Trigonometry

1. In the right-angled triangle $\mathrm{△}ABC$,   $A\hat{C}B={90}^{\circ },A\hat{B}C={37}^{\circ }{25}^{\prime }$ and $a=BC=14\mathrm{cm}$.
   

   (a)   Find the lengths of the other two sides $b=AC$ and $c=AB$.

   (b)   Calculate the perimeter $P$ of the triangle.

   Solutions:    (a)  $b\approx 10.7\mathrm{cm},c\approx 17.6\mathrm{cm}$;     (b)  $P=a+b+c\approx 42.3\mathrm{cm}$
2. In the right-angled triangle $\mathrm{△}ABC$,   $A\hat{C}B={90}^{\circ },A\hat{B}C={30}^{\circ }$ and $a=BC=12\mathrm{cm}$.
   

   (a)   Find the lengths of the other two sides $b=AC$ and $c=AB$.

   (b)   Calculate the perimeter $P$ of the triangle.

   Solutions:    (a)  $b=4\sqrt{3}\mathrm{cm},c=8\sqrt{3}\mathrm{cm}$;     (b)  $P=a+b+c=12+12\sqrt{3}\mathrm{cm}$
3. In the isosceles triangle $\mathrm{△}ABC$,   $a=b=7\mathrm{cm}$ and $\alpha =B\hat{A}C={68}^{\circ }$.
   

   (a)   Find the length of the side $c$.

   (b)   Find the length of the height $h_c$.

   Solutions:    (a)  $c\approx 5.24\mathrm{cm}$;     (b)  $h_c\approx 6.49\mathrm{cm}$
4. The isosceles triangle $\mathrm{△}ABC$ $(a=b)$ has the base $c=AB=24\mathrm{cm}$ and the height $h_c=35\mathrm{cm}$
   

   (a)   Find the perimeter of this triangle.

   (b)   Calculate the angles of this triangle. Write them in degrees and minutes.

   Solutions:    (a)  $P=24+37+37=98\mathrm{cm}$;     (b)  $\alpha =\beta \approx {71}^{\circ }{5}^{\prime },\gamma \approx {37}^{\circ }{51}^{\prime }$
5. The rectangle $ABCD$ has the diagonal $e=AC=20\mathrm{cm}$. The angle between this diagonal and the side $a=AB$ is $C\hat{A}B={27}^{\circ }{45}^{\prime }$.
   

   (a)   Find the sides of this rectangle.

   (b)   Calculate the acute angle between the diagonals.

   Solutions:    (a)  $a\approx 17.7\mathrm{cm},b\approx 9.31\mathrm{cm}$;     (b)  $\phi ={55}^{\circ }{30}^{\prime }$
6. The rectangle $ABCD$ has the side $a=AB=18\mathrm{cm}$. The angle between the diagonal $d=AC$ and the side $a$ is $C\hat{A}B={30}^{\circ }$. Find the side $b=BC$ and the diagonal $d=AC$. Write your results in the exact form.
   Solutions:    $b=6\sqrt{3}\mathrm{cm},d=12\sqrt{3}\mathrm{cm}$
7. The diagonals of the rhombus $ABCD$ have lengths $e=AC=30\mathrm{cm}$ and $f=BD=16\mathrm{cm}$.
   

   (a)   Find the side $a=AB$.

   (b)   Calculate the angle $\alpha =D\hat{A}B$.

