answer:To solve the given problem, we start by expressing the given equation in terms of sine and cosine: [8 tan theta = 3 cos theta.] This can be rewritten using the definition of tan theta = frac{sin theta}{cos theta}, leading to: [frac{8 sin theta}{cos theta} = 3 cos theta.] By cross-multiplying, we obtain: [8 sin theta = 3 cos^2 theta.] We know that cos^2 theta = 1 - sin^2 theta from the Pythagorean identity, so substituting this into our equation gives: [8 sin theta = 3(1 - sin^2 theta).] Expanding and rearranging terms to form a quadratic equation in sin theta, we get: [3 sin^2 theta + 8 sin theta - 3 = 0.] Factoring this quadratic equation yields: [(3 sin theta - 1)(sin theta + 3) = 0.] For sin theta to be real and within its domain [-1, 1], we solve for sin theta from each factor: - From 3 sin theta - 1 = 0, we find sin theta = frac{1}{3}. - The equation sin theta + 3 = 0 would give sin theta = -3, which is not possible since sin theta must be between -1 and 1. Therefore, the only valid solution within the given domain 0 < theta < pi is: [sin theta = boxed{frac{1}{3}}.]

answer:Given that (z^2 = 15 - 20i), we need to find (|z|). 1. First, find (|z^2|): [ |z^2| = |15 - 20i| ] The modulus of (15 - 20i) can be calculated as: [ |15 - 20i| = sqrt{15^2 + (-20)^2} = sqrt{225 + 400} = sqrt{625} = 25 ] 2. Since (|z|^2 = |z^2|), we have: [ |z|^2 = 25 ] 3. Taking the square root to find (|z|), we get: [ |z| = sqrt{25} = boxed{5} ] Conclusion: The problem makes sense and is solved correctly following the standard method for computing the modulus of a complex number given its square. The new problem maintains logical consistency without any contradictions, and the modulus found is unique and valid in all mathematical contexts.

answer:To find out how much each CD cost, we first need to calculate the total cost of the films and books that Josh bought. The cost of the films is: 9 films * 5 per film = 45 The cost of the books is: 4 books * 4 per book = 16 Now we add the cost of the films and books together to find the total cost of these items: 45 (films) + 16 (books) = 61 Josh spent 79 in total, so to find out how much he spent on CDs, we subtract the total cost of the films and books from the total amount spent: 79 (total spent) - 61 (cost of films and books) = 18 Josh spent 18 on CDs. Since he bought 6 CDs, we divide the total cost of the CDs by the number of CDs to find the cost per CD: 18 (total cost of CDs) / 6 CDs = 3 per CD Therefore, each CD cost boxed{3} .

answer:**Step 1: Understanding the problem** This problem requires us to find the value of a given that two lines are perpendicular to each other. The key to solving this problem is to recall that the product of the slopes of two perpendicular lines is -1. **Step 2: Finding the slopes** First, we find the slope of the first line, ax+2y-1=0. By rearranging the equation into slope-intercept form (y = mx + b), we get y = -frac{a}{2}x + frac{1}{2}. Therefore, the slope of this line is m_1 = -frac{a}{2}. Next, we find the slope of the second line, 2x+y-1=0. By rearranging the equation into slope-intercept form, we get y = -2x + 1. Therefore, the slope of this line is m_2 = -2. **Step 3: Applying the perpendicular condition** Since the two lines are perpendicular, their slopes' product should equal -1. Hence, we have m_1 times m_2 = -1, which implies (-frac{a}{2}) times (-2) = -1. **Step 4: Solving for a** Solving the above equation for a, we obtain a = -1. Therefore, the value of a is boxed{a = -1}.