UK A-Level: Trigonometric Functions and Graphs

Understanding trigonometric functions and their graphs is not just an abstract exercise; it is the language of oscillating systems, from sound waves and alternating current to the cyclical patterns in nature and data. Mastering these graphs, their transformations, and their inverses provides you with a powerful toolkit for modelling periodic behaviour and solving a wide range of mathematical problems, which is a cornerstone of the A-Level Mathematics curriculum.

The Core Trio: Sine, Cosine, and Tangent

The journey begins with the three primary trigonometric functions, defined initially using the unit circle. The sine function, $y=\sin x$, maps an angle to the y-coordinate of a point on the unit circle. Its graph is a smooth, continuous wave that starts at the origin (0,0), oscillates between -1 and 1, and has a period of $360^{\circ }$ or $2\pi$ radians. The cosine function, $y=\cos x$, maps to the x-coordinate. Its graph is identical in shape to the sine wave but is shifted horizontally; it starts at a maximum value of 1 when $x=0$. The cosine graph is often described as a "shifted sine wave," specifically $\cos x=\sin (x+90^{\circ })$.

In contrast, the tangent function, $y=\tan x$, defined as $\tan x=\frac{\sin x}{\cos x}$, has a fundamentally different graph. It is discontinuous, with asymptotes (vertical lines the graph approaches but never touches) wherever $\cos x=0$ (i.e., at $x=\pm 90^{\circ },\pm 270^{\circ }...$). Its period is $180^{\circ }$ or $\pi$ radians, and it ranges from $-\infty$ to $+\infty$. A useful analogy is to think of sine and cosine as describing the smooth, circular motion of a Ferris wheel car, while tangent relates to the slope of the line from the wheel's centre to that car.

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

Each primary function has a reciprocal. These are defined as:

• Secant: $\sec x=\frac{1}{\cos x}$
• Cosecant: $\mathrm{cosec}x=\frac{1}{\sin x}$
• Cotangent: $\cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}$

Their graphs are best understood by first sketching the corresponding primary function. Where $\cos x=\pm 1$, $\sec x$ will also be $\pm 1$. However, as $\cos x$ approaches zero, $\sec x$ races off towards positive or negative infinity, creating vertical asymptotes at the same locations as the zeros of the cosine graph. Consequently, the graph of $y=\sec x$ consists of a series of U-shaped and inverted U-shaped curves, each located between a pair of asymptotes, with a range of $y\le -1$ or $y\ge 1$. The graph of $y=\mathrm{cosec}x$ behaves similarly but is based on the sine graph. The graph of $y=\cot x$, the reciprocal of tangent, also has asymptotes (where $\sin x=0$) and a period of $\pi$.

Inverse Trigonometric Functions and Their Restricted Domains

The standard trigonometric functions are not one-to-one—they fail the "horizontal line test" because their values repeat every period. Therefore, to define their inverse functions, we must restrict their domains to an interval where they are one-to-one. These restricted functions are denoted $\arcsin$, $\arccos$, and $\arctan$ (or ${\sin }^{-1},{\cos }^{-1},{\tan }^{-1}$).

The standard domain restrictions are:

• For $y=\arcsin x$: The domain of $\sin x$ is restricted to $[-\frac{\pi }{2},\frac{\pi }{2}]$. This yields a range for $\arcsin x$ of $[-\frac{\pi }{2},\frac{\pi }{2}]$.
• For $y=\arccos x$: The domain of $\cos x$ is restricted to $[0,\pi ]$. This yields a range for $\arccos x$ of $[0,\pi ]$.
• For $y=\arctan x$: The domain of $\tan x$ is restricted to $(-\frac{\pi }{2},\frac{\pi }{2})$. This yields a range for $\arctan x$ of $(-\frac{\pi }{2},\frac{\pi }{2})$.

Their graphs are the reflections of the restricted part of the original trig graph in the line $y=x$. Crucially, for a given input like $x=0.5$, $\arcsin (0.5)$ will return the principal value—the single answer within the restricted range, which in this case is $\frac{\pi }{6}$ or $30^{\circ }$.

Transformations of Trigonometric Graphs

You can apply the standard function transformations to trigonometric graphs. For a function $y=a\sin (b(x+c))+d$:

• $|a|$ is the amplitude (vertical stretch). It controls the peak deviation from the centre line. If $a$ is negative, the graph is also reflected in the x-axis.
• The period is calculated as $\frac{360^{\circ }}{|b|}$ or $\frac{2\pi }{|b|}$ radians (horizontal stretch/squash). The factor $b$ affects the frequency of the wave.
• $c$ is the phase shift (horizontal translation). The graph shifts left if $c$ is positive, and right if $c$ is negative.
• $d$ is the vertical translation. This moves the central axis of the wave up or down.

