What are the main differences between planets and dwarf planets in our solar system?The main differences between planets and dwarf planets in our solar system lie in their ability to clear their orbital paths. While both orbit the Sun and are spherical due to their own gravity, planets have cleared their orbits of other debris, whereas dwarf planets have not. Additionally, dwarf planets are typically smaller and may share their space with other objects of similar size.
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How do you use matrix algebra to solve a system of linear equations and what are the practical applications of this method?
How do you use matrix algebra to solve a system of linear equations and what are the practical applications of this method?Matrix algebra is used to solve systems of linear equations by representing the system as a matrix equation Ax = b, where A is the coefficient matrix, x is the column vector of variables, and b is the column vector of constants. By finding the inverse of matrix A (if it exists), we can solve for x using x = A^(-1)b. Practical applications include engineering, computer graphics, economics, and optimization problems.
What are the different layers of the Earth and what are their characteristics?
What are the different layers of the Earth and what are their characteristics?The Earth is composed of four main layers: the crust, mantle, outer core, and inner core. The crust is the outermost layer, thin and solid. The mantle lies beneath the crust, composed of semi-solid rock. The outer core is liquid iron and nickel, while the inner core is solid iron and nickel.
What are the main functions of the human respiratory system?
What are the main functions of the human respiratory system?The main functions of the human respiratory system include the intake of oxygen and removal of carbon dioxide through the process of gas exchange, regulation of blood pH, protection against pathogens and irritants, and vocalization. The system also helps maintain homeostasis and supports cellular respiration by supplying oxygen to and removing carbon dioxide from the bloodstream.
How do you prove that the angle subtended by an arc in a circle is equal to half the angle subtended by the same arc when measured at the center of the circle?
How do you prove that the angle subtended by an arc in a circle is equal to half the angle subtended by the same arc when measured at the center of the circle?To prove that the angle subtended by an arc at the circumference of a circle is half the angle subtended by the same arc at the center, consider a circle with center O. Let points A, B, and C lie on the circle such that arc AC subtends angle ∠AOC at the center and angle ∠ABC at the circumference. By the Inscribed Angle Theorem, ∠ABC = 1/2 ∠AOC. This is because the angle at the center is formed by two radii, while the angle at the circumference is formed by a chord and a secant, making the central angle double the inscribed angle.
What are the three branches of the United States Government and their primary functions?
What are the three branches of the United States Government and their primary functions?The three branches of the United States Government are the Legislative, Executive, and Judicial branches. The Legislative branch makes laws, the Executive branch enforces laws, and the Judicial branch interprets laws. This system ensures a balance of power through checks and balances.
If the expression 6x + 9y – 14 = 5y + 13 is solved for x, what are the steps to write x as a function of y?
If the expression 6x + 9y – 14 = 5y + 13 is solved for x, what are the steps to write x as a function of y?To solve the equation 6x + 9y – 14 = 5y + 13 for x, follow these steps: 1. Subtract 5y from both sides to get 6x + 4y – 14 = 13. 2. Add 14 to both sides to get 6x + 4y = 27. 3. Subtract 4y from both sides to isolate 6x, giving 6x = 27 – 4y. 4. Divide both sides by 6 to solve for x, resulting in x = (27 – 4y)/6. Therefore, x as a function of y is x = (27 – 4y)/6.
How can I determine the exact values for the sine, cosine, and tangent of a 45-degree angle?
How can I determine the exact values for the sine, cosine, and tangent of a 45-degree angle?To determine the exact values for sine, cosine, and tangent of a 45-degree angle, consider a right triangle with equal legs. The hypotenuse is √2 times the leg length. Thus, sin(45°) = cos(45°) = 1/√2 or √2/2, and tan(45°) = 1.
How do you prove that the sum of the angles of any triangle always equals 180 degrees using trigonometric functions and identities?
How do you prove that the sum of the angles of any triangle always equals 180 degrees using trigonometric functions and identities?To prove the sum of the angles of any triangle equals 180 degrees using trigonometric functions and identities, consider a triangle with angles A, B, and C. Using the identity for the tangent of the sum of two angles, tan(A + B) = (tan A + tan B) / (1 – tan A tan B). Since tan(C) = tan(180° – (A + B)) and tan(180° – x) = -tan(x), it follows that tan(A + B) = -tan(C). This implies that A + B + C = 180°.
What is Newton’s first law of motion and can you provide an example demonstrating it?
What is Newton’s first law of motion and can you provide an example demonstrating it?Newton’s first law of motion states that an object at rest will stay at rest, and an object in motion will stay in motion at a constant velocity, unless acted upon by an external force. For example, a book on a table will remain stationary until someone pushes it.
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Find the arc length of a sector given angle θ and radius r
Answer 1 To find the arc length of a sector given angle $\theta$ and radius $r$, use the formula: $ L = r \cdot \theta $ In this formula, $L$ is the arc length of the sector, $r$ is the radius of the circle, and $\theta$ is the central angle in...
Find the value of cos(π/3) using the unit circle on a graphing calculator
Answer 1 On the unit circle, the angle $\frac{\pi}{3}$ corresponds to 60 degrees. The coordinates of this point are $(\frac{1}{2}, \frac{\sqrt{3}}{2})$. The x-coordinate of this point is $\cos(\frac{\pi}{3})$.Therefore,$ \cos(\frac{\pi}{3}) =...
Find the exact values of sine and cosine for an angle of 225 degrees using the unit circle
Answer 1 To find the exact values of $ \sin $ and $ \cos $ for an angle of 225 degrees using the unit circle, we first convert the angle to radians:$ 225^\circ = 225 \times \frac{\pi}{180} = \frac{5\pi}{4} $The angle \( \frac{5\pi}{4} \) is in the...
Find the sine and cosine of \( \frac{\pi}{4} \) using the unit circle
Answer 1 To find the sine and cosine of $ \frac{\pi}{4} $ using the unit circle:On the unit circle, the angle $ \frac{\pi}{4} $ corresponds to the coordinates $ \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) $.Therefore,$ \sin\left(...
Determine the coordinates of a point on the unit circle at an angle of \( \frac{\pi}{4} \)
Answer 1 To find the coordinates of a point on the unit circle at an angle of \( \frac{\pi}{4} \), we use the unit circle definition:\n The unit circle is defined as all points (x, y) such that:\n $ x^2 + y^2 = 1 $\n For an angle \( \theta \), the...
Find the tangent of angle θ on a unit circle
Answer 1 To find the tangent of the angle $ \theta $ on a unit circle, one must understand that the tangent of an angle is defined as the ratio of the sine to the cosine of that angle: $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $ For example,...