   Solutions:    (a)  $a=17\mathrm{cm}$;     (b)  $\alpha \approx {56}^{\circ }{9}^{\prime }$
8. A regular pentagon $ABCDE$ is inscribed in the circle with the radius $r=10\mathrm{cm}$. Find the side $a=AB$ of this pentagon.
   Solutions:    $a\approx 11.8\mathrm{cm}$
9. A regular hexagon $ABCDEF$ has the side $a=7\mathrm{cm}$. Find the length of the diagonal $d=AE$.
   Solutions:    $d=7\sqrt{3}\mathrm{cm}$
10. In the triangle $\mathrm{△}ABC$, $\alpha =C\hat{A}B={60}^{\circ },\beta =A\hat{B}C={45}^{\circ }$ and the height $h_c=4\sqrt{3}\mathrm{cm}$. Find the lengths of the sides $a,b$ and $c$. Write the exact values.
    Solutions:    $a=4\sqrt{6}\mathrm{cm},b=8\mathrm{cm},c=(4+4\sqrt{3})\mathrm{cm}$
11. In the triangle $\mathrm{△}ABC$, $b=AC=6\mathrm{cm},c=AB=10\mathrm{cm}$ and $\alpha =C\hat{A}B={70}^{\circ }$. The height $h_c$ has the endpoints $C$ and $H$.
    

    (a)   Find the height $h_c$.

    (b)   Find the distances $AH$ and $HB$.

    (c)   Hence, calculate the side $a=BC$.

    Solutions:    (a)  $h_c\approx 5.64\mathrm{cm}$;     (b)  $AH\approx 2.05\mathrm{cm},HB\approx 7.95\mathrm{cm}$;     (c)  $a\approx 9.74\mathrm{cm}$
12. In the triangle $\mathrm{△}ABC$, $b=AC=5\mathrm{cm},c=AB=8\mathrm{cm}$ and $\alpha =C\hat{A}B={60}^{\circ }$. Find the length of the side $a$.
    Solutions:    $a=7\mathrm{cm}$
13. In the triangle $\mathrm{△}ABC$, $a=BC=13\mathrm{cm},b=AC=19\mathrm{cm}$ and $\gamma =A\hat{C}B={64}^{\circ }{30}^{\prime }$. Find the length of the side $c$.
    Solutions:    $c\approx 17.8\mathrm{cm}$
14. In the triangle $\mathrm{△}ABC$, $a=BC=9\mathrm{cm},b=AC=14\mathrm{cm}$ and $c=AB=13\mathrm{cm}$. Calculate the angles $\alpha ,\beta$ and $\gamma$. Round the results to the nearest minute.
    Solutions:    $\alpha \approx {38}^{\circ }{43}^{\prime },\beta \approx {76}^{\circ }{39}^{\prime },\gamma \approx {64}^{\circ }{37}^{\prime }$
15. In the triangle $\mathrm{△}ABC$, $a=BC=13\mathrm{cm},b=AC=8\mathrm{cm}$ and $c=AB=7\mathrm{cm}$. Calculate the angles $\alpha ,\beta$ and $\gamma$. Round the results to the nearest minute.
    Solutions:    $\alpha ={120}^{\circ },\beta \approx {32}^{\circ }{12}^{\prime },\gamma \approx {27}^{\circ }{48}^{\prime }$
16. In the triangle $\mathrm{△}ABC$, $a=BC=2\mathrm{cm},b=AC=7\mathrm{cm}$ and $c=AB=3\sqrt{3}\mathrm{cm}$. Calculate the angle $\beta$.
    Solutions:    $\beta ={150}^{\circ }$
17. In the triangle $\mathrm{△}ABC$, $a=23\mathrm{cm},c=17\mathrm{cm}$ and $\beta ={107}^{\circ }$.
    

    (a)   Find the side $b$.

    (b)   Calculate the angles $\alpha$ and $\gamma$.

    Solutions:    (a)  $b\approx 32.4\mathrm{cm}$;     (b)  $\alpha \approx {42}^{\circ }{50}^{\prime },\gamma \approx {30}^{\circ }{10}^{\prime }$
18. In the triangle $\mathrm{△}ABC$, $\alpha ={30}^{\circ },\beta ={71}^{\circ }$ and $b=16\mathrm{cm}$.
    

    (a)   Find the height $h_c$.

    (b)   Calculate the side $a$.

    Solutions:    (a)  $h_c=8\mathrm{cm}$;     (b)  $a\approx 8.46\mathrm{cm}$
19. In the triangle $\mathrm{△}ABC$, $\alpha ={57}^{\circ },\beta ={71}^{\circ }$ and $c=15\mathrm{cm}$.
    

