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.

By definition, \(\cot\theta=\dfrac{1}{\tan\theta}\). Therefore, the reciprocal of \(\tan\theta\) is \(\cot\theta\).

(\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.

The correct answer is option D: \\(1\\). Start with the fundamental identity \\(1+\tan^2\theta=\sec^2\theta\\). Rearranging gives \\(\sec^2\theta-\tan^2\theta=1\\). Therefore option D is correct. Option A, \\(\tan\theta\\), is not a constant and is not what the identity gives. Option B, \\(\sec\theta\\), is also just one trigonometric function, whereas the squared difference simplifies to a constant. Option C, 0, would incorrectly assume the two squared terms are equal; in fact they differ by 1 wherever both are defined. Option D matches the identity directly. The expression is defined wherever \\(\cos\theta\\neq0\\), because secant and tangent must exist. Memory cue: \\(\sec^2\theta-\tan^2\theta=1\\), just as \\(1+\tan^2\theta=\sec^2\theta\\).

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 governing concept is the fundamental period: the least positive number T for which f(θ + T) = f(θ) for every θ. For the sine function, one complete revolution on the unit circle has angular measure 2π, and returning to the same point gives sin(θ + 2π) = sin θ. Thus the fundamental period is 2π, so option B is correct. Although 4π also reproduces the sine value, it is not the smallest positive period. A shift of π gives sin(θ + π) = −sin θ, so π is not a period of sine. A shift of π/2 changes the function to cosine in general, so it also cannot be the fundamental period. These distinctions establish both repetition and minimality.

A function’s fundamental period is the least positive interval after which all its values repeat. Cosine is the horizontal coordinate of a point moving around the unit circle, and a complete revolution has measure 2π. Consequently, cos(θ + 2π) = cos θ for every θ. A shift of π gives cos(θ + π) = −cos θ, so π is not the period of the original cosine values; π/2 also does not reproduce every value, while 4π repeats the function only after two complete revolutions and is therefore not fundamental. Thus the correct answer is option D, 2π. Sine and cosine have the same fundamental period even though their graphs are phase-shifted.

A period is the smallest positive angle by which a function repeats its values. For tangent, the identity \\(\tan(\theta+\pi)=\tan\theta\\) holds because both sine and cosine change sign when \\(\pi\\) is added, so their quotient remains unchanged. The tangent graph therefore repeats after every \\(\pi\\) radians.

The word “fundamental” means the smallest positive period. Although \\(2\pi\\) also returns the same value, it is not the fundamental period because it is twice as large as necessary. The value \\(\pi/2\\) is not a period; it shifts a tangent graph toward a point where the function is undefined. Hence option C, \\(\pi\\), is correct. Sine has fundamental period \\(2\pi\\), which should not be confused with tangent.

The fundamental period is the smallest positive T that leaves a function unchanged wherever it is defined. Since cot θ = cos θ/sin θ, adding π changes both numerator and denominator signs: cot(θ + π) = cos(θ + π)/sin(θ + π) = (−cos θ)/(−sin θ) = cot θ. Thus π is a period. It is fundamental because no smaller positive shift works for every angle; in particular, a shift of π/2 changes cotangent into a related but generally different expression. A shift of 2π or 3π also repeats the values, but those are larger than the least positive period. Therefore option A, π, is correct. The same period applies to tan θ because both functions repeat after a half-turn.

\(sin \theta\) 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.

\(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\). For values of \(\theta\) where \(\cos\theta\ne0\), \(\tan\theta\) can take every real value. Hence its range is \(( -\infty,\infty )\). The interval \([-1,1]\) is the range of \(\sin\theta\) and \(\cos\theta\), not \(\tan\theta\). Exam tip: the range of \(\tan\theta\) is all real numbers, although it is undefined at \(\theta=\frac{\pi}{2}+n\pi\).

Since (\sec \theta=\frac{1}{\cos \theta}), its values do not lie inside ([-1,1]). Its range is ((-\infty,-1]\cup[1,\infty)).

\(\cosec\theta=\frac{1}{\sin\theta}\). Since \(-1\leq\sin\theta\leq1\) and \(\sin\theta\neq0\), the value of \(\cosec\theta\) is either less than or equal to \(-1\), or greater than or equal to 1. At \(\sin\theta=-1\) and \(1\), we get \(\cosec\theta=-1\) and \(1\), so both endpoints are included. Option C incorrectly excludes these endpoints. Exam tip: for reciprocal trigonometric functions, check values for which the denominator becomes zero.

(\tan \theta=\frac{\sin \theta}{\cos \theta}), so it is undefined when (\cos \theta=0). The denominator must not be zero.

(\sec \theta=\frac{1}{\cos \theta}), so it is undefined when (\cos \theta=0). Check the denominator in reciprocal functions.