Mathematics 11 (Punjab Board)
Chapter 11: Trigonometric Functions and their Graphs — Theory
11.1 Trigonometric Functions of Real Numbers
Take the unit circle with centre at the origin. For any real number \(\theta\), let \(P(x, y)\) be the point on the unit circle reached by measuring an angle \(\theta\) in standard position from the positive \(x\)-axis. Then we define \(\cos \theta = x\) and \(\sin \theta = y\). Roman Urdu mein: yeh definition angle ko kisi point ki coordinates se jorti hai, isi liye \(\sin \theta\) aur \(\cos \theta\) har real \(\theta\) ke liye defined hain.
Key point: The other four functions are defined from these two: \(\tan \theta = \dfrac{\sin \theta}{\cos \theta}\), \(\cot \theta = \dfrac{\cos \theta}{\sin \theta}\), \(\sec \theta = \dfrac{1}{\cos \theta}\), \(\csc \theta = \dfrac{1}{\sin \theta}\), each wherever its denominator is non-zero.
11.2 Domain and Range
Since every point \(P\) on the unit circle has coordinates between \(-1\) and \(1\), sine and cosine accept every real number but only produce values in \([-1, 1]\).
\(\tan \theta\) is undefined where \(\cos \theta = 0\), i.e. at \(\theta = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \dots\); its range is all of \(\mathbb{R}\). \(\sec \theta\) shares the domain of \(\tan \theta\), while \(\csc \theta\) and \(\cot \theta\) are undefined where \(\sin \theta = 0\), i.e. at \(\theta = 0, \pm\pi, \pm 2\pi, \dots\)
Key point: For exams, remember the restricted values: \(-1 \le \sin \theta \le 1\) and \(-1 \le \cos \theta \le 1\) — most max/min questions of this chapter start exactly here.
11.3 Even and Odd Functions
A function \(f\) is even if \(f(-x) = f(x)\) for every \(x\) in its domain; its graph is symmetric about the \(y\)-axis. It is odd if \(f(-x) = -f(x)\); its graph is symmetric about the origin. Agar dono mein se koi condition poori na ho, to function neither even nor odd hota hai.
Quick rules: an even power of an odd function is even (for example \(\sin^2 x\)); a sum of an even and an odd function is generally neither.
Key point: Always test with \(f(-x)\) and compare with both \(f(x)\) and \(-f(x)\) — checking only one side is the most common error.
11.4 Periodic Functions
A function is periodic if its values repeat after a fixed interval; the smallest such positive interval is the period. Sine and cosine repeat every \(2\pi\) radians, while tangent and cotangent repeat every \(\pi\) radians.
Vertical shifts (adding a constant \(d\)) and phase shifts (replacing \(x\) by \(x + c\)) do not change the period — sirf \(x\) ke saath multiply hone wala number \(b\) period ko affect karta hai.
Key point: In \(y = a\sin(bx + c) + d\), read the period from \(b\) alone; \(a\), \(c\) and \(d\) are irrelevant to it.
11.5 Graphs of the Sine, Cosine and Tangent Functions
The sine curve starts at the origin, rises to \(1\) at \(\frac{\pi}{2}\), returns to \(0\) at \(\pi\), falls to \(-1\) at \(\frac{3\pi}{2}\) and completes one full wave at \(2\pi\). The cosine curve is the same wave shifted left by \(\frac{\pi}{2}\): it starts at its maximum \((0, 1)\). Tangent has period \(\pi\), passes through the origin, and has vertical asymptotes where cosine is zero.
Key point: To sketch any of these on paper, first write down the period and the quarter-points \(0, \frac{T}{4}, \frac{T}{2}, \frac{3T}{4}, T\) — the graph is then fixed.
11.6 Amplitude, Period and Phase Shift of \(y = a\sin(bx + c) + d\)
For a sinusoidal function written in the standard form \(y = a\sin(bx + c) + d\) (the same applies with cosine):
The graph oscillates between \(d - |a|\) and \(d + |a|\); \(d\) is the average or midline of the oscillation. Frequency (cycles per unit \(x\)) equals \(\dfrac{1}{T} = \dfrac{|b|}{2\pi}\). Negative \(a\) simply flips the wave upside down — start at the minimum instead of the maximum.
Key point: Amplitude is always a positive number \(|a|\); students often forget the modulus sign when \(a\) is negative.
11.7 Maximum and Minimum Values
Because \(-1 \le \sin u \le 1\) and \(-1 \le \cos u \le 1\) for every \(u\), the function \(f(x) = a + b\sin u\) attains its largest value when \(\sin u\) has the same sign as \(b\) and its smallest value in the opposite case. Yahi principle har max/min sawal ki buniyad hai.
For a reciprocal such as \(\dfrac{1}{a + b\sin u}\) (with \(a + b\sin u > 0\) throughout), the maximum occurs where the denominator is smallest and the minimum where the denominator is largest — reciprocal max aur min ko ulat deta hai.
Key point: Before taking a reciprocal, confirm the denominator never reaches zero or changes sign on the interval in question.
11.8 Applications: Modelling Periodic Phenomena
Anything that repeats in time — temperature over a day, a Ferris wheel, tides, a swing — can be modelled by \(y = a\sin(bx + c) + d\). The modelling steps are: (1) amplitude \(|a|\) = half the total swing (radius for a wheel); (2) midline \(d\) = average value (centre height of the wheel); (3) \(b = \dfrac{2\pi}{T}\) from the observed period \(T\); (4) choose sine or cosine and the sign so that the model starts at the correct point (use \(-\cos\) if it starts at the lowest point).
Angle-of-elevation/depression problems are solved by drawing the right triangles, writing \(\tan(\text{angle}) = \dfrac{\text{height}}{\text{base}}\) for each, and solving the resulting equations — this is the standard "two ships / two observers" pattern of Exercise 11.3.
Key point: In elevation/depression problems always label the unknown height and the unknown horizontal distances first; the two tangent equations then solve themselves.