Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
tan θ / sec θ​

Answer:

  x = 3/8

Step-by-step explanation:

You want the solution to 3/2 = 4x.

You solve this one-step linear equation by multiplying both sides by the inverse of the x-coefficient:

  [tex]\dfrac{3}{2}=4x\qquad\text{given}\\\\\\\dfrac{1}{4}\times\dfrac{3}{2}=\dfrac{1}{4}\times4x\\\\\\\dfrac{3}{8}=\dfrac{4}{4}x\\\\\\\boxed{x=\dfrac{3}{8}}[/tex]

__

Additional comment

The "one step" is multiplying both sides by 1/4. The rest is simplifying the result.

Of course, this is the same as dividing both sides by 4. The result is x = (3/2)÷4 = 3/(2·4) = 3/8.

The length of the extension ladder that is against the outside wall of the house is 22.58 feet.

What is the length of the ladder?

The ladder forms a right triangle when it is placed up against a home. The hypotenuse of a ladder is its length. The base of the ladder is where the distance from the house to the base begins. The length is the distance from the home.

Here, we have

Given: An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 22 feet up. The ladder makes an angle of 77

To find the length of the ladder, Sin would be used:

Sin = opposite / hypotenuse

Sin77 = 22 / h

h = 22 / 0.9743 = 22.58 feet

Hence, the length of the extension ladder that is against the outside wall of the house is 22.58 feet.

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By answering the presented question, we may conclude that tan(X): trigonometry Fraction 3/4, Decimal 0.750

The study of the connection between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their possible applications in calculations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle characteristics, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric forms.

Using the Pythagorean theorem,

hypotenuse² = opposite² + adjacent²

hypotenuse² = 4² + 3²

hypotenuse² = 16 + 9

hypotenuse² = 25

hypotenuse = √25

hypotenuse = 5

Now we can find the trigonometric ratios:

sin(X) = opposite / hypotenuse = 3/5

sin(X) = 0.600 (rounded to the nearest thousandth)

cos(X) = adjacent / hypotenuse = 4/5

cos(X) = 0.800 (rounded to the nearest thousandth)

tan(X) = opposite / adjacent = 3/4

tan(X) = 0.750 (rounded to the nearest thousandth)

Therefore, the trigonometric ratios for angle X are:

sin(X): Fraction 3/5, Decimal 0.600

cos(X): Fraction 4/5, Decimal 0.800

tan(X): Fraction 3/4, Decimal 0.750

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The cost of Veronica's supper was $11.92, plus an additional 8% tax of $0.954 and a 15% gratuity on top of tax of $2.0681, for a total price of $14.94.

We first need to figure out the tax and tip amounts and add them to the initial cost to get the final price of Veronica's dinner.

Let's first figure out the tax amount. The cost of the meal is multiplied by the 8% tax rate to determine the tax:

Tax = 11.92 * 0.08 = 0.954

Hence, the meal's tax is $0.954.

Next, let's figure out how much to tip. We multiply the price of the meal plus the tax by the 15% tip percentage to determine the tip:

Tip = (11.92 + 0.954) x 0.15 = 2.0681

Thus, $2.0681 is the total tip for the lunch, tax included.

The original price of the meal, the tax, and the tip can now be added to determine the final price:

Total Cost: 11.92 plus 0.9544 plus 2.0681

Veronica's total bill, rounded to the nearest cent, is therefore $14.94.

In conclusion, Veronica's dinner cost $11.92, plus an additional 8% tax of $0.954 and a 15% tip on top of the meal plus tax of $2.0681, for a total of $14.94.

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Answer:

A

Step-by-step explanation:

I added a photo of my notes (the angles I've marked as equal are cross angles)

[tex]w = 180° - 30° - x°[/tex]

[tex]w = 150° - x°[/tex]

The cannonball will land approximately 20.4 seconds after it was fired.

A round shot is a solid, spherical projectile fired from a gun that is also known as a decent shot or simply a ball. Its diameter is just a little bit smaller than the diameter of the barrel that it is fired from.

An iron anti-personnel bullet having a hollow internal cavity filled with leads or iron round pellets and a tiny explosive charge that explodes with just enough power to crack open the iron projectile's thin walls. A time fuse was put into a receptacle at the projectile's outer edge after a powder trains in a tiny iron sleeve.

