Math

How do you derive and apply the law of cosines to solve non-right triangles, especially when given one angle and two sides?The Law of Cosines is derived from the Pythagorean theorem and is used to solve non-right triangles. It states that for any triangle with sides a, b, and c, and angle C opposite side c: c^2 = a^2 + b^2 – 2ab*cos(C). To solve a triangle given one angle and two sides, use this formula to find the unknown side, then apply the Law of Sines or other trigonometric principles to find the remaining angles and sides.

How do you use the unit circle to determine the exact values of sin, cos, and tan for common angles such as 30°, 45°, and 60°?To determine the exact values of sin, cos, and tan for 30°, 45°, and 60° using the unit circle, locate these angles on the circle. For 30° (π/6), sin(30°) = 1/2, cos(30°) = √3/2, tan(30°) = 1/√3. For 45° (π/4), sin(45°) = cos(45°) = √2/2, tan(45°) = 1. For 60° (π/3), sin(60°) = √3/2, cos(60°) = 1/2, tan(60°) = √3.

Can you explain how to use the sum and difference formulas for sine, cosine, and tangent to solve complex trigonometric expressions?The sum and difference formulas for sine, cosine, and tangent are vital tools in trigonometry. They allow us to simplify complex expressions by breaking them into manageable parts. For sine: sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). For cosine: cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b). For tangent: tan(a ± b) = (tan(a) ± tan(b)) / (1 ∓ tan(a)tan(b)). These formulas are particularly useful in solving equations, proving identities, and evaluating trigonometric functions at specific angles.

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 you solve the system of nonlinear equations using the method of substitution or elimination: x^2 + y^2 = 25 and xy = 12?To solve the system of nonlinear equations x^2 + y^2 = 25 and xy = 12, use substitution. Express y in terms of x from xy = 12 (y = 12/x). Substitute y in x^2 + y^2 = 25 to get x^2 + (12/x)^2 = 25. Solve this equation to find x, then use it to find y.

How do you solve systems of linear equations using the substitution method?The substitution method for solving systems of linear equations involves isolating one variable in one equation and substituting this expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved. Finally, the value is substituted back into the original equation to find the other variable.

What is the difference between a derivative and an integral in Calculus?In Calculus, a derivative represents the rate of change of a function with respect to a variable, essentially measuring how a function changes as its input changes. An integral, on the other hand, represents the accumulation of quantities, such as areas under a curve. While derivatives focus on instantaneous rates of change, integrals focus on total accumulation over an interval.

If two cars travel in opposite directions starting from the same point, one car going 40 miles per hour and the other going 60 miles per hour, how long will it take for them to be 300 miles apart?To find the time it takes for the two cars to be 300 miles apart, we add their speeds together: 40 mph + 60 mph = 100 mph. Then, we divide the distance by their combined speed: 300 miles ÷ 100 mph = 3 hours. Therefore, it will take 3 hours for the cars to be 300 miles apart.

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°.

Can you explain how to use the method of Lagrange multipliers to find the maximum and minimum values of a function subject to a constraint?The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints. Suppose we have a function f(x, y, …) that we want to maximize or minimize subject to a constraint g(x, y, …) = 0. The method involves introducing a new variable, λ (the Lagrange multiplier), and studying the Lagrange function L(x, y, …, λ) = f(x, y, …) – λ(g(x, y, …) – c). We then find the stationary points of L by solving the system of equations given by the partial derivatives of L with respect to all variables (including λ) being equal to zero. These points give the candidates for the extrema of f subject to the constraint g.