Math

What is the sine function used for, and how do you calculate it for a given angle in a right triangle?The sine function is used to relate the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. For an angle θ, sine (θ) is calculated as the length of the opposite side divided by the length of the hypotenuse.

How do you calculate the standard deviation of a data set, and what does it explain about the distribution of the data?To calculate the standard deviation of a data set, first find the mean (average) of the data. Then, subtract the mean from each data point and square the result. Find the average of these squared differences, and finally, take the square root of this average. This value represents the standard deviation. It measures the amount of variation or dispersion in a data set, indicating how spread out the data points are around the mean. A low standard deviation means data points are close to the mean, while a high standard deviation indicates a wide range of values.

What is the least common multiple (LCM) of 6 and 9?The least common multiple (LCM) of 6 and 9 is the smallest positive integer that is divisible by both numbers. To find the LCM, we can use the prime factorization method or the greatest common divisor (GCD) method. The LCM of 6 and 9 is 18.

How do you determine the domain and range of a composite function, specifically f(g(x))?To determine the domain of f(g(x)), first find the domain of g(x). Then, identify the set of values for which g(x) lies within the domain of f(x). The range of f(g(x)) is found by evaluating f at all points in the range of g(x) that fall within the domain of f.

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.

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.

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?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?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°.

How do you solve trigonometric equations that involve multiple angles, such as 2sin(x)cos(x) = sin(x), within the interval [0, 2π]?To solve 2sin(x)cos(x) = sin(x) within [0, 2π], first use the identity 2sin(x)cos(x) = sin(2x). The equation becomes sin(2x) = sin(x). This implies two cases: 2x = x + 2kπ or 2x = π – x + 2kπ. Solving these gives x = 0, π, 2π, π/3, 5π/3.