We can use the equations of motion for projectile motion to solve this problem. Let's assume that the cannonball is fired at an angle θ above the horizontal and lands at a distance of x = 2000 meters from the cannon. We also know that the initial speed of the cannonball is 200 m/s.

The horizontal and vertical components of the velocity can be expressed as:

vₓ = v₀ cos(θ)

vₐ = v₀ sin(θ) - gt

where v₀ is the initial velocity (200 m/s), g is the acceleration due to gravity (9.8 m/s²), and t is the time taken for the cannonball to land.

Using the equation for the horizontal motion, we can find the time taken for the cannonball to travel a distance of 2000 meters:

x = v₀ cos(θ) * t

t = x / (v₀ cos(θ))

Using the equation for the vertical motion, we can find the time taken for the cannonball to reach the ground:

y = v₀ sin(θ) * t - (1/2) * g * t²

0 = v₀ sin(θ) * t - (1/2) * g * t²

Solving for t in the second equation gives:

t = 2 * v₀ sin(θ) / g

Substituting this expression for t into the first equation gives:

x = (v₀² / g) * sin(2θ)

We can now solve for θ:

sin(2θ) = (g * x) / v₀²

θ = 0.5 * arcsin((g * x) / v₀²)

Plugging in the given values for g, x, and v0, we get:

θ = 0.5 * arcsin((9.8 m/s² * 2000 m) / (200 m/s)²) ≈ 20.3 degrees

Therefore, the cannon fired the shot at an angle of approximately 20.3 degrees above the horizontal.

To find the time taken for the cannonball to land, we can use the expression for t derived earlier:

t = 2 * v₀ sin(θ) / g

t = 2 * 200 m/s * sin(20.3°) / 9.8 m/s² ≈ 20.4 seconds

Therefore, the cannonball will land approximately 20.4 seconds after it was fired.

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Answer:

x=2

Step-by-step explanation:

4 x1,2 =2

The correct statement which compares the graph of function g with the graph of function f is,

⇒ The graph of function g is a vertical compression of the graph of function f.

A relation between a set of inputs having one output each is called a function. and an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).

Given that;

Functions are,

f (x) = eˣ - 4

g (x) = 1/2eˣ - 4

Now, We have to find that;

The graph of function g is a vertical compression of the graph of function f.

Hence, The correct statement which compares the graph of function g with the graph of function f is,

⇒ The graph of function g is a vertical compression of the graph of function f.

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Tammy ran 2.25 miles on Sunday.

A linear equation is a mathematical expression that describes a straight line on a graph. It is of the form y = mx + b, where "m" is the slope of the line and "b" is the y-intercept.

Let's start by figuring out how far Tammy ran on Sunday. We know that she ran for 1/2 of the distance she ran on Saturday. Therefore, if we let "x" be the distance Tammy ran on Sunday, we can set up the following equation:

x = 1/2(4.5)

Here, we used the fact that Tammy ran 4.5 miles on Saturday.

To solve for "x", we simply need to simplify the right-hand side of the equation:

x = 1/2(4.5)

x = 2.25

Therefore, Tammy ran 2.25 miles on Sunday.

In summary, we used the equation x = 1/2(4.5) to represent the distance Tammy ran on Sunday, where "x" is the unknown distance. We then solved for "x" by simplifying the equation and found that Tammy ran 2.25 miles on Sunday.

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Answer:

To calculate the amount in the IRA at retirement, we can use the formula for the future value of an annuity:

S = PMT * (((1 + r)^n - 1) / r)

Where:

PMT = the monthly payment (in dollars)

r = the monthly interest rate (APR/12)

n = the number of months (45 years * 12 months/year)

Substituting the given values, we get:

PMT = $50

r = 0.06/12 = 0.005

n = 45 * 12 = 540

S = $50 * (((1 + 0.005)^540 - 1) / 0.005) = $204,055.57

So the IRA will contain $204,055.57 at retirement.

The total deposits made over the time period is:

Total deposits = 540 * $50 = $27,000

Therefore, the amount in the IRA at retirement ($204,055.57) is significantly higher than the total deposits made over the time period ($27,000). This demonstrates the power of compound interest over a long time horizon

Step-by-step explanation:

If u have other qstns too, feel free to ask me on my sn ap

the side CD will be 27 units.

What is parallelogram?

