Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Periodic Relationships: Trigonometric Functions, Exercise 29: Chapter review

Let $S\left(x\right)={\left(\sin 2x+\cos 2x\right)}^2$. State the range $S$.

Let $S\left(x\right)={\left(\sin 2x+\cos 2x\right)}^2$. Sketch the graph of the function $C$ defined by $C\left(x\right)=\cos \left(2x\right)-1, \text{for} 0\le x\le 2\pi$.

Let $S\left(x\right)={\left(\sin 2x+\cos 2x\right)}^2$. The graph of $C$ can be obtained from the graph of $S$ under a horizontal stretch of scale factor $K$ followed by a translation by the vector $\left(\begin{array}{c}p\\ q\end{array}\right)$. Write down the value of $k$.

Let $S\left(x\right)={\left(\sin 2x+\cos 2x\right)}^2$. The graph of $C$ can be obtained from the graph of $S$ under a horizontal stretch of scale factor $K$ followed by a translation by the vector $\left(\begin{array}{c}p\\ q\end{array}\right)$. Write down a possible vector $\left(\begin{array}{c}p\\ q\end{array}\right)$.

The diagram shows a circle with centre $O$ and radius $r$. The points $A \text{and} B$ lie on the circumference of the circle. Let $\theta =A\stackrel{⏜}{O}B$. If $\theta =\frac{2\pi }{3} \text{and} r=2 \mathrm{cm}$, state, in terms of $\pi$ the area of the sector $AOB$.

The diagram shows a circle with centre $O$ and radius $r$. The points $A \text{and} B$ lie on the circumference of the circle. Let $\theta =A\stackrel{⏜}{O}B$. If $\theta =\frac{2\pi }{3} \text{and} r=2 \mathrm{cm}$, state, in terms of $\pi$ the length of the arc $AB$.

The diagram shows a circle with centre $O$ and radius $r$. The points $A \text{and} B$ lie on the circumference of the circle. Let $\theta =A\stackrel{⏜}{O}B$. Given that the area of the sector $AOB \text{is} \pi {\mathrm{cm}}^2$ and the length of the arc $ABC \text{is} \frac{\pi }{3} \mathrm{cm}$, find the exact value of $r$.

The diagram shows a circle with centre $O$ and radius $r$. The points $A \text{and} B$ lie on the circumference of the circle. Let $\theta =A\stackrel{⏜}{O}B$. Given that the area of the sector $AOB \text{is} \pi {\mathrm{cm}}^2$ and the length of the arc $ABC \text{is} \frac{\pi }{3} \mathrm{cm}$, find the exact value of $\theta$.