Math

How do you solve trigonometric equations involving both sine and cosine within specific intervals and verify the solutions using unit circle principles?To solve trigonometric equations involving both sine and cosine within specific intervals, isolate one trigonometric function, use identities to simplify, and solve for the angle. Verify solutions by checking them on the unit circle, ensuring they lie within the given interval.

How do you solve the equation 3x – 4 = 11?To solve the equation 3x – 4 = 11, first add 4 to both sides to get 3x = 15. Then, divide both sides by 3 to find x = 5.

How do you compute the limit of a multivariable function using L’Hopital’s Rule when approaching the origin?To compute the limit of a multivariable function using L’Hopital’s Rule when approaching the origin, first confirm the limit is in an indeterminate form. Then, apply partial derivatives to each variable iteratively, simplifying the function. Repeat until the limit can be evaluated directly.

How do you solve systems of nonlinear equations using substitution and elimination methods?To solve systems of nonlinear equations using substitution, solve one equation for one variable and substitute into the other. For elimination, manipulate equations to cancel one variable. Both methods simplify the system to solve for all variables. Verify solutions by substituting back into original equations.

How do you prove that the diagonals of a parallelogram bisect each other using coordinate geometry?To prove that the diagonals of a parallelogram bisect each other using coordinate geometry, consider a parallelogram ABCD with vertices A(x1, y1), B(x2, y2), C(x3, y3), and D(x4, y4). The midpoint of diagonal AC is ((x1+x3)/2, (y1+y3)/2) and the midpoint of diagonal BD is ((x2+x4)/2, (y2+y4)/2). Since ABCD is a parallelogram, opposite sides are equal and parallel, leading to the conclusion that these midpoints are the same, thus proving that the diagonals bisect each other.

What is the power rule for differentiation, and can you provide a simple example of how it is used?The power rule for differentiation states that if you have a function f(x) = x^n, where n is a constant, then the derivative of the function is f'(x) = nx^(n-1). For example, if f(x) = x^3, then f'(x) = 3x^2.

How do I solve the inequality 3x – 2 ≤ 7?To solve the inequality 3x – 2 ≤ 7, first add 2 to both sides to get 3x ≤ 9. Then, divide both sides by 3 to isolate x, resulting in x ≤ 3. Therefore, the solution to the inequality is x ≤ 3.

How do you solve for x in a linear equation like 2x + 5 = 15?To solve the linear equation 2x + 5 = 15, first subtract 5 from both sides to get 2x = 10. Then, divide both sides by 2 to isolate x, resulting in x = 5.

How do you use the Pythagorean theorem to find the unknown side length of a right triangle in a real-world problem?To use the Pythagorean theorem in a real-world problem, identify the right triangle’s sides: the two legs (a and b) and the hypotenuse (c). Apply the formula a² + b² = c². Solve for the unknown side by rearranging the equation and taking the square root if necessary. For example, in construction, you can determine the length of a ladder needed to reach a certain height by knowing the distance from the wall.

How do you find the exact values of trigonometric functions for angles that aren’t standard, like 75 degrees or 15 degrees?To find the exact values of trigonometric functions for non-standard angles like 75° or 15°, use angle sum and difference identities. For example, sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°). Similarly, cos(75°) and tan(75°) can be calculated using the same identities.