For example, $y=3\cos (2x-90^{\circ })+1$ has an amplitude of 3, a period of $180^{\circ }$, a phase shift of $45^{\circ }$ to the right (rewritten as $y=3\cos (2(x-45^{\circ }))+1$), and is shifted up by 1 unit. These rules apply identically to the graphs of cosine, tangent, and their reciprocals.

Solving Equations Using Principal Values

When asked to solve an equation like $\sin \theta =0.4$, your calculator will give you the principal value—the answer within the restricted domain of the inverse function (for sine, between $-90^{\circ }$ and $90^{\circ }$). This is your starting point. To find all solutions within a given range (e.g., $0^{\circ }\le \theta <360^{\circ }$), you must use the symmetry of the trig graphs.

The systematic approach is:

1. Use your calculator (in degree/radian mode as required) to find the principal value, $\alpha$.
2. Use the CAST diagram or sketch the relevant trig graph to identify other angles in the required range that have the same sine, cosine, or tangent value.

• For $\sin \theta =k$, solutions are found in the first and second quadrants: $\theta =\alpha ,180^{\circ }-\alpha$.
• For $\cos \theta =k$, solutions are in the first and fourth quadrants: $\theta =\alpha ,360^{\circ }-\alpha$.
• For $\tan \theta =k$, solutions are in the first and third quadrants: $\theta =\alpha ,180^{\circ }+\alpha$.

1. Add or subtract multiples of the function's period ($360^{\circ }$ for sin/cos, $180^{\circ }$ for tan) to these fundamental solutions to generate further solutions if a wider range is specified.

For example, to solve $2\cos (3x)=1$ for $0\le x<180^{\circ }$, you would first rearrange to $\cos (3x)=0.5$, find the principal value for the angle $3x$ (which is $60^{\circ }$), then find all solutions for $3x$ in the range $0\le 3x<540^{\circ }$, and finally divide each by 3 to solve for $x$.

Common Pitfalls

1. Misapplying the period to all solutions: Remember that while sine and cosine have a period of $360^{\circ }$, tangent has a period of $180^{\circ }$. A common error is to use $360^{\circ }$ when finding additional solutions to a $\tan$ equation, which will cause you to miss half the required answers or include incorrect ones.
2. Confusing the properties of reciprocal functions: It is incorrect to state that $\sec x$ is always greater than 1. It is $\ge 1$ or $\le -1$. Similarly, thinking the amplitude of $y=\sec x$ is 1 is a mistake; the concept of amplitude doesn't strictly apply to these unbounded functions.
3. Forgetting the restricted domain for inverse functions: The equation ${\sin }^{-1}(\sin 150^{\circ })=150^{\circ }$ is false. You must apply the inverse function's range: $\sin 150^{\circ }=0.5$, and ${\sin }^{-1}(0.5)=30^{\circ }$. The inverse function "outputs" the principal value, not the original input angle if it lies outside the restricted range.
4. Incorrect phase shift with a coefficient of $x$: When a transformation is written as $\sin (bx+c)$, the phase shift is not $c$ units. You must factor out the coefficient to see the transformation applied to $x$ directly: $\sin (b(x+\frac{c}{b}))$. The phase shift is $\frac{c}{b}$.

Summary

• The graphs of $y=\sin x$ and $y=\cos x$ are periodic waves with amplitude 1 and period $2\pi$ radians, while $y=\tan x$ has asymptotes and a period of $\pi$ radians.
• The reciprocal functions ($\sec$, $\mathrm{cosec}$, $\cot$) have graphs defined by the behaviour of their corresponding primary functions, featuring vertical asymptotes where the primary function equals zero.
• Inverse trigonometric functions ($\arcsin$, $\arccos$, $\arctan$) require a restricted domain on the original function to be one-to-one, and they return only the principal value within a specific range.
• Transformations of the form $y=a\sin (b(x+c))+d$ affect the amplitude, period, horizontal shift, and vertical shift of the graph, respectively.
• Solving trigonometric equations involves finding the principal value first, then using graph symmetry (via CAST or sketches) and the correct period to find all solutions within a specified interval.