A unique variety of quadrilateral made up of parallel lines is known as a parallelogram. A parallelogram can have any angle between its adjacent sides as long as its opposite sides are parallel. If two opposite sides of a quadrilateral are parallel and congruent, it will be a parallelogram. Consequently, a quadrilateral in which both pairs of opposite sides are parallel and equal is known as a parallelogram.

Here CD=AB

2x+13=5x-8

3x = 13+8

x= 21/3

x=7

CD= 2*7+13=14+13=27

Hence the side CD will be 27 units.

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The limits [tex]\lim_{x \to 0 } f(x)[/tex] exists and the function f(x) is continuous at x = 0

Other solutions are shown below

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

From the graph, we have the domain and the range to be:

From the graph, we have the values of the function to be

f(-4) = 1 and f(-6) = 2

From the graph, we have the values of the limits to be

[tex]\lim_{x \to 4^{-} } f(x) = -2[/tex]

[tex]\lim_{x \to 4^{+} } f(x) = 2[/tex]

Yes, the limit  [tex]\lim_{x \to 0 } f(x)[/tex]

To check if a function is continuous at x=0, you need to check three conditions:

Hence, the function is continuous at x = 0

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By looking at the 0 and 1 column, 200 people own less than two smartphones

Triangle DEF as shown ion figure D(-4, 6), E(2,6), F(0,8) translate by a factor of 1/2, So the new vertex are: D' (-2, 3), E' (1, 3), F' (0, 4)

Triangle is commonly represented by a graph in which the three points are labeled as A, B, and C, and the three line segments are labeled as AB, AC, and BC. The graph of a triangle typically looks like below figure.

The new coordinates of each vertex of the triangle after scaling by 1/2.

For vertex D:

x-coordinate = -4, y-coordinate = 6

New x-coordinate = (1/2) × (-4) = -2

New y-coordinate = (1/2) × 6 = 3

So the new coordinates for D' are (-2, 3)

For vertex E:

E (2, 6) ⇒ 1/2 × (2, 6) ⇒ (1, 3)

So the new coordinates for E' are (1, 3)

For vertex F:

F (0, 8) ⇒ 1/2 × (0, 8) ⇒ (0, 4)

So the new coordinates for F' are (0, 4)

D(-4, 6), E(2,6), F(0,8)  ⇒   D' (-2, 3), E' (1, 3), F' (0, 4)

Finally, plot the new points on the same coordinate plane and connect them to form the new triangle. You should see that the new triangle is a smaller version of the original triangle, with all sides and angles scaled down by a factor of 1/2.

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The resulting values for r will be sqrt(140).

Let's assume that the bridge is a straight line segment that connects the top of the mountain (point A) to a point on the ground (point B).

Let's also assume that the distance from the top of the mountain to point B is a fixed value d.

If we attach the higher end of the bridge to the top of the mountain at a height y, the distance between point A and the attachment point can be calculated using the Pythagorean theorem as:

x = sqrt([tex]d^2[/tex] - [tex]y^2[/tex])

The length of the bridge, which is also the hypotenuse of the right triangle formed by points A, B, and the attachment point, can then be calculated as:

r = sqrt([tex]x^2[/tex] + [tex](d-y)^2[/tex])

To find the values of r for different heights y, we can simply substitute different values of y into these equations and calculate the resulting values of r.

For example, if we use the values d=10 and y=3, we get:

To calculate the values for "r" given 5 possible values for "y," we need to use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the bridge is the hypotenuse, and we know the height of the mountain (x) and the distance from the mountain to the lower end of the bridge (d). So we can write:

r^2 = [tex]x^2[/tex] + [tex](d + y)^2[/tex]

x = sqrt([tex]10^2[/tex] - [tex]3^2[/tex])

x = sqrt(91)

r = sqrt((sqrt([tex]91))^2[/tex] + [tex](10-3)^2[/tex])

r = sqrt(91 + 49)

r =  sqrt(140)

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a. The recursive formula to represent the sequence is aₓ = aₓ₋₁ + 10.

b. The explicit formula to represent the sequence is aₓ = 50 + 10x.

c. We have $310 in savings after 26 weeks.

What is sequence?

A progression or sequence of numbers known as an arithmetic sequence keeps the difference between any subsequent term and its preceding term constant throughout the entire sequence. In that arithmetic progression, the constant difference is known as the common difference.

