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
Step 1: Identify the trigonometric identity. The expression inside the inverse cosine is in the form A x + B x. We can relate this to the cosine addition formula (A-B) = A B + A B. Let's try to find an angle such that = (3)/(5) and = (4)/(5). We check if ^2 + ^2 = 1: ((3)/(5))^2 + ((4)/(5))^2 = (9)/(25) + (16)/(25) = (25)/(25) = 1 Since the sum of squares is 1, such an angle exists. Step 2: Substitute the trigonometric values. Let = ^-1((3)/(5)). This implies = (3)/(5). From ^2 + ^2 = 1, we have = sqrt(1 - ^2 ) = sqrt(1 - ((3)/(5))^2) = sqrt(1 - (9)/(25)) = sqrt((16)/(25)) = (4)/(5). So, we can rewrite the expression as: ^-1( x + x) Step 3: Apply the cosine subtraction formula. Using the identity (A-B) = A B + A B, we can simplify the argument of the inverse cosine: ^-1(( - x)) Step 4: Simplify using the inverse function property. Since ^-1( ) = for in the principal range, we get: x Step 5: Substitute back the value of . Substitute = ^-1((3)/(5)) back into the expression: ^-1((3)/(5)) - x Send me the next one 📸