Mathematics Lab Activity-05 Class XII

Mathematics Laboratory Activities on Inverse Trigonometric Functions for class XI Non-Medical students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 02

inverse trigonometric functions

Activity - 05

Objective

To draw the graph of sin^-1x, using the graph of sinx and demonstrate the concept of mirror reflection (about the line y = x)

Material Required

Cardboard, white chart paper, ruler, colored pens, adhesive, pencil, eraser, cutter and thin wires.

Procedure

1. Take a cardboard of suitable dimensions, say, 30cm x 30cm.

2. On the cardboard, paste a white chart paper of size 25cm x 25cm (say).

3. On the paper, draw two lines, perpendicular to each other and name them as X’OX and YOY’ as rectangular axis as shown in figure 5.1

4. Graduate the axis approximately as shown in Fig. 5.1 by taking unit on X-axis =1.25 times the unit of y-axis.

Figure 5.1

5. Mark approximately the points $\left(\frac{\pi}{6},\:sin\frac{\pi}{6}\right)$ , $\left(\frac{\pi}{4},\:sin\frac{\pi}{4}\right)$ , ………., $\left(\frac{\pi}{2},\:sin\frac{\pi}{2}\right)$ in the coordinate plane and at each point N₁, N₂, N₃, N₄ fix a nail.

6. 1. Repeat the above process on the other side of the x-axis, marking the points $\left(\frac{-\pi}{6},\:sin\frac{-\pi}{6}\right)$ , $\left(\frac{-\pi}{4},\:sin\frac{-\pi}{4}\right)$ , ………., $\left(\frac{-\pi}{2},\:sin\frac{-\pi}{2}\right)$ approximately and fix nails on these points as N’₁, N’₂, N’₃, N’₄ Also fix a nail at O.

7. Join the nails with the help of a tight wire on both sides of x-axis to get the graph of sinx from $\frac{-\pi}{2}$ to $\frac{\pi}{2}$ .

8. Draw the graph of the line y = x by plotting the points (1,1), (2, 2), (3, 3), (4, 4), ……….

9. From the nails N₁, N₂, N₃, N₄ draw perpendicular on the line y = x and produce these lines such that length of perpendicular on both sides of the line y = x are equal. At these points fix nails I₁, I₂, I₃, I₄.

10. Repeat the above activity on the other side of X-axis and fix nails at I’₁, I’₂, I’₃, I’₄.

11. Join the nails on both sides of the line y = x by a tight wire that will show the graph of y = sin^-1x.

12. Put a mirror on the line y = x. The image of the graph of sinx in the mirror will represent the graph of is mirror reflection of sinx and vice versa.

Observations

1. The image of point N₁ in the mirror (the line y = x) is I₁.

2. The image of point N₂ in the mirror (the line y = x) is I₂.

3. The image of point N₃ in the mirror (the line y = x) is I₃.

4. The image of point N₄ in the mirror (the line y = x) is I₄.

5. The image of point N’₁ in the mirror (the line y = x) is.I’₁.

6. The image of point N’₂ in the mirror (the line y = x) is.I’₂.

7. The image of point N’₃ in the mirror (the line y = x) is.I’₃.

8. The image of point N’₄ in the mirror (the line y = x) is.I’₄.

Result

The image of the graph of sinx in y = x is the graph of sin^-1x , and the image of the graph of sin^-1x in y = x is the graph of sinx.

Applications

Similar activity can be used to find the graph of cos^-1x and tan^-1x etc.

VIVA – VOICE

Q1. What is the inverse of sin x ?

Ans. Sin^-1x

Q2. Is $sin^{-1}x=\frac{1}{sinx}$ ?

Ans. No, $sin^{-1}x\neq\frac{1}{sinx}$

Q3. Express cos^-1x into sin^-1x ?

Ans. $cos^{-1}x=sin^{-1}\sqrt{1-x^{2}}$

Q4. Express sin^-1x into tan^-1x ?

Ans. $sin^{-1}x=tan^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right)$

Q5. If $sin\left(sin^{-1}x\right)=x,$ then what is the domain of x ?

Ans. $x\:\varepsilon\left[\frac{-\pi}{2},\frac{\pi}{2}\right].$

Q6. If $sin^{-1}\left(sinx\right)=x,$ then what is the domain of x ?

Ans. x∈[-1,1].

Q7. What is the domain of sin^-1x ?

Ans. Domain of sin^-1x is = [-1,1].

Q8. What is the range of sin^-1x ?

Ans. Range of sin^-1x = $\left[\frac{-\pi}{2},\frac{\pi}{2}\right].$

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