Consider the following statements:
(A) The sine function restricted to \(\left[\displaystyle\frac{\pi}{2}, \displaystyle\frac{3\pi}{2}\right]\) is one-one and onto \([-1, 1]\).
(B) The cosine function restricted to \([-\pi, 0]\) is one-one and onto \([-1, 1]\).
(C) The cosine function restricted to \(\left[-\displaystyle\frac{\pi}{2}, \displaystyle\frac{\pi}{2}\right]\) is one-one.
(D) The tangent function restricted to \(\left(\displaystyle\frac{\pi}{2}, \displaystyle\frac{3\pi}{2}\right)\) is one-one and onto \(\mathrm{R}\).
Which of the above statements are correct?
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A), (B) and (D) only
A restriction is one-one when the function only rises or only falls on the interval (no value repeats), and onto a set when it takes every value of that set.
\( \displaystyle \sin\frac{\pi}{2} = 1, \quad \sin\frac{3\pi}{2} = -1, \quad \cos(-\pi) = -1, \quad \cos 0 = 1 \)
So sine falls from 1 to \(-1\) on \(\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]\), cosine rises from \(-1\) to 1 on \([-\pi, 0]\), and tangent rises through all real values on \(\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)\).
Checking the statements
(A) Sine restricted to \(\left[\frac{\pi}{2}, \frac{3\pi}{2}\right]\) is one-one and onto \([-1, 1]\). — Correct. It falls steadily from 1 to \(-1\); this gives another branch of \(\sin^{-1}\).
(B) Cosine restricted to \([-\pi, 0]\) is one-one and onto \([-1, 1]\). — Correct. It rises steadily from \(-1\) to 1.
(C) Cosine restricted to \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) is one-one. — Incorrect. \(\cos\left(-\frac{\pi}{3}\right) = \cos\frac{\pi}{3} = \frac{1}{2}\): two angles give the same value.
(D) Tangent restricted to \(\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)\) is one-one and onto \(\mathrm{R}\). — Correct. Tangent has period \(\pi\), so this piece is a copy of \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) moved by \(\pi\), rising through every real value.