From a point A due south of a tower, the angle of elevation of the top of the tower T is 23° - HSC - SSCE Mathematics Extension 1 - Question 6 - 2008 - Paper 1

From point A, we can use the tangent of angle 23° to find the height (h) of the tower:

$h = AB \times \tan(23°)$

Calculate the height using the distance AB (200 m):

$h = 200 \times \tan(23°) \approx 200 \times 0.4245 \approx 84.9 ext{ m}$. Therefore, the height of the tower is approximately 84.9 metres.

Utilize the equation sin 3θ = 3 sin θ - 4 sin³ θ:

1. Substitute into the equation, giving: $3 sin θ - 4 sin³ θ + sin 2θ - sin θ = 0$
2. Rearranging yields: $-4 sin³ θ + 2 sin θ + sin 2θ = 0$
3. Solve for sin 2θ using the identity sin 2θ = 2 sin θ cos θ: $-4 sin³ θ + 2 sin θ + 2 sin θ cos θ = 0$
4. Factor to find zeros and solve for values of θ.

The angles satisfying this equation within the range can be found through numerical methods or further analysis.

Expanding using the binomial theorem:

$(1 + x)^{p + q} = \sum_{k=0}^{p + q} {p + q \choose k} x^k$

To find the term independent of x, set k = 0. The required term in the combined expansion will be the constant term found in the overall expansion.

From the earlier part, we know how to expand (1 + x)ᵖ:

1. Rewrite the new expression: $(1 + x)^p (1 + x^q) = (1 + x)^p + q * x^2$
2. Apply the binomial expansion to the right side: $= (1 + x)^p + q * x^2$
3. Combine terms from both expansions and isolate those involving x and independent terms to simplify further.