Review exercises  —  trigonometry

1. A right-angled triangle has sides \(b=12~\mathrm{cm},~ c=37~\mathrm{cm}\). Angle \(B\hat{C}A\) is the right angle.
   

   (a)   Find the side \(a\).

   (b)   Calculate angles \(\alpha=C\hat{A}B\) and \(\beta=A\hat{B}C\).

   Solutions:    (a)  \(a=35~\mathrm{cm}\);     (b)  \(\alpha\approx71.1^\circ,~ \beta\approx18.9^\circ\)
2. An isosceles triangle has sides \(a=b=33~\mathrm{cm},~ c=18~\mathrm{cm}\).
   

   (a)   Find the height \(h_c\) giving your result in exact form.

   (b)   Calculate area. Round your result to three significant figures.

   (c)   Calculate all three angles.

   Solutions:    (a)  \(h_c=12\sqrt{7}~\mathrm{cm}\);     (b)  \(A\approx286~\mathrm{cm}^2\);     (c)  \(\alpha=\beta\approx74.2^\circ,~ \gamma\approx31.7^\circ\)
3. Triangle \(ABC\) has sides \(b=17~\mathrm{cm},~ c=22~\mathrm{cm}\) and angle \(\hat{A}=52^\circ40'\).
   

   (a)   Calculate side \(a\).

   (b)   Calculate angle \(\hat{B}\). Give its value rounded to the nearest minute.

   (c)   Calculate the area of this triangle.

   Solutions:    (a)  \(a\approx17.9~\mathrm{cm}\);     (b)  \(\hat{B}\approx49^\circ9'\);     (c)  \(A\approx~149~\mathrm{cm}^2\)
4. Triangle \(ABC\) has side \(b=8~\mathrm{cm}\) and angles \(\hat{A}=45.6^\circ,~ \hat{C}=76.5^\circ\).
   

   (a)   Calculate side \(a\).

   (b)   Calculate the area.

   (c)   Hence calculate the altitude \(h_a\).

   Solutions:    (a)  \(a\approx6.75~\mathrm{cm}\);     (b)  \(A\approx26.2~\mathrm{cm}^2\);     (c)  \(h_a=\frac{2A}{a}\approx7.78~\mathrm{cm}\)
5. Triangle \(ABC\) has sides \(a=11~\mathrm{cm},~ b=20~\mathrm{cm},~ c=13~\mathrm{cm}\).
   

   (a)   Calculate the largest angle in this triangle.

   (b)   Hence or otherwise calculate the area.

   The midpoint of the side \(AB\) is labelled \(P\).

   (c)   Calculate \(PC\).

   Solutions:    (a)  \(A\hat{B}C\approx113^\circ\);     (b)  \(A=66~\mathrm{cm}^2\);     (c)  \(PC\approx14.8~\mathrm{cm}\)
6. Function \(f\) has the equation \(f(x)=2\sin 3x\).
   

   (a)   Draw the graph of \(f\), for \(-3\leqslant x \leqslant 3\).

   Let \(P\) be the \(x\)-axis intercept with the smallest positive \(x\).

   (b)   Find the equation of the tangent at \(P\).

   (c)   Calculate the area of the region enclosed by \(f\) and the horizontal axis between the origin \(O\) and \(P\).

   Solutions:    \(P(\frac{\pi}{3},0)\);     (b)  \(y=-6x+2\pi\);     (c)  \(A=\frac{4}{3}\approx1.33\)
7. Consider the function \(f(x)=\cos\frac{\textstyle x+\pi}{\textstyle 2}\). This function has a tangent \(L\) at point \(A(2\pi,0)\).
   

   (a)   Find the equation of the tangent \(L\).

   (b)   Calculate the area of the triangular region enclosed by \(L\) and both coordinate axes. Write the result in exact form.

   Solutions:    (a)  \(y=\frac{1}{2}x-\pi\);     (b)  \(A=\pi^2\)
8. Function \(f(x)=a\sin bx+c\) has a maximum at \(A(\frac{1}{2},5)\). The first next minimum is at \(B(\frac{3}{2},1)\). Find values of constants \(a,~ b\) and \(c\).
   Solutions:    \(a=2,~ b=\pi,~ c=3\)
9. Consider the functions \(f(x)=\sin^2 x\) and \(g(x)=\frac{1}{8}x^3\), for \(-2\leqslant x\leqslant 4\).
   

   (a)   Draw graphs of \(f\) and \(g\) on the given interval.

   (b)   Find intersection points of \(f\) and \(g\).

   (c)   Calculate the area of the region enclosed by \(f\) and \(g\).

   Solutions:    (b)  \(P(0,0),~ Q(1.92, 0.884)\);     (c)  \(A\approx0.696\)
10. Function \(f\) is a piecewise defined function: \(f(x)=\left\{\begin{array}{ll} 4-x^2; & x\leqslant 0 \\ 2\cos x+2; & x\gt 0 \end{array}\right.\)
    

    (a)   Draw graph of \(f\).

    (b)   Find the equation of the tangent to \(f\) at \(x=-1\).

    (c)   Calculate the area of the region enclosed by \(f\) and \(x\)-axis, for \(-2\leqslant x\leqslant \pi\).

    Solutions:    (b)  \(y=2x+5\);     (c)  \(A\approx11.6\)

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