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 $36 0^{\circ}$ or $2 π$ 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 $cos x = sin (x + 9 0^{\circ})$ .

In contrast, the tangent function, $y = tan x$ , defined as $tan x = \frac{s i n x}{c o s x}$ , 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 $x = \pm 9 0^{\circ}, \pm 27 0^{\circ} ...$ ). Its period is $18 0^{\circ}$ or $π$ 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: $sec x = \frac{1}{c o s x}$
Cosecant: $cosec x = \frac{1}{s i n x}$
Cotangent: $cot x = \frac{1}{t a n x} = \frac{c o s x}{s i n x}$

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 $y \leq - 1$ or $y \geq 1$ . The graph of $y = cosec x$ 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 $π$ .

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

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 $\frac{π}{6}$ or $3 0^{\circ}$ .

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 $\frac{36 0 ^{\circ}}{∣ b ∣}$ or $\frac{2 π}{∣ b ∣}$ 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, $y = 3 cos (2 x - 9 0^{\circ}) + 1$ has an amplitude of 3, a period of $18 0^{\circ}$ , a phase shift of $4 5^{\circ}$ to the right (rewritten as $y = 3 cos (2 (x - 4 5^{\circ})) + 1$ ), 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 $sin θ = 0.4$ , your calculator will give you the principal value—the answer within the restricted domain of the inverse function (for sine, between $- 9 0^{\circ}$ and $9 0^{\circ}$ ). This is your starting point. To find all solutions within a given range (e.g., $0^{\circ} \leq θ < 36 0^{\circ}$ ), you must use the symmetry of the trig graphs.

The systematic approach is:

Use your calculator (in degree/radian mode as required) to find the principal value, $α$ .
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 $sin θ = k$ , solutions are found in the first and second quadrants: $θ = α, 18 0^{\circ} - α$ .
For $cos θ = k$ , solutions are in the first and fourth quadrants: $θ = α, 36 0^{\circ} - α$ .
For $tan θ = k$ , solutions are in the first and third quadrants: $θ = α, 18 0^{\circ} + α$ .

Add or subtract multiples of the function's period ( $36 0^{\circ}$ for sin/cos, $18 0^{\circ}$ for tan) to these fundamental solutions to generate further solutions if a wider range is specified.

For example, to solve $2 cos (3 x) = 1$ for $0 \leq x < 18 0^{\circ}$ , you would first rearrange to $cos (3 x) = 0.5$ , find the principal value for the angle $3 x$ (which is $6 0^{\circ}$ ), then find all solutions for $3 x$ in the range $0 \leq 3 x < 54 0^{\circ}$ , and finally divide each by 3 to solve for $x$ .

Common Pitfalls

Misapplying the period to all solutions: Remember that while sine and cosine have a period of $36 0^{\circ}$ , tangent has a period of $18 0^{\circ}$ . A common error is to use $36 0^{\circ}$ when finding additional solutions to a $tan$ equation, which will cause you to miss half the required answers or include incorrect ones.
Confusing the properties of reciprocal functions: It is incorrect to state that $sec x$ is always greater than 1. It is $\geq 1$ or $\leq - 1$ . 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.
Forgetting the restricted domain for inverse functions: The equation $sin^{- 1} (sin 15 0^{\circ}) = 15 0^{\circ}$ is false. You must apply the inverse function's range: $sin 15 0^{\circ} = 0.5$ , and $sin^{- 1} (0.5) = 3 0^{\circ}$ . The inverse function "outputs" the principal value, not the original input angle if it lies outside the restricted range.
Incorrect phase shift with a coefficient of $x$ : When a transformation is written as $sin (b x + c)$ , 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 $2 π$ radians, while $y = tan x$ has asymptotes and a period of $π$ radians.
The reciprocal functions ( $sec$ , $cosec$ , $cot$ ) 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.

UK A-Level: Trigonometric Functions and Graphs

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: $\cosec x = \frac{1}{\sin x}$
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 $y \leq -1$ or $y \geq 1$ . The graph of $y = \cosec x$ 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., $0^\circ \leq \theta < 360^\circ$ ), you must use the symmetry of the trig graphs.

The systematic approach is:

Use your calculator (in degree/radian mode as required) to find the principal value, $\alpha$ .
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: $\theta = \alpha, 180^\circ + \alpha$ .

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 $0 \leq x < 180^\circ$ , 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 $0 \leq 3x < 540^\circ$ , and finally divide each by 3 to solve for $x$ .

Common Pitfalls

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.
Confusing the properties of reciprocal functions: It is incorrect to state that $\sec x$ is always greater than 1. It is $\geq 1$ or $\leq -1$ . 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.
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.
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 ( $\sec$ , $\cosec$ , $\cot$ ) 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.

UK A-Level: Trigonometric Functions and Graphs

UK A-Level: Trigonometric Functions and Graphs

The Core Trio: Sine, Cosine, and Tangent

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

Inverse Trigonometric Functions and Their Restricted Domains

Transformations of Trigonometric Graphs

Solving Equations Using Principal Values

Common Pitfalls

Summary

UK A-Level: Trigonometric Functions and Graphs

The Core Trio: Sine, Cosine, and Tangent

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

Inverse Trigonometric Functions and Their Restricted Domains

Transformations of Trigonometric Graphs

Solving Equations Using Principal Values

Common Pitfalls

Summary

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UK A-Level: Trigonometric Functions and Graphs

The Core Trio: Sine, Cosine, and Tangent

Reciprocal Trigonometric Functions: Secant, Cosecant, and Cotangent

Inverse Trigonometric Functions and Their Restricted Domains

Transformations of Trigonometric Graphs

Solving Equations Using Principal Values

Common Pitfalls

Summary