Math

How do you prove that the sum of the interior angles of a triangle equals 180 degrees using parallel lines and transversals?To prove the sum of the interior angles of a triangle equals 180 degrees using parallel lines and transversals, extend one side of the triangle and draw a line parallel to the opposite side through the third vertex. The alternate interior angles formed are congruent to the triangle’s angles, and the linear pair formed with the exterior angle sums to 180 degrees.

How do you solve a quadratic equation using the quadratic formula?To solve a quadratic equation using the quadratic formula, follow these steps: For a quadratic equation in the form ax^2 + bx + c = 0, the quadratic formula is x = (-b ± √(b² – 4ac)) / (2a). Substitute the values of a, b, and c into the formula, then calculate the discriminant (b² – 4ac). Finally, solve for x by completing the operations indicated in the formula.

How do you find the vertex of a parabola given in standard form?To find the vertex of a parabola in standard form y = ax^2 + bx + c, use the vertex formula: x = -b/(2a). Substitute this x-value back into the equation to find the y-coordinate. The vertex is at (x, y).

How do you find the derivative of a function using the chain rule when dealing with composite functions?To find the derivative of a composite function using the chain rule, identify the outer function f(u) and the inner function u=g(x). The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). Differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.

Prove that the sum of the interior angles of a regular polygon can be calculated with the formula (n-2)×180°, where n is the number of sides.To prove that the sum of the interior angles of a regular polygon is (n-2)×180°, consider dividing the polygon into (n-2) triangles. Each triangle has an angle sum of 180°. Thus, the total interior angle sum is (n-2)×180°.

What is the difference between the derivative and the integral of a function?The derivative of a function measures the rate at which the function’s value changes with respect to a change in its input value, often interpreted as the slope of the function. The integral of a function, on the other hand, measures the total accumulation of the function’s values over an interval, often interpreted as the area under the curve of the function. While the derivative focuses on local behavior and instantaneous rates of change, the integral focuses on global behavior and cumulative quantities.

How do you calculate the height of a triangle given its base length and area?To calculate the height of a triangle given its base length and area, use the formula: height = (2 * area) / base. This formula derives from the basic area formula for a triangle, which is area = 0.5 * base * height. Rearrange it to solve for height.

What are the basic trigonometric functions and how are they related to the sides of a right triangle?The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan). They relate to a right triangle as follows: sine is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side.

How do you find the area of a parallelogram when the base is 8 units and the height is 5 units?To find the area of a parallelogram, you use the formula: Area = base × height. Given that the base is 8 units and the height is 5 units, the area is calculated as follows: Area = 8 units × 5 units = 40 square units. Therefore, the area of the parallelogram is 40 square units.

How do you find the domain and range of a quadratic function?To find the domain of a quadratic function, identify the set of all possible input values (x-values). For any quadratic function, the domain is all real numbers (-∞, ∞). To find the range, determine the vertex of the parabola, which is either the maximum or minimum point. If the parabola opens upwards (a > 0), the range is [k, ∞). If it opens downwards (a < 0), the range is (-∞, k], where k is the y-coordinate of the vertex.