Connecting Angle between Two Lines and Trigonometric Ratios
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.
Note : Use the checkboxes to turn on and off the display of lengths and other lines/objects shown.
Carry out the two approaches below.
We start with the angle = 40o .
Case 1
1. Show the line through D, perpendicular to AC
which intersects AB at F, and the triangle AFD formed (Click on the checkbox for this).
2. Drag the point D so that AD is (i) 5 (ii) 10 or (iii) 15 units long.
(It may hard to drag D to achieve exactly integer values depending on your device)
3. Note down the following length ratios for each of the values of AD (in (i), (ii) and (iii))
(a)
(b)
(c)
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))
Repeat step 1 to 4 for Case 2 as shown below
Case 2
1. Show the line through D, perpendicular to AB, which intersects AB at E, and the triangle AED formed.
2. Drag the point D so that AD is (i) 5 (ii) 10 or (iii) 15 units long.
3. Note down the following length ratios for each of the values of AD (in (i), (ii) and (iii))
(a)
(b)
(c)
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))
Observations
5. What do you observe about the length ratios, for both cases (using different values of AD) in
(i) step 3 (a)
(ii) step 3 (b)
(iii) step 3 (c)
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.
Show the "positional relationship between angle and side of triangle" by clicking on the checkbox.
Which of the ratios in step 3 (a), (b) or (c) correspond to the sine(sin), cosine(cos) or tangent(tan) of angle ?
Use a calculator to obtain the value of
sin 40o , cos 40o and tan 40o .
Do they correspond closely to the values in 3(a), (b) and (c) for all cases?
Repeat steps 1 to 5 by changing the value of angle from 40o to 70o .
Then repeat again for = 20o
7. Do the sine, cosine and tangents of this angle change when the angle changes?
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)
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?