    (a)   Find the angle $\gamma$.

    (b)   Find the sides $a$ and $b$.

    Solutions:    (a)  $\gamma ={52}^{\circ }$;     (b)  $a\approx 16.0\mathrm{cm},b\approx 18.0\mathrm{cm}$
20. In the triangle $\mathrm{△}ABC$, $a=23\mathrm{cm},b=39\mathrm{cm}$ and $\beta ={83}^{\circ }$.
    

    (a)   Find the angles $\alpha$ and $\gamma$.

    (b)   Find the side $c$.

    Solutions:    (a)  $\alpha \approx {35}^{\circ }{50}^{\prime },\gamma \approx {61}^{\circ }{10}^{\prime }$;     (b)  $c\approx 34.4\mathrm{cm}$
21. In the triangle $\mathrm{△}ABC$, $b=6\mathrm{cm},c=8\mathrm{cm}$ and $\alpha ={30}^{\circ }$.
    

    (a)   Find the height $h_c$.

    (b)   Find the area $A$.

    Solutions:    (a)  $h_c=3\mathrm{cm}$;     (b)  $A=12{\mathrm{cm}}^2$
22. In the triangle $\mathrm{△}ABC$, $a=14\mathrm{cm},c=11\mathrm{cm}$ and $\beta ={77}^{\circ }{33}^{\prime }$.
    

    (a)   Find the side $b$.

    (b)   Find the area $A$.

    Solutions:    (a)  $b\approx 15.8\mathrm{cm}$;     (b)  $A\approx 75.2{\mathrm{cm}}^2$
23. In the triangle $\mathrm{△}ABC$, $a=5\mathrm{cm},b=2\mathrm{cm}$ and $\gamma ={58}^{\circ }$. Find the area of this triangle.
    Solutions:    $A\approx 4.24{\mathrm{cm}}^2$
24. In the triangle $\mathrm{△}ABC$, $a=7\mathrm{cm},b=8\mathrm{cm}$ and $c=5\mathrm{cm}$.
    

    (a)   Find the angle $\alpha =C\hat{A}B$.

    (b)   Find the area $A$.

    Solutions:    (a)  $\alpha ={60}^{\circ }$;     (b)  $A=10\sqrt{3}\mathrm{cm}\approx 17.3{\mathrm{cm}}^2$
25. The triangle $\mathrm{△}ABC$ has sides: $a=26\mathrm{cm},b=15\mathrm{cm}$ and $c=37\mathrm{cm}$. Find the area of this triangle.
    Solutions:    $A=156{\mathrm{cm}}^2$
26. The triangle $\mathrm{△}ABC$ has sides: $a=14\mathrm{cm},b=19\mathrm{cm}$ and $c=7\mathrm{cm}$. Find the area of this triangle.
    Solutions:    $A\approx 39.5{\mathrm{cm}}^2$
27. The triangle $\mathrm{△}ABC$ has sides: $a=6\mathrm{cm},b=8\mathrm{cm}$ and $c=17\mathrm{cm}$. Find the area of this triangle.
    Solutions:    A triangle with these sides doesn't exist.
28. The triangle $\mathrm{△}ABC$ has the sides: $a=17\mathrm{cm},b=10\mathrm{cm}$ and $c=21\mathrm{cm}$.
    

    (a)   Find the area.

    (b)   Find the height $h_c$.

    Solutions:    (a)  $A=84{\mathrm{cm}}^2$;     (b)  $h_c=8\mathrm{cm}$
29. In the triangle $\mathrm{△}ABC$: $\alpha ={39}^{\circ },\beta ={68}^{\circ }$ and $c=14\mathrm{cm}$.
    

    (a)   Find the angle $\gamma$.

    (b)   Find the sides $a$ and $b$.

    (c)   Find the perimeter and area.

