Trigonometric functions and their properties

For the standard angle, \(\sin 0^\circ=0\). On the unit circle, the point corresponding to \(0^\circ\) is \((1,0)\), and sine is the \(y\)-coordinate of that point. Therefore, the correct value is 0. Since \(\sin 90^\circ=1\), 1 is not correct here. Exam tip: Remember the sine values from \(0^\circ\) to \(90^\circ\) in the order \(0,\frac{1}{2},\frac{1}{\sqrt{2}},\frac{\sqrt{3}}{2},1\).

For the standard angle, \(\cos 0^\circ=1\). On the unit circle, the point corresponding to \(0^\circ\) is \((1,0)\), and its x-coordinate represents cosine. Hence, the correct value is 1. Option 0 is the value of \(\sin 0^\circ\), not \(\cos 0^\circ\). Exam tip: Remember that at \(0^\circ\), \(\sin\theta=0\) and \(\cos\theta=1\).

In a \(45^\circ\)–\(45^\circ\)–\(90^\circ\) right triangle, the two perpendicular sides are equal. Therefore, \(\tan 45^\circ=\frac{\text{perpendicular}}{\text{base}}=1\). The value \(\frac{1}{\sqrt{3}}\) is \(\tan 30^\circ\), so it is not correct here. Exam tip: remember \(\tan 30^\circ,\tan 45^\circ,\tan 60^\circ\) as \(\frac{1}{\sqrt{3}},1,\sqrt{3}\), respectively.

The reciprocal of \(\sin \theta\) is \(\dfrac{1}{\sin \theta}\). By definition, \(\cosec \theta=\dfrac{1}{\sin \theta}\), so \(\cosec \theta\) is the correct option. \(\sec \theta\) is the reciprocal of \(\cos \theta\), whereas \(\cot \theta\) is the reciprocal of \(\tan \theta\). Exam tip: Remember the reciprocal pairs: sin–cosec, cos–sec, and tan–cot.

(\sec \theta=\frac{1}{\cos \theta}). Do not confuse (\sec \theta) with (\cosec \theta) in exams.

(\cot \theta=\frac{1}{\tan \theta}). Check reciprocal functions by writing them as formulas.

(\tan \theta=\frac{\sin \theta}{\cos \theta}). Recognizing quotient relations is useful in easy questions.

(\cot \theta=\frac{\cos \theta}{\sin \theta}). The relations of (\tan \theta) and (\cot \theta) are reciprocal.

(\sin^2 \theta+\cos^2 \theta=1) is a basic identity. Use it as a base formula throughout the chapter.

From (\sec^2 \theta=1+\tan^2 \theta), the value is (1). Rearrange the formula to get the answer.

Using the trigonometric identity \(cosec^2\theta=1+\cot^2\theta\), we obtain \(cosec^2\theta-\cot^2\theta=(1+\cot^2\theta)-\cot^2\theta=1\). Hence, option A is correct. Option B is incorrect because the constant term 1 remains after the \(\cot^2\theta\) terms cancel. For exams, remember the direct identity \(cosec^2\theta-\cot^2\theta=1\).

The fundamental period of (\sin \theta) is (2\pi). In period questions, identify the function first.

The fundamental period of (\cos \theta) is (2\pi). (\sin \theta) and (\cos \theta) have the same period.

The fundamental period of (\tan \theta) is (\pi). The period of (\tan \theta) differs from that of (\sin \theta).

The fundamental period of (\cot \theta) is (\pi). (\tan \theta) and (\cot \theta) have the same period.

\(sin \theta\) is an odd function, so \(sin(-\theta)=-\sin \theta\). For negative angles, identify even-odd properties.

\(cos \theta\) is an even function, so \(cos(-\theta)=\cos \theta\). In an even function, the negative sign does not change the value.

(\sec \theta) is an even function, so (\sec(-\theta)=\sec \theta). Both (\cos \theta) and (\sec \theta) are even.

(\cosec \theta) is an odd function, so (\cosec(-\theta)=-\cosec \theta). It behaves like the reciprocal of (\sin \theta).

The value of (\sin \theta) always lies in ([-1,1]). For range questions, check the maximum and minimum values.