In a 30-60-90 triangle, the length of the hypotenuse is 30. Find the length of the longer leg.
In a 30-60-90 triangle, the sides are in a specific ratio. The lengths of the sides opposite the 30°, 60°, and 90° angles are in the ratio of 1 : √3 : 2. Given that the hypotenuse (the side opposite the 90° angle) is 30, we can use the ratio to find the lengths of […]
Solve This Quadratic Equation Using the Quadratic Formula: 3x² + 5x + 1 = 0
To solve the quadratic equation 3x² + 5x + 1 = 0, we can use the quadratic formula, which is given by: x = (-b ± √(b² – 4ac)) / 2a In this formula, a, b, and c represent the coefficients of the quadratic equation ax² + bx + c = 0. For our equation, […]
What is the point-slope form of the line passing through (2, 12) and perpendicular to y = 3x?
To find the point-slope form of a line, we need two key pieces of information: a point through which the line passes and the slope of the line. First, we are given a point, which is (2, 12). Next, we need to determine the slope of the line perpendicular to the line described by the […]
How to Solve the Differential Equation xy 4y x4 Using Variation of Parameters
To solve the differential equation of the form xy * 4y = x^4 using the method of variation of parameters, we start by rewriting the equation in a standard form. First, we need to identify the associated homogeneous equation and solve it. The homogeneous part derived from the equation will typically be of the form: […]
How to Factor the Expression csc²x – 1?
To factor the expression csc²x – 1, we can recognize that this is a difference of squares. The difference of squares can be factored using the formula: a² – b² = (a – b)(a + b) In this case, we can let: a = csc x and b = 1. Applying the difference of squares […]
How to Express a Percent as a Fraction
To express a percent as a fraction, you start by understanding that “percent” means “per hundred.” This means that any percentage can be converted into a fraction with a denominator of 100. For example, if you have 25%, you can think of this as 25 per 100, which gives you the fraction: 25% = 25/100 […]
The point P(9, 12) is on the terminal side of angle θ; find the exact values of each trigonometric function.
To find the exact values of the trigonometric functions for the angle θ whose terminal side passes through the point P(9, 12), we will first determine the radius r from the origin to the point P. The radius r can be calculated using the Pythagorean theorem: r = √(x² + y²) Where x = 9 […]
How to Solve the System of Equations: x + 2y = 6 and 2x + 4y = 10
To solve the system of equations given by x + 2y = 6 and 2x + 4y = 10, we can use either substitution or elimination method. Here, we will opt for the substitution method. First, let’s solve the first equation for x: x = 6 – 2y Now, we can substitute this expression for […]
By How Much Does A3 2A Exceed A2 A6?
To determine how much A3 2A exceeds A2 A6, we first need to understand the values represented by these terms. Typically, in contexts such as mathematics or measurements, A3 2A and A2 A6 could denote specific quantities or numerical values. Assuming A3 2A represents a quantity or value of 2A (where ‘A’ could be a […]
How to Solve the Triangle with Sides b = 36, a = 42, c = 18?
To solve the triangle with the given sides, we can use the Law of Cosines. The Law of Cosines states that for any triangle with sides a, b, and c, the following relationships hold: c2 = a2 + b2 – 2ab * cos(C) In our case, we need to solve for the angles given sides […]