    Solutions:    (a)  $\gamma ={73}^{\circ }$;     (b)  $a\approx 9.21\mathrm{cm},b\approx 13.6\mathrm{cm}$;     (c)  $P\approx 36.8\mathrm{cm},A\approx 59.8{\mathrm{cm}}^2$
30. In the parallelogram $ABCD$: $a=AB=15\mathrm{cm},b=BC=7\mathrm{cm}$ and $\alpha =B\hat{A}D={60}^{\circ }$.
    

    (a)   Find the diagonal $f=BD$.

    (b)   Find the area of this parallelogram.

    Solutions:    (a)  $f=13\mathrm{cm}$;     (b)  $A\approx 90.9{\mathrm{cm}}^2$
31. The rhombus $ABCD$ has the side $a=7\mathrm{cm}$ and the angle $\alpha ={69}^{\circ }$.
    

    (a)   Find the diagonal $f=BD$.

    (b)   Find the height $h$.

    (c)   Find the area of this rhombus.

    Solutions:    (a)  $f\approx 7.93\mathrm{cm}$;     (b)  $h\approx 6.54\mathrm{cm}$;     (c)  $A\approx 45.7{\mathrm{cm}}^2$
32. A regular pentagon is inscribed in the circle with the radius $r=8\mathrm{cm}$.
    

    (a)   Find the side of this pentagon.

    (b)   Find the diagonal.

    (c)   Find the area.

    Solutions:    (a)  $a\approx 9.40\mathrm{cm}$;     (b)  $d\approx 15.2\mathrm{cm}$;     (c)  $A\approx 152{\mathrm{cm}}^2$
33. A regular octagon $ABCDEFGH$ has the side $a=10\mathrm{cm}$.
    

    (a)   Find the diagonal $AC$.

    (b)   Find the inradius $r$ (radius of the inscribed circle).

    (c)   Find the area of this octagon.

    Solutions:    (a)  $AC\approx 18.5\mathrm{cm}$;     (b)  $r\approx 12.1\mathrm{cm}$;     (c)  $A\approx 483{\mathrm{cm}}^2$
34. A regular nonagon has the side $a=6\mathrm{cm}$.
    

    (a)   Find the circumradius $R$ (radius of the circumscribed circle).

    (b)   Find the length of the longest diagonal.

    (c)   Find the area of this nonagon.

    Solutions:    (a)  $R\approx 8.77\mathrm{cm}$;     (b)  $d\approx 17.3\mathrm{cm}$;     (c)  $A\approx 223{\mathrm{cm}}^2$
35. In the triangle $\mathrm{△}ABC$: $a=2\mathrm{cm},b=7\mathrm{cm}$ and $c=3\sqrt{3}\mathrm{cm}$.
    

    (a)   Find the angle $\alpha$.

    (b)   Find the angle $\beta$. Try calculating $\beta$ using the cosine rule and using the sine rule.

    Solutions:    (a)  $\alpha \approx 8^{\circ }{13}^{\prime }$;     (b)  $\beta ={150}^{\circ }$ (and not ${30}^{\circ }$)
36. In the triangle $\mathrm{△}ABC$: $a=19\mathrm{cm},b=21\mathrm{cm}$ and $\alpha ={60}^{\circ }$.
    

    (a)   Find the angle $\beta$. Write down both possible values of $\beta$.

    (b)   Find the side $c$. Write down both possible values of $c$.

