To solve this equation, we can use trigonometric identities to simplify the cosine term. One such identity is cos(2x) = 1 - 2sin²(x), which allows us to rewrite the equation as 1 - 2sin²(x) - √2 sin(x) = 1. From there, we subtract 1 from both sides, thus isolating the sine terms on the left: - 2sin²(x) - √2 sin(x) = 0. Factoring out the common term sin(x), we get sin(x)(-2sin(x) - √2) = 0. Setting each factor equal to zero gives us two possible solutions: sin(x) = 0 and sin(x) = -√2/2. In the context of a right triangle, these solutions correspond to specific angles where the sine value is 0 and -√2/2, respectively. The solutions can be found using standard trigonometric unit circle values or by calculating the inverse sine for -√2/2.
3 is 1 percent of what amount
The value of the solution is, 3 is 1 percent of 300.
We have to give that,
To find the amount for 1 percent is 3.
Let us assume that,
3 is 1 percent of x.
Hence, It can be written as,
3 = 1% of x
Solve for x,
3 = 1/100 × x
3 × 100 = x
x = 300
Therefore, 3 is 1 percent of 300.
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Find the area of a parallelogram with sides of 12 inches and 8 inches if one of the angles is 120
degrees
Answer:
48√3 sq. in.
Step-by-step explanation:
I know this is correct bc I just had this question and this was the correct answer
Simplify. Show your work.
5 1/3 + (-3 9/18)
Answer: Simplified form is
[tex]1\frac{5}{6}[/tex]
Step-by-step explanation:
Since we have given that
[tex]5\frac{1}{3}-3\frac{9}{18}[/tex]
Now, we will simplify it with the following steps :
[tex]5\frac{1}{3}-3\frac{9}{18}\\\\=\frac{16}{3}-3\frac{1}{2}\\\\=\frac{16}{3}-\frac{7}{2}\\\\=\frac{32-21}{6}\\\\=\frac{11}{6}\\\\=1\frac{5}{6}[/tex]
Hence, simplified form is
[tex]1\frac{5}{6}[/tex]
Triangle ABC is similar to triangle DEF.
What is the scale factor of triangle DEF to triangle ABC?
Triangle A B C and triangle D E F are drawn. Side A B is labeled 9. Side A C is labeled 12. Side D E is labeled 3. Side D F is labeled x.
3
1/3
4
1/4
Triangle ABC is similar to triangle DEF.
What is the scale factor of triangle DEF to triangle ABC?
Triangle A B C and triangle D E F are drawn. Side A B is labeled 9. Side A C is labeled 12. Side D E is labeled 3. Side D F is labeled x.
3
1/3
4
1/4
What are the factors of the expression? 3⋅(4r+y) Drag the factors of the term into the box.
{4x+3y=6
{2x -5y=16
Which of the following points is the solution to the system?
A recipe calls for 32 ounces of orange juice. How many cups of juice would you need for the recipe?
When it is 2 hours after 2 o'clock, then it is 4 o'clock (2 + 2 = 4). When it is 10 hours after 10 o'clock, then it is 8 o'clock. In this kind of "clock arithmetic," 10 + 10 = 8.
When a clock time gets bigger than 12, you subtract 12 and take the answer as the actual clock time. For example, if you subtract 12 from 20, the answer is 8, so 20 o'clock is really 8 o'clock.
Brad has a certain medication that he needs to take every 5 hours without fail, starting at 1 o'clock on a certain day. The sequence of clock times that he takes his pills is 1, 6, 11, 4, 9, ...
What is the clock time when Brad takes his 16th pill?