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)  $\alpha \approx {71.1}^{\circ },\beta \approx {18.9}^{\circ }$
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)  $A\approx 286{\mathrm{cm}}^2$;     (c)  $\alpha =\beta \approx {74.2}^{\circ },\gamma \approx {31.7}^{\circ }$
3. Triangle $ABC$ has sides $b=17\mathrm{cm},c=22\mathrm{cm}$ and angle $\hat{A}={52}^{\circ }{40}^{\prime }$.
   

   (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)  $a\approx 17.9\mathrm{cm}$;     (b)  $\hat{B}\approx {49}^{\circ }{9}^{\prime }$;     (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)  $a\approx 6.75\mathrm{cm}$;     (b)  $A\approx 26.2{\mathrm{cm}}^2$;     (c)  $h_a=\frac{2A}{a}\approx 7.78\mathrm{cm}$
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)  $A\hat{B}C\approx {113}^{\circ }$;     (b)  $A=66{\mathrm{cm}}^2$;     (c)  $PC\approx 14.8\mathrm{cm}$
6. Function $f$ has the equation $f(x)=2\sin 3x$.
   

   (a)   Draw the graph of $f$, for $-3⩽x⩽3$.

   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)  $A=\frac{4}{3}\approx 1.33$
7. Consider the function $f(x)=\cos \frac{x+\pi }{2}$. 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 $-2⩽x⩽4$.
   

   (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)  $A\approx 0.696$
10. Function $f$ is a piecewise defined function: $f(x)=\{\begin{array}{ll}4-x^2;& x⩽0\\ 2\cos x+2;& x>0\end{array}$
    

    (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 $-2⩽x⩽\pi$.

    Solutions:    (b)  $y=2x+5$;     (c)  $A\approx 11.6$

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