This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the solutions to the structured questions: Q1. Given A = (3)/(5) and B = (12)/(13). First, we find the missing trigonometric values for angles A and B using the Pythagorean identity ^2 + ^2 = 1. Assuming A and B are acute angles (in the first quadrant): For angle A: ^2 A = 1 - ^2 A = 1 - ((3)/(5))^2 = 1 - (9)/(25) = (16)/(25) A = sqrt((16)/(25)) = (4)/(5) For angle B: ^2 B = 1 - ^2 B = 1 - ((12)/(13))^2 = 1 - (144)/(169) = (25)/(169) B = sqrt((25)/(169)) = (5)/(13) So we have: A = (4)/(5), A = (3)/(5), B = (12)/(13), B = (5)/(13). a) Find (A + B) Step 1: Use the cosine addition formula: (A + B) = A B - A B. Step 2: Substitute the known values into the formula. (A + B) = ((3)/(5))((5)/(13)) - ((4)/(5))((12)/(13)) Step 3: Perform the multiplication and subtraction. (A + B) = (15)/(65) - (48)/(65) = (15 - 48)/(65) = (-33)/(65) a) (A + B) = -(33)/(65) b) Find (A - B) Step 1: Use the sine subtraction formula: (A - B) = A B - A B. Step 2: Substitute the known values into the formula. (A - B) = ((4)/(5))((5)/(13)) - ((3)/(5))((12)/(13)) Step 3: Perform the multiplication and subtraction. (A - B) = (20)/(65) - (36)/(65) = (20 - 36)/(65) = (-16)/(65) b) (A - B) = -(16)/(65) c) Find 2A Step 1: Use the sine double angle formula: 2A = 2 A A. Step 2: Substitute the known values for A and A. 2A = 2 ((4)/(5))((3)/(5)) Step 3: Perform the multiplication. 2A = 2 × (12)/(25) = (24)/(25) c) 2A = (24)/(25) Q2. a) Solve the equation 2^2 x - x - 1 = 0 Step 1: Let y = x. The equation becomes a quadratic equation in terms of y. 2y^2 - y - 1 = 0 Step 2: Solve the quadratic equation for y by factoring. (2y + 1)(y - 1) = 0 Step 3: Set each factor to zero to find the possible values for y. 2y + 1 = 0 2y = -1 y = -(1)/(2) y - 1 = 0 y = 1 Step 4: Substitute back y = x and find the values of x in the range 0^ x < 360^. For x = 1: The angle x where x = 1 is 90^. For x = -(1)/(2): The reference angle for x = (1)/(2) is 30^. Since x is negative, x lies in the third and fourth quadrants. In the third quadrant: x = 180^ + 30^ = 210^. In the fourth quadrant: x = 360^ - 30^ = 330^. Step 5: List all solutions for x. x = 90^, 210^, 330^ a) x = 90^, 210^, 330^ That's 2 down. 3 left today — send the next one.