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.
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Question 1 Given x = (3)/(5), find x. Step 1: Start with the Pythagorean identity. ^2 x + ^2 x = 1 Step 2: Substitute the given value. ( (3)/(5) )^2 + ^2 x = 1 (9)/(25) + ^2 x = 1 Step 3: Isolate ^2 x. ^2 x = 1 - (9)/(25) = (16)/(25) Step 4: Take the square root (principal value for acute angle). x = ± (4)/(5) Assuming x is acute, x = (4)/(5). x = (4)/(5) Question 2 Given x = (4)/(5), find x. Step 1: Start with the Pythagorean identity. ^2 x + ^2 x = 1 Step 2: Substitute the given value. ^2 x + ( (4)/(5) )^2 = 1 ^2 x + (16)/(25) = 1 Step 3: Isolate ^2 x. ^2 x = 1 - (16)/(25) = (9)/(25) Step 4: Take the square root (principal value for acute angle). x = ± (3)/(5) Assuming x is acute, x = (3)/(5). x = (3)/(5) Question 4.1 Simplify (90^ - ). Step 1: In a right triangle, (90^ - ) is opposite over hypotenuse for angle (90^ - ). The opposite side to (90^ - ) is the adjacent side to . Step 2: Thus, (90^ - ) = adjacent to hypotenuse = . (90^ - ) = Question 4.2 Simplify (90^ - ). Step 1: (90^ - ) is adjacent over hypotenuse for angle (90^ - ). The adjacent side to (90^ - ) is the opposite side to . Step 2: Thus, (90^ - ) = opposite to hypotenuse = . (90^ - ) = Question 4.3 Simplify (90^ - ). Step 1: (90^ - ) = opposite to (90^ - )adjacent to (90^ - ). Step 2: Opposite to (90^ - ) is adjacent to , adjacent to (90^ - ) is opposite to . Step 3: Thus, (90^ - ) = adjacent to opposite to = . (90^ - ) = Question 4.4 Prove ^2 + ^2 = 1 using the right triangle. Step 1: Label the right triangle with hypotenuse r, opposite side to is y, adjacent side is x. = (y)/(r), = (x)/(r) Step 2: By Pythagorean theorem in the triangle, r^2 = x^2 + y^2 Step 3: Divide both sides by r^2. (r^2)/(r^2) = (x^2)/(r^2) + (y^2)/(r^2) 1 = ^2 + ^2 ^2 + ^2 = 1