    Solutions:    (a)  ${\beta }_1\approx {73}^{\circ }{10}^{\prime },{\beta }_2\approx {106}^{\circ }{50}^{\prime }$;     (b)  $c_1=16\mathrm{cm},c_2=5\mathrm{cm}$
37. In the triangle $\mathrm{△}ABC$: $a=7\mathrm{cm},b=13\mathrm{cm}$ and $\alpha ={105}^{\circ }$. Find the angle $\beta$ and the side $c$.
    Solutions:    Such a triangle can't exist.
38. In the triangle $\mathrm{△}ABC$: $a=7\mathrm{cm},b=13\mathrm{cm}$ and $\alpha ={68}^{\circ }$. Find the angle $\beta$ and the side $c$.
    Solutions:    Such a triangle can't exist.
39. In the triangle $\mathrm{△}ABC$: $a=7\mathrm{cm},b=13\mathrm{cm}$ and $\alpha ={29}^{\circ }$. Find the angle $\beta$ and the side $c$.
    Solutions:    Two solutions: (1)  ${\beta }_1\approx {64}^{\circ }{12}^{\prime },c_1\approx 14.4\mathrm{cm}$;     (2)  ${\beta }_2\approx {115}^{\circ }{48}^{\prime },c_2\approx 8.32\mathrm{cm}$
40. The triangle $\mathrm{△}ABC$ has the sides $a=5\mathrm{cm},b=8\mathrm{cm}$ and the area $A=8{\mathrm{cm}}^2$. Find the side $c$.
    Solutions:    Two solutions: $c_1\approx 3.96\mathrm{cm},c_2\approx 12.7\mathrm{cm}$
41. The circle has the radius $r=5\mathrm{cm}$. Find the area and perimeter of this circle.
    Solutions:    $A=25\pi {\mathrm{cm}}^2\approx 78.5{\mathrm{cm}}^2,P=10\pi \mathrm{cm}\approx 31.4\mathrm{cm}$
42. The circle has the circumference $100\mathrm{cm}$. Find the area of this circle.
    Solutions:    $r\approx 15.9\mathrm{cm},A\approx 796{\mathrm{cm}}^2$
43. Write the following angles in radians:
    

    (a)   ${30}^{\circ }$

    (b)   ${45}^{\circ }$

    (c)   ${120}^{\circ }$

    (d)   ${135}^{\circ }$

    Solutions:    (a)  ${30}^{\circ }=\frac{\pi }{6}$;     (b)  ${45}^{\circ }=\frac{\pi }{4}$;     (c)  ${120}^{\circ }=\frac{2\pi }{3}$;     (d)  ${135}^{\circ }=\frac{3\pi }{4}$
44. Convert the following angles to degrees:
    

    (a)   $\frac{\pi }{3}$

    (b)   $\frac{\pi }{2}$

    (c)   $\frac{5\pi }{6}$

    (d)   $\frac{5\pi }{4}$

    Solutions:    (a)  $\frac{\pi }{3}={60}^{\circ }$;     (b)  $\frac{\pi }{2}={90}^{\circ }$;     (c)  $\frac{5\pi }{6}={150}^{\circ }$;     (d)  $\frac{5\pi }{4}={225}^{\circ }$
45. The circle has the radius $r=12\mathrm{cm}$. Find the length of the arc which subtends the angle $\theta ={40}^{\circ }$ at the centre of this circle.
    Solutions:    $L=\frac{8}{3}\pi \mathrm{cm}\approx 8.38\mathrm{cm}$
46. The circle has the radius $r=9\mathrm{cm}$. Find the length of the arc which subtends the angle $\theta =2\mathrm{radians}$.
    Solutions:    $L=18\mathrm{cm}$
47. Find the area of the sector of the circle with the radius $r=18\mathrm{cm}$ and the central angle $\theta ={10}^{\circ }$.
    Solutions:    $A=9\pi {\mathrm{cm}}^2\approx 28.3{\mathrm{cm}}^2$
48. Given the radius $r=6\mathrm{cm}$ and central angle $\theta =\frac{1}{8}\pi$,
    

    (a)   find the area of the circular sector,

    (b)   find the length of the arc.

    Solutions:    (a)  $A\approx 7.07{\mathrm{cm}}^2$;     (b)  $L\approx 2.36\mathrm{cm}$
49. Given the radius $r=8\mathrm{cm}$ and central angle $\theta ={90}^{\circ }$,
    

    (a)   find the area of the circular sector,

    (b)   find the area of the corresponding circular segment.

    Solutions:    (a)  $A_1\approx 50.3{\mathrm{cm}}^2$;     (b)  $A_2\approx 18.3{\mathrm{cm}}^2$
50. Find the area of the circular segment of the radius $r=14\mathrm{cm}$ and central angle $\theta =1$.
    Solutions:    $A\approx 15.5{\mathrm{cm}}^2$

 Index