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
The problem states that = (4)/(5). Step 1: Determine the value of . We know that = (1)/( ). = (1)/(4/5) = (5)/(4) The value = (5)/(4) = 1.25 is outside the possible range for the cosine function, which is [-1, 1]. Therefore, no real angle exists for which = (4)/(5). It is highly probable that there is a typo in the question, and it was intended to be = (5)/(4) (or = (4)/(5)). We will proceed by assuming the intended value was = (5)/(4). Assumption: We assume the question intended = (5)/(4). Step 1: Find using the reciprocal identity. Given = (5)/(4). = (1)/( ) = (1)/(5/4) = (4)/(5) Step 2: Find using the Pythagorean identity. The Pythagorean identity is ^2 + ^2 = 1. Substitute the value of : ^2 + ((4)/(5))^2 = 1 ^2 + (16)/(25) = 1 ^2 = 1 - (16)/(25) ^2 = (25)/(25) - (16)/(25) ^2 = (9)/(25) Take the square root of both sides: = ± sqrt((9)/(25)) = ± (3)/(5) Since the quadrant of is not specified, can be positive or negative. Step 3: Calculate . The identity for is = ( )/( ). Case 1: If = (3)/(5) = (4/5)/(3/5) = (4)/(3) Case 2: If = -(3)/(5) = (4/5)/(-3/5) = -(4)/(3) Therefore, based on the assumption that = (5)/(4), there are two possible values for . The final answer is ± (4)/(3).