Mathematics Lab Activity-8 Class XI

Mathematics Laboratory Activities on Trigonometric Functions for class XI students Non-Medical. These activities are strictly according to the CBSE syllabus

Chapter - 3 | trigonometric functions

Activity - 8

Objective

To find the values of sine and cosine functions in second, third and fourth quadrant.

Material Required

Cardboard, white chart paper, ruler, coloured pens, adhesive, steel wires and needle.

Theory

1.  Conversion of degree measure into radian measure. 

Angle in degree	0o	30o	45o	60o	90o	180o	270o	360o
Angle in radian	0o	π/6	π/4	π/3	π/2	π	3π/2	2π

2. Values of trigonometric functions :

Angle in radian	0o	π/6	π/4	π/3	π/2	π	3π/2	2π
Sinθ	0	1/2	1/√2	√3/2	1	0	-1	0
Cosθ	1	√3/2	1/√2	1/2	0	-1	0	1
Tanθ	0	1/√3	1	√3	Not defined	0	Not defined	0

Values of cotθ, secθ and cosec θ are the reciprocal of the above given values.

3.  Sign of trigonometric functions in different quadrants:

Quadrants	I	II	III	IV
Sinθ	+	+	-	-
cosθ	+	-	-	+
tanθ	+	-	+	-
cotθ	+	-	+	-
secθ	+	-	-	+
cosecθ	+	+	-	-

Procedure

1. Paste a white chart paper on a card board of a convenient size.

2. Draw a circle of unit radius, with O as centre on the chart paper.

3. Through the centre of the circle, draw two lines X’OX and YOY’ perpendicular to each other , representing x-axis and y-axis as shown in figure 8.1

Figure 8.1

4. Mark the points A, B, C and D, where the circle cuts the x-axis and y-axis, respectively.

5. Through O, draw angles ∠P₁ OX = (π/6) radian or 30^o, ∠P₂ OX = (π/4) radian or 45^o, ∠P₃ OX = (π/3) radian or 60^o

6.       Coordinates of point A are (1, 0)

           Coordinates of point B are (0, 1)

Coordinates of point C are (-1, 0)

Coordinates of point D are (0, -1)

Coordinates of point P₁ are  (cos π/6, sin π/6)  i.e.  (√3/2 , 1/2)

           Coordinates of point P₂ are (cos π/4, sin π/4)  i.e.  (1/√2 , 1/√2)

Coordinates of point P₃ are  (cos π/3, sin π/3)  i.e.  (1/2, √3/2)

7. To find the value of sine or cosine at  2π/3 or 120^o  in the second quadrant, draw an ∠P₄ OX = 2π/3 or 120^o  as shown in fig. 8.2.   

Fig.8.2 

8. OP₄ is the mirror image of OP₃ with respect to the y-axis. i.e. the point p₄ is the mirror image of p₃. So the coordinates of point P₄ are  = 

⇒ cos(2π/3) = Cos 120^o = -1/2

⇒ sin(2π/3) = sin120^o = √3/2

9. To find the value of sine or cosine at 4π/3 or 240^o  in the third quadrant, draw an ∠P₅OX = 4π/3 or 240^o   as shown in fig. 8.3.

Fig. 8.3

10. OP₅ is the mirror image of OP₄ with respect to the x-axis. i.e. the point p₅ is the mirror image of p₄. So the coordinates of point P₅ are  = 

⇒ cos θ = cos 240^o = -1/2

⇒ sin θ = sin240^o = -√3/2

11. To find the value of sine or cosine at 5π/3 or 300^o  in the fourth quadrant, draw an ∠P₆OX = 5π/3 or 300^o as shown in fig. 8.4.

Fig. 8.4

12. OP₆ is the mirror image of OP₂ with respect to the x-axis. i.e. the point p₆ is the mirror image of p₂. So the coordinates of point P₆ are  =  (1/2 , - √3/2)  or (cos5π/3, sin5π/3)

⇒ cos 5π/3 = cos 300^o = 1/2

⇒  sin5π/3 = sin300^o = - √3/2

13. Here we find the value of sine and cosine function in all the four quadrant at 60^o , 120^o , 240^o , 300^o . We can also extend this knowlege to find the value of all trigonometric functions with other standard angles.

14. Proceeding in the same manner, we can find the sine and cosine of any other angle using their given values in the first quadrant.

observations

Values sine and cosine functions in II quadrant are

Cos (2π/3) = Cos 120^o = -1/2

sin(2π/3) = sin120^o = √3/2

Values sine and cosine functions in III quadrant are

cos θ = cos 240^o = -1/2

sin θ = sin240^o = -√3/2

Values sine and cosine functions in IV quadrant are

cos 5π/3 = cos 300^o = 1/2

 sin5π/3 = sin300^o = - √3/2

Result

Sine function is negative in III and IV quadrants.

Cosine function is negative in II and III quadrants.

Applications

This activity can be used to obtain the values of other trigonometric functions in different quadrants.

VIVA – VOICE

Q. 1. What do you mean by an angle ?

Ans. An angle is the measure of rotation of a given ray about its initial point.

Q. 2. When is the angle said to be positive.

Ans. If the direction of rotation of the given angle is anticlockwise , then the angle is said to be positive.

Q. 3. When is the angle said to be negative?

Ans. If the direction of rotation of the given angle is clock-wise , then the angle is said to be negative.

Q. 4. Define radian ?

Ans. One radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

Q. 5. How is radian related to degree measure of an angle ?

Ans.   1 Radian = 57^o16’21’’ = 57.27^o

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