Aim:
\[1.\ \text{To draw the graphs of A sin (Bx + C) and A cos (Bx + C) for some fixed finite}\ \hspace{10cm}\\ \text{ real values of A, B and C.}\hspace{10cm}\]
\[2.\ \text{To find their domain, range, maximum value, minimum value, amplitude,}\ \hspace{10cm}\\ \text{ period and phase shift.}\hspace{10cm}\]
Procedure
\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{8cm}\]
\[\color{green}{Step\ 2:}\ \text{To draw the graphs of A sin (Bx + C) and A cos (Bx + C) → change the x – axis →}\hspace{5cm}\\ \hspace{1cm}\ \text{values to multiples of}\ \frac{\pi}{2},\ \text{do the following → right click on Graphics view →}\\ \text{Graphics → x Axis → Distance →}\ \frac{\pi}{2}\]
\[\color{green}{Step\ 3:}\ \text{To draw graphs of A sin (Bx + C) and A cos (Bx + C) →}\ \hspace{5cm}\ \\ \hspace{1cm}\ \text{press the Enter Key→ the pop-up window shows → }\\ \text{Create sliders for A, B,C}\]
\[\color{green}{Step\ 4:}\ \text{To create an input box for sliders A, B and C → link with A, B and C.}\ \hspace{6cm}\]
\[\color{green}{Step\ 5:}\ \text{To create an input box for Function f(x) = → link with f(x)}\ \hspace{7cm}\]
\[\color{green}{Step\ 6:}\ \text{To set the values A = 5, B = 3 and C = 2 by using input box}\ \hspace{7cm}\\ \text{or using the sliders A, B and C.}\]
\[\color{green}{Step\ 7:}\ \text{Domain is the set of values of x for which the curve exists. By using }\ \hspace{7cm}\\ \text{input bar to type Domain = – ∞ < x < ∞ and press Enter Key}\]
\[\color{green}{Step\ 8:}\ \text{Range is the set of values of y for which the curve exists. By using }\ \hspace{7cm}\\ \text{input bar to type Range = – 5 ≤ y ≤ 5 and press Enter Key}\]
\[\color{green}{Step\ 9:}\ \text{To calculate the period of the function by using the input bar to type}\hspace{5cm}\\ \hspace{1cm}\ \text{Period = 2 pi/abs(B). One cycle of f(x) = A sin (Bx + C)}\ \text{is given by the interval}\\ [ 0, \frac{2\pi}{|B|}]\ \text{and press Enter Key}\]
\[\color{green}{Step\ 10:}\ \text{To find the maximum value in the interval}\hspace{10cm}\\ \hspace{1cm}\ 0, \frac{2\pi}{|B|}]\ \text{is by using the input bar to type M = Max{f,0,2 pi/abs(B)}}\\ \text{and press Enter Key}\]
\[\color{green}{Step\ 11:}\ \text{To obtain the function value using the input command}\ \hspace{7cm}\\ \text{Maximum = y(M) and press Enter Key}\]
\[\color{green}{Step\ 12:}\ \text{To find the minimum value in the interval}\hspace{9cm}\\ \hspace{1cm}\ [0, \frac{2\pi}{|B|}]\ \text{is by using the input bar to type}\ \text{m = Min(f,0,2 pi/abs(B)}\]
\[\color{green}{Step\ 13:}\ \text{To obtain the function value using the input command}\ \hspace{7cm}\\ \text{Minimum = y(m) and press Enter Key}\]
\[\color{green}{Step\ 14:}\ \text{To find the Amplitude of the function by using the}\ \hspace{8cm}\\ \hspace{1cm}\ \text{by using the input bar to type Amplitude = (1/2)(Maximum-Minimum)}\\ \text{and press Enter Key}\]
\[\color{green}{Step\ 15:}\ \text{The phase shift is found by using the input bar to type}\ \hspace{7cm}\\ \text{Phase Shift = -C/abs(B) and press Enter Key}\]
\[\color{green}{Step\ 16:}\ \text{To draw the graph of f(x) = 4 cos(2x – 1)}\ \hspace{10cm}\\ \text{by using the input boxes of A, B, C and f(x)}\]
Output:
Output table for y = 5 sin (3x + 2) and y = 4 cos (2x – 1)
| Function | y = 5 sin (3x + 2) | y = 4 cos (2x – 1) |
| Domain | -∞ <= x <= ∞ | -∞ <= x <= ∞ |
| Range | -5 <= y <= 5 | -5 <= y <= 5 |
| Maximum value | (1.95 , 5) | (0.5 , 4) |
| Minimum value | (0.9, – 5) | (2.07 , – 4) |
| Amplitude | 5 | 4 |
| Period | 2.09 | 3.14 |
| Phase Shift | – 2/3 | 1 / 2 |
INVERSE TRIGONOMETRIC FUNCTIONS
Aim:
\[\hspace{1cm}\ \color{green}{1.}\ \text{To draw the graph of sin⁻¹(x) and cos⁻¹(x)}\ \hspace{10cm}\]
\[\hspace{1cm}\ \color{green}{2.}\ \text{To find their domain, range, maximum value and minimum value}\ \hspace{5cm}\]
Procedure:
\[\color{green}{Step\ 1:}\ \text{Open Geogebra classic 5 (by double clicking on the icon)}\ \hspace{8cm}\]
\[\color{green}{Step\ 2:}\ \text{To change the y – axis → right click on Graphics view →}\ \hspace{10cm}\\ \text{Graphics → y Axis → Distance →}\ \frac{\pi}{2}\]
\[\color{green}{Step\ 3:}\ \text{To draw the graph of f(x) = sin⁻¹(x) by using the input bar to type}\ \hspace{7cm}\\ \text{Asin(x) or arcsin(x) and press Enter Key}\]
\[\color{green}{Step\ 4:}\ \text{To set of values of x for which the graph of sin⁻¹(x) exists is called the domain.}\ \hspace{7cm}\\ \text{by using input bar to type Domain = – 1 ≤ x ≤ 1 and press Enter Key}\]
\[\color{green}{Step\ 5:}\ \text{To set of values of y for which the graph of sin⁻¹(x) exists is called the range,}\ \hspace{7cm}\\ \text{by using input bar to type Range = – pi/2 ≤ y ≤ pi/2 and press Enter Key}\]
\[\color{green}{Step\ 6:}\ \text{To find the maximum value of the function sin⁻¹(x)}\hspace{10cm}\\ \hspace{1cm}\ \text{in the interval [-1,1] by using the input bar to type M = Max{f,-1,1}}\\ \text{and press Enter Key}\]
\[\color{green}{Step\ 7:}\ \text{To find the minimum value of the function sin⁻¹(x)}\hspace{10cm}\\ \hspace{1cm}\ \text{in the interval [-1,1] by using the input bar to type m = Min{f,-1,1}}\\ \text{and press Enter Key}\]
\[\color{green}{Step\ 8:}\ \text{To create an input box for Function, link with f(x) = sin⁻¹(x) }\ \hspace{7cm}\]
\[\color{green}{Step\ 9:}\ \text{To change the function f(x) = cos⁻¹(x) }\ \hspace{12cm}\\ \text{by using the input box to type arccos(x) and press Enter Key}\]
Output:
Output for Inverse Trigonometric Function
| Function | f(x) = sin⁻¹(x) | f(x) = cos⁻¹(x) |
| Domain | ||
| Range | ||
| Maximum Value | ||
| Minimum Value |
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