Substitute the given values of P and Q into the expression (P)/(Q).

Another one vostaomondi747 — let's solve it. 1. Finding the values of a, b, and c Step 1: Substitute the given values of P and Q into the expression (P)/(Q). (P)/(Q) = 4 + sqrt(2)2 + sqrt(2) Step 2: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator, which is 2 - sqrt(2). (P)/(Q) = 4 + sqrt(2)2 + sqrt(2) × 2 - sqrt(2)2 - sqrt(2) Step 3: Expand the numerator and the denominator. Numerator: (4 + sqrt(2))(2 - sqrt(2)) = 4(2) - 4sqrt(2) + 2sqrt(2) - (sqrt(2))^2 = 8 - 4sqrt(2) + 2sqrt(2) - 2 = 6 - 2sqrt(2) Denominator: (2 + sqrt(2))(2 - sqrt(2)) = 2^2 - (sqrt(2))^2 = 4 - 2 = 2 Step 4: Combine the simplified numerator and denominator. (P)/(Q) = 6 - 2sqrt(2)2 = (6)/(2) - 2sqrt(2)2 = 3 - sqrt(2) Step 5: Compare the result with a + bsqrt(c). We have 3 - sqrt(2) = a + bsqrt(c). By comparing the terms, we find: a = 3 b = -1 c = 2 The values are a=3, b=-1, c=2. 2. Solving the trigonometric equation Step 1: Use the identity ^2 x = 1 - ^2 x to rewrite the equation in terms of x. 4^2 x + 4 x = 5 4(1 - ^2 x) + 4 x = 5 Step 2: Expand and rearrange the equation into a quadratic form. 4 - 4^2 x + 4 x = 5 -4^2 x + 4 x - 1 = 0 Multiply by -1 to make the leading coefficient positive: 4^2 x - 4 x + 1 = 0 Step 3: Solve the quadratic equation for x. This is a perfect square trinomial. (2 x - 1)^2 = 0 Take the square root of both sides: 2 x - 1 = 0 2 x = 1 x = (1)/(2) Step 4: Find the values of x in the range 0^ ≤ x ≤ 360^. The principal value for x = (1)/(2) is x = 60^. Since cosine is positive in the first and fourth quadrants, the other solution in the given range is: x = 360^ - 60^ = 300^ The solutions are x = 60^, 300^. 3. Making a the subject of the formula Step 1: The given formula is R = ksqrt(y^2 - a^2). First, isolate the square root term by dividing by k. (R)/(k) = sqrt(y^2 - a^2) Step 2: Square both sides of the equation to eliminate the square root. ((R)/(k))^2 = y^2 - a^2 (R^2)/(k^2) = y^2 - a^2 Step 3: Rearrange the equation to isolate a^2. Add a^2 to both sides and subtract (R^2)/(k^2) from both sides. a^2 = y^2 - (R^2)/(k^2) Step 4: Take the square root of both sides to find a. a = ±sqrt(y^2 - (R^2)/(k^2)) The subject of the formula is a = ±sqrt(y^2 - (R^2)/(k^2)). Send me the next one 📸