Sine, Cosine, Tangent, and Cotangent

9000038902

Level: 
B
Consider the function \(f\colon y = A\cdot \sin (B\cdot x + C)\), with real nonzero parameters \(A\), \(B\) and \(C\). Which of the following operations makes the amplitude of the function five times bigger?
Decrease \(A\) by a factor \(5\).
Increase \(A\) by a factor \(5\).
Increase \(B\) by a factor \(5\).
Decrease \(B\) by a factor \(5\).
Increase \(C\) by a factor \(5\).
Decrease \(C\) by a factor \(5\).

9000038909

Level: 
B
Consider the function \(f\colon y =\sin \left (\frac{x} {2} + \frac{\pi } {2}\right )\). In the following list identify the function which has the same graph as the graph of the function \(f\).
\(g\colon y =\cos \frac{x} {2} \)
\(k\colon y =\cos \left (\frac{x} {2} + \frac{\pi } {2}\right )\)
\(b\colon y =\cos \left (\frac{x} {2} - \frac{\pi } {2}\right )\)
\(h\colon y =\cos \left (\frac{x} {2} -\pi \right )\)
\(m\colon y =\cos 2x\)

9000038906

Level: 
B
Consider the function \(f\colon y =\mathop{\mathrm{tg}}\nolimits x\). In the following list identify the nonnegative function.
None of the given functions is nonnegative.
\(A\cdot f(x)\), where \(A\in (-\infty ;0)\)
\(A\cdot f(x)\), where \(A\in (0;+\infty )\)
\(f(B\cdot x)\), where \(B\in (0;+\infty )\)
\(f(x + C)\), where \(C\in (-\infty ;0)\)

9000038908

Level: 
B
Consider the function \(f\colon y =\mathop{\mathrm{tg}}\nolimits x\) with domain restricted to the interval \(\mathop{\mathrm{Dom}}(f) = \left ( \frac{\pi }{2}; \frac{3\pi } {2}\right )\). In the following list identify the function with domain \((0;\pi )\).
\(f\left (x + \frac{\pi } {2}\right )\)
\(\left ( \frac{\pi }{2}\right )\cdot f(x)\)
\(f\left (x - \frac{\pi } {2}\right )\)
\(f(x) + \frac{\pi } {2}\)
\(f(x) - \frac{\pi } {2}\)

9000038905

Level: 
B
Identify the transformation which transforms the graph of the function \(g(x) =\sin 3x\) to the graph of the function \(f(x) =\sin (3x + 5)\).
Shift of graph of \(g\) by \(\frac{5} {3}\) of a unit to the left.
Shift of graph of \(g\) by \(5\) units to the right.
Shift of graph of \(g\) by \(5\) units to the left.
Shift of graph of \(g\) by \(3\) units to the right.
Shift of graph of \(g\) by \(3\) units to the left.
Shift of graph of \(g\) by \(\frac{5} {3}\) of a unit to the right.

9000038907

Level: 
B
Consider the function \(f(x) =\mathop{\mathrm{cotg}}\nolimits x\) with domain restricted to the interval \(\mathop{\mathrm{Dom}}(f) = (0;\pi )\). In the following list identify the function with domain \(\left (0; \frac{\pi } {3}\right )\).
\(f(3\cdot x)\)
\(f(x - 3)\)
\(f(x + 3)\)
\(f\left (\frac{x} {3} \right )\)
\(3\cdot f(x)\)

9000033805

Level: 
B
In the following list identify a true statement for the function \(h\colon y =\mathop{\mathrm{cotg}}\nolimits x\), \(x\in \left (-\frac{\pi }{2};0\right )\cup \left (0; \frac{\pi } {2}\right )\).
The function \(h\) is neither increasing nor decreasing.
The function \(h\) is increasing.
The function \(h\) is decreasing.

9000033806

Level: 
B
In the following list identify a true statement for the function \(i\colon y =\mathop{\mathrm{tg}}\nolimits x\), \(x\in \left ( \frac{\pi }{2}; \frac{3\pi } {2}\right )\).
The function \(i\) is increasing.
The function \(i\) is decreasing.
The function \(i\) is neither increasing nor decreasing.