We are given that there are $50 in a savings account and each week additional $10 are deposited.

a. Let x be the number of weeks

a₀ = $50

aₓ = aₓ₋₁ + 10

So, the recursive formula to represent the sequence is aₓ = aₓ₋₁ + 10.

b. Let x be the number of weeks.

So, the explicit formula is given by

aₓ = 50 + 10x

c. Now, we are given x = 26.

So, by substituting this, we get

⇒ a₂₆ = 50 + 10 * 26

⇒ a₂₆ = 50 + 260

⇒ a₂₆ = $310

Hence, the required solutions have been obtained.

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Answer:

Area = 100 units^2

Step-by-step explanation:

This shape is a trapezoid. The area of a trapezoid is:

A = 1/2(Base+Base)•h

What you do is average the bases and times by the height. The only thing is in your picture the bases are the 12 and 8. And the height is the 10.

Area = 1/2(12+8)•10

= 1/2(20)•10

= 10•10

= 100

The area is 100 square units.

The value of n = 12, and the area of the trapezium can be written as 12√12.

Let's call the diameter of the circle (and one of the parallel sides of the trapezium) "AB", and let's call the non-parallel sides "CD" and "EF". We know that CD and EF are both equal to the radius of the circle, which is 2 units.

We know that AB is a diameter of the circle, which means that its length is twice the radius, or 4 units.

We also know that CD and EF are each 2 units long.

To find the height of the trapezium, we can draw a perpendicular line from point C to line AB:

We can see that the height of the trapezium is the distance from point H to line AB. Since point H is directly above the center of the circle (point O), we can use the Pythagorean theorem to find its distance from point A (or point B, since AB is a diameter):

The length of the hypotenuse (OB) is 2 units (the radius of the circle), and the length of one of the legs (OA) is 2 units (half of AB). To find the length of the other leg (HB), we can use the Pythagorean theorem:

HB^2 + OA^2 = OB^2

HB^2 + 2^2 = 2^2

HB^2 = 0

This tells us that HB (and therefore the height of the trapezium) is 0 units.

Since the height is 0, the area of the trapezium is simply the average of the two parallel sides (AB and CD) multiplied by the distance between them (which is also AB).

Area = (AB + CD) / 2 * AB

= (4 + 2) / 2 * 4

= 3 * 4

= 12

Therefore, n = 12, and the area of the trapezium can be written as 12√12.

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Putting all these values together, we get the equation: h(t) = 4 cos(2π/3 (t - 2)) + 1 with Mike's height above the ground as a function of time t in seconds, where h(t) is measured in feet.

An equation is a mathematical statement that shows that two expressions are equal. It contains an equals sign (=) and at least two expressions on either side of the equals sign. The expressions on either side of the equals sign can be numbers, variables, or a combination of both, and the equation represents a relationship between them. Equations are used in mathematics to solve problems and find unknown values by manipulating the expressions within the equation while keeping the equality true.

Here,

Assuming that Mike's motion on the swing set follows a periodic pattern, we can model his height above the ground using a sinusoidal function. Let's use the cosine function, since it reaches its maximum value when the input is 0 and decreases to its minimum value at π or 180°.

The general form of a cosine function is:

y = A cos(Bx + C) + D

where:

A = amplitude (half the distance between the maximum and minimum values)

B = frequency (the number of cycles per unit of x)

C = phase shift (horizontal shift of the graph)

D = vertical shift (vertical shift of the graph)

We are given that Mike's highest point is at 9 feet and his lowest point is at 1 foot. Therefore, the amplitude A = (9 - 1) / 2 = 4.

The frequency is determined by the time it takes for Mike to complete one cycle. We know that it takes him 1.5 seconds to go from his highest point to his lowest point and back up again. Therefore, the period of the function is 3 seconds (2 x 1.5), and the frequency is 1/3 cycles per second. Hence, B = 2π/3.

The phase shift is the horizontal displacement of the graph from the origin. We know that at t = 2 seconds, Mike is at his highest point. Therefore, we need to shift the cosine curve 2 seconds to the right to match this point. Thus, C = -2.

Finally, the vertical shift D is the position of the center of the curve. Since Mike's lowest point is at 1 foot, we need to shift the entire curve up by 1 foot. Thus, D = 1.

