<ol class="abc-list"><li><div class="list-item-paragraph">Consider a function \(f\)  such that&nbsp;\(f(x)=3\sin{(2x+\alpha)}\) where&nbsp;\(0 \leq x \leq 2\pi\) and \(0 \leq \alpha \leq \frac{\pi}{2}\).</div><div class="list-item-paragraph">Write down the amplitude of \(f\)<span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The graph of \(y=f(x)\) passes through the point \(\left(\frac{\pi}{3},\frac{3}{2}\right)\).</div><ol class="i-list"><li><div class="list-item-paragraph">Find the value of \(\alpha\).</div></li><li><div class="list-item-paragraph">Find the \(x\)-coordinates of the four other pints where the graph of \(f(x)=\frac{3}{2}\).<span class="inline-marks">[4]</span></div></li></ol></li><li><div class="list-item-paragraph">The function \(g\) is given by \(g(x)=pf(x)+q\) where&nbsp;\(0\leq x \leq 2\pi\) and \(p,q \in \mathbb{R}\).</div><div class="list-item-paragraph">The graph of \(y=g(x)\) passes through the origin and intercepts the graph of \(y=f(x)\) at the points where&nbsp;\(f(x)\) is at its maximum value.</div><div class="list-item-paragraph">Find the values of \(p\) and \(q\).<span class="inline-marks">[5]</span></div></li><li><div class="list-item-paragraph">In the case where&nbsp;\(\alpha=0\) and \(p\) and \(q\) have the same values found in part (c), show that&nbsp;\(g\) can be written in the form \(g(x)=k(m\sin{x}\cos{x}+n)\), where&nbsp;\(k, m\) and \(n\) are integers to be found and where&nbsp;\(k&gt;0, k\neq 1\).<span class="inline-marks">[3]</span></div></li></ol>

Consider a function \(f\)  such that \(f(x)=3\sin{(2x+\alpha)}\) where \(0 \leq x \leq 2\pi\) and \(0 \leq \alpha \leq \frac{\pi}{2}\).

The graph of \(y=f(x)\) passes through the point \(\left(\frac{\pi}{3},\frac{3}{2}\right)\).

The function \(g\) is given by \(g(x)=pf(x)+q\) where \(0\leq x \leq 2\pi\) and \(p,q \in \mathbb{R}\).

The graph of \(y=g(x)\) passes through the origin and intercepts the graph of \(y=f(x)\) at the points where \(f(x)\) is at its maximum value.

In the case where \(\alpha=0\) and \(p\) and \(q\) have the same values found in part (c), show that \(g\) can be written in the form \(g(x)=k(m\sin{x}\cos{x}+n)\), where \(k, m\) and \(n\) are integers to be found and where  \(k>0, k\neq 1\).

Trigonometric Equations and Identities

International Baccalaureate Standard Level Mathematics