Expand R (x + alpha) using the angle addition formula.

Here's the solution for Item 6: a) Express 3 x + 4 x in the form R (x + ). Step 1: Expand R (x + ) using the angle addition formula. R (x + ) = R( x + x ) = (R ) x + (R ) x Step 2: Compare coefficients with 3 x + 4 x. We have R = 3 and R = 4. Step 3: Find R by squaring and adding the two equations. (R )^2 + (R )^2 = 3^2 + 4^2 R^2 (^2 + ^2 ) = 9 + 16 R^2 (1) = 25 R = sqrt(25) = 5 Since R represents amplitude, it must be positive. Step 4: Find by dividing the two equations. (R )/(R ) = (4)/(3) = (4)/(3) = ((4)/(3)) ≈ 53.13^ Since R = 3 > 0 and R = 4 > 0, is in the first quadrant. Therefore, 3 x + 4 x = 5 (x + 53.13^). b) Hence solve the equation 3 x + 4 x = 5 for 0^ ≤ x ≤ 360^. Step 1: Substitute the result from part (a) into the equation. 5 (x + 53.13^) = 5 Step 2: Isolate the sine function. (x + 53.13^) = 1 Step 3: Find the principal value for x + 53.13^. The general solution for = 1 is = 90^ + n · 360^, where n is an integer. So, x + 53.13^ = 90^. Step 4: Solve for x. x = 90^ - 53.13^ x = 36.87^ To check for other solutions within 0^ ≤ x ≤ 360^, consider x + 53.13^ = 90^ + 360^ = 450^. Then x = 450^ - 53.13^ = 396.87^, which is outside the given range. The only solution in the given range is x = 36.87^. c) Using double angle identities, simplify 2 ^2 x - 1. Step 1: Recall the double angle identity for cosine. (2x) = ^2 x - ^2 x Step 2: Use the Pythagorean identity ^2 x = 1 - ^2 x. Substitute this into the double angle identity: (2x) = (1 - ^2 x) - ^2 x (2x) = 1 - 2 ^2 x Step 3: Rearrange to match the given expression. The expression to simplify is 2 ^2 x - 1. This is the negative of 1 - 2 ^2 x. 2 ^2 x - 1 = -(1 - 2 ^2 x) 2 ^2 x - 1 = -(2x) d) Explain how such transformations are useful in analysing oscillatory systems. Transforming an expression like a x + b x into the form R (x + ) (or R (x - )) is crucial for analyzing oscillatory systems because it directly reveals the amplitude (R) and phase shift () of the oscillation. The amplitude R represents the maximum displacement or intensity of the wave, while the phase shift indicates the horizontal shift of the wave relative to a standard sine or cosine function. This simplified form makes it easier to understand the overall behavior, predict maximum and minimum values, and analyze the timing of the oscillation, which is vital in fields like physics and engineering. That's 2 down. 3 left today — send the next one.