[size=150][size=200]Let's investigate and study the connection between the angle formed by the two lines AB and AC and the trigonometric rations sine, cosine and tangent of that angle.[br][br]Note : Use the checkboxes to turn on and off the display of lengths and other lines/objects shown.[br][br]Carry out the two approaches below.[br]We start with the angle [math]\theta[/math] = 40[sup]o[/sup] .[br][br]Case 1[br]1. Show the line through D, perpendicular to AC [br] which intersects AB at F, and the triangle AFD formed (Click on the checkbox for this).[br][br][br]2. Drag the point D so that AD is (i) 5 (ii) 10 or (iii) 15 units long. [br](It may hard to drag D to achieve exactly integer values depending on your device)[br][br]3. Note down the following length ratios for each of the values of AD (in (i), (ii) and (iii))[br] (a) [math]\frac{FD}{AF}[/math][/size][/size] [br] [size=200] (b)[/size] [math]\frac{AD}{AF}[/math] [br] [size=200](c)[/size] [math]\frac{FD}{AD}[/math] [br][br][size=200]4. What do you observe about the ratios in 3(a), (b) and (c) for the 3 different values of AD (in (i), (ii) and (iii))[br][br]Repeat step 1 to 4 for Case 2 as shown below[br][size=150][size=200][br]Case 2[br][br]1. Show the line through D, perpendicular to AB, which intersects AB at E, and the triangle AED formed. [br][br]2. Drag the point D so that AD is (i) 5 (ii) 10 or (iii) 15 units long. [br][br]3. Note down the following length ratios for each of the values of AD (in (i), (ii) and (iii))[br] (a) [math]\frac{ED}{AD}[/math][/size][/size][br]          (b) [math]\frac{AE}{AD}[/math] [br] (c) [math]\frac{ED}{AE}[/math][br][br][size=200]4. What do you observe about the ratios in 3(a), (b) and (c) for the 3 different values of AD (in (i), (ii) and (iii))[br][br]Observations [br]5. What do you observe about the length ratios, for both cases (using different values of AD) in [br][br](i) step 3 (a) [br](ii) step 3 (b)[br](iii) step 3 (c)[br][br]6. Search the internet about the basic definitions of the trigonometric ratios, sine (sin), cosine (cos) and tangent (tan) of an angle in a right angled triangle.[br][br]Show the "positional relationship between angle and side of triangle" by clicking on the checkbox.[br][br]Which of the ratios in step 3 (a), (b) or (c) correspond to the sine(sin), cosine(cos) or tangent(tan) of angle[/size][/size] [size=200][math]\theta[/math] ? [br]Use a calculator to obtain the value of [br]sin 40[sup]o[/sup] , cos 40[sup]o[/sup] and tan 40[sup]o[/sup] . [br]Do they correspond closely to the values in 3(a), (b) and (c) for all cases?[br][br]Repeat steps 1 to 5 by changing the value of angle [math]\theta[/math] from 40[sup]o[/sup] to 70[sup]o [/sup].[br]Then repeat again for [math]\theta[/math] = 20[sup]o [/sup][/size][size=200][br][br]7. Do the sine, cosine and tangents of this angle change when the angle changes?[br][br]8. Does the sine, cosine or tangent of a particular value of the angle change when we change the length of sides (step 2) or position of the right angled triangle (case 1 or 2)[br][br]9. Can we conclude that the sine, cosine or tangent of a particular size of angle is constant (not dependant on any particular way of drawing a right angled triangle ) and that different size of angle will give different values of sine (sin) , cosine (cos) and tangent (tan) of the angle?[br][br][/size]