Putting all these values together, we get the equation:

h(t) = 4 cos(2π/3 (t - 2)) + 1

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Answer:

The population is the total number of people who go to the movie theater, and the sample is the first 150 people at the theater.

Step-by-step explanation:

The population is the potential people that could be sampled, and the sample is the people who are being asked the question.

Answer:ccccccccccccccccccccccccccc

Step-by-step explanation:

Step-by-step explanation:

[tex] \frac{ \frac{1}{7} }{ \frac{2}{5} } = \frac{1}{7} \times \frac{5}{2 } = \frac{5}{14} \\ [/tex]

Answer: 5/14

Step-by-step explanation:

It would help if you used the KCF (Keep, Change, Flip) method. First, you write 1/7 as a fraction, then change the division sign to a multiplication sign, and flip the fraction 2/5 to 5/2. Then multiply 1/7 by 5/2 to get 5/14.

Answer:

68°

Step-by-step explanation:

because the inscribed angle (at B) is half of the arc angle (at A).

angle B = 1/2 × angle A

34 = 1/2 × angle A

angle A = 34×2 = 68°

Answer:

Two

Step-by-step explanation:

Trangles are labled n named alphabetically therefore on the bigger triangle D comes first than E and E comes forsr than R wich means that itvis triangle DER not any ather way around.

On the second triangle A comes first than N and N comes first then T thetefore it is triangle ANT

Answer:

Option A is the correct answer

Triangle RED IS SIMILAR TO TRIANGLE TAN

Answer:

Each side length of the cube will be a polynomial of degree 3.

The number of units a given figure would be translated using the given notation are shown below.

In Mathematics, the translation of a graph to the left is a type of transformation that simply means subtracting a digit from the value on the x-coordinate of the pre-image while the translation of a graph to the right is a type of transformation that simply means adding a digit to the value on the x-coordinate of the pre-image.

Based on each of the transformation rule, the number of units a given figure would be translated using the given notation include the following:

(x, y)   →   (x + 7, y - 6)

7 units right.

6 units down.

(x, y)   →   (x - 9, y + 2)

9 units left.

2 units up.

(x, y)   →   (x - 11, y)

11 units left.

0 units up/down.

(x, y)   →   (x - 8, y - 5)

8 units left.

5 units down.

(x, y)   →   (x + 13, y + 3)

7 units right.

3 units up.

(x, y)   →   (x, y + 9)

0 units right/left.

9 units up.

(x, y)   →   (x - 14, y + 12)

14 units left.

12 units up.

(x, y)   →   (x, y - 4)

0 units right/left.

4 units down.

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The domain of the relation is the set {-5, -2, 0, 1, 3, 5}.

The domain of a relation is the set of all possible input values (independent variable) that correspond to an output (dependent variable).

The domain of a function or relation is the set of all possible input values (independent variable) that correspond to an output (dependent variable). It represents the values for which the function or relation is defined.

Looking at the given points, we can see that the x-coordinates of the points are -5, -2, 0, 1, 3, and 5. Therefore, the possible input values for this relation (i.e., the domain) are:

Domain: {-5, -2, 0, 1, 3, 5}

So the domain of the relation is the set {-5, -2, 0, 1, 3, 5}.

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The length of the ladder is approximately 30.6 feet. The height of the building is 90 degrees.

Let "θ" be the angle between the ladder and the ground. Then, we have:

sin(θ) = 30 ÷ x

Solving for "x", we get:

x = 30 ÷ sin(θ)

Since the ladder is the hypotenuse, we know that the angle opposite the height of the building is 90 degrees. Therefore, the angle between the ladder and the ground is the complement of this angle, which is:

θ = 90 - arcsin(30 ÷ x)

Substituting this expression for "θ" into the equation for "x", we get:

x = 30 ÷ sin(90 - arcsin(30 ÷ x))

Using the identity sin(90 - θ) = cos(θ), we can simplify this to:

x = 30 / cos(arcsin(30 ÷ x))

Using the identity cos(arcsin(x)) = [tex]\sqrt[] 1-x^{2}[/tex] , we can simplify this further to:

x = [tex]\frac{30}{\sqrt{(1-(30/x^{2}))\\} }[/tex]

Squaring both sides and simplifying, we get:

x² = 30² + 6²

x² = 900 + 36

x² = 936

x = [tex]\sqrt{936}[/tex]

x = 30.6 feet is height of ladder

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