Inverse trigonometric functions and their graphs

8.1The inverse trigonometric functions

You already know how to go from an angle to a ratio, for example $\cos 60^{\circ }=0.5$. An inverse trigonometric function goes the other way. It starts from the value of a ratio and gives back the angle. If the cosine of an angle is $0.5$, the inverse cosine function gives the angle $60^{\circ }$, that is $\frac{\pi }{3}$ radians.

There is one inverse function for each of the six trigonometric functions. Each has two common names:

Why the domain must be restricted

A function has an inverse only if it is one-to-one: each output comes from exactly one input. The trigonometric functions are periodic, so they repeat their values again and again. For example, $\cos x=0.5$ has infinitely many solutions:

The horizontal line $y=0.5$ cuts the cosine curve many times, so cosine fails the horizontal line test and has no inverse on the whole real line. To fix this we keep only one piece of the curve on which every value from $-1$ to $1$ occurs exactly once. The function on this piece is called a principal function, and its inverse is the inverse trigonometric function. For cosine the piece used is $0\le x\le \pi$, so the only answer to $\cos x=0.5$ there is $x=\frac{\pi }{3}$.

Restricted (principal) domain

The interval to which a trigonometric function is limited so that it becomes one-to-one, for example $[0,\pi ]$ for cosine.

Principal function

The trigonometric function defined only on its restricted domain, written with a capital letter: $\mathrm{Cos}x$, $\mathrm{Sin}x$, $\mathrm{Tan}x$, $\mathrm{Cot}x$, $\mathrm{Sec}x$, $\mathrm{Csc}x$.

Principal value

The value of an inverse trigonometric function: the one angle in the fixed range that has the given ratio. For example, the principal value of ${\mathrm{Cos}}^{-1}(0.5)$ is $\frac{\pi }{3}$.

8.1.1 to 8.1.3Principal cosine function and inverse cosine functionOften examined

Principal cosine function (8.1.1)

We need an interval on which $\cos x=k$ has only one solution for every $k$ in $[-1,1]$. On $[0,\pi ]$ the cosine curve falls steadily from $1$ (at $x=0$) to $-1$ (at $x=\pi$), so it takes every value in $[-1,1]$ exactly once. Other intervals, such as $[\pi ,2\pi ]$, would also work, but by convention we use $[0,\pi ]$.

The cosine function defined on $[0,\pi ]$ is called the principal cosine function. For each $k\in [-1,1]$ the equation $\mathrm{Cos}x=k$ has a unique solution. We write it as $x={\mathrm{Cos}}^{-1}k$, read as '$x$ is the real number in $[0,\pi ]$ whose cosine is $k$'.

Inverse cosine function (8.1.2)

Here $x$ is a real number (a value of cosine) and $y$ is an angle in radians. Three values follow straight from the definition:

Domain and range of the inverse cosine function (8.1.3)

An inverse function swaps inputs and outputs, so the domain and the range of the principal function swap:

The domain of an inverse trigonometric function is a set of real numbers (values of a ratio). Its range is a set of angles, measured in radians or degrees.

Graph by reflection

The graph of an inverse function is the reflection of the original graph in the line $y=x$, because every point $(a,b)$ becomes $(b,a)$. So we draw $y=\mathrm{Cos}x$ on $[0,\pi ]$ and reflect it in $y=x$.

8.1.4 to 8.1.6Principal sine function and inverse sine functionOften examined

Principal sine function (8.1.4)

On $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ the sine curve rises steadily from $-1$ to $1$, so it takes every value in $[-1,1]$ exactly once. On this interval, $\sin x=k$ has a unique solution for each $k\in [-1,1]$.

The sine function defined on $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ is called the principal sine function. The unique solution of $\mathrm{Sin}x=k$ is written $x={\mathrm{Sin}}^{-1}k$: '$x$ is the real number in $\left[-\frac{\pi }{2},\frac{\pi }{2}\right]$ whose sine is $k$'.

Inverse sine function (8.1.5)

Domain and range of the inverse sine function (8.1.6)

As for cosine, the graph of $y={\mathrm{Sin}}^{-1}x$ is the reflection of the principal sine curve in the line $y=x$.

8.1.7 to 8.1.9Principal tangent function and inverse tangent functionOften examined

Principal tangent function (8.1.7)

The tangent function is defined for every real x except the odd multiples of \frac{\pi}{2}, where \cos x=0. So its domain is \mathbb{R}-\left\{(2n+1)\frac{\pi}{2}:n\in\mathbb{Z}\right\} and its range is all of \mathbb{R}. Because \tan x repeats every \pi, the equation \tan x=k has infinitely many solutions for every real k.

Between -\frac{\pi}{2} and \frac{\pi}{2} the tangent curve rises from -\infty to +\infty and takes every real value exactly once. The end points are left out (an open interval) because \tan x is not defined there.

This is the principal tangent function. The unique solution of \operatorname{Tan} x=k is written x=\operatorname{Tan}^{-1} k: 'x is the real number in \left(-\frac{\pi}{2},\frac{\pi}{2}\right) whose tangent is k'.

Inverse tangent function (8.1.8)

Domain and range of the inverse tangent function (8.1.9)

8.1.10 to 8.1.12Principal cotangent function and inverse cotangent functionOften examined

Principal cotangent function (8.1.10)

Like tangent, \cot x repeats every \pi, so \cot x=k has infinitely many solutions for each real k. On the open interval (0,\pi) the cotangent falls steadily from +\infty to -\infty and takes each real value exactly once. The end points 0 and \pi are left out because \cot x=\frac{\cos x}{\sin x} is not defined where \sin x=0.

This is the principal cotangent function. The unique solution of \operatorname{Cot} x=k is written x=\operatorname{Cot}^{-1} k: 'x is the real number in (0,\pi) whose cotangent is k'.

Inverse cotangent function (8.1.11)

Domain and range of the inverse cotangent function (8.1.12)

(The book's heading line here says 'inverse tangent function y=\operatorname{Cot}^{-1} x'; it means the inverse cotangent function.)

8.1.13 to 8.1.15Principal secant function and inverse secant functionOften examined

Principal secant function (8.1.13)

\sec x=\dfrac{1}{\cos x} is defined wherever \cos x\ne0, that is for x\ne(2n+1)\frac{\pi}{2}, n\in\mathbb{Z}. Since |\cos x|\le1, its values satisfy |\sec x|\ge1, so the range of secant is \mathbb{R}-(-1,1). The equation \sec x=k has no solution when -1<k<1 and infinitely many solutions for every other k; for example \sec x=1 at x=0,\pm2\pi,\pm4\pi,\dots

Cosine is one-to-one on [0,\pi], so its reciprocal is one-to-one there too. We must only remove x=\frac{\pi}{2}, where \cos x=0. Other choices such as (\pi,2\pi) are possible, but by convention we take [0,\pi] without \frac{\pi}{2}.

This is the principal secant function. The unique solution of \operatorname{Sec} x=k, for k\in\mathbb{R}-(-1,1), is written x=\operatorname{Sec}^{-1} k: 'x is the real number in \left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right] whose secant is k'.

Inverse secant function (8.1.14)

Domain and range of the inverse secant function (8.1.15)

(The book writes the range of \operatorname{Sec}^{-1} x as 'x\in[0,\pi],\ x\ne\frac{\pi}{2}'; the variable should be y, the output.)

8.1.16 to 8.1.18Principal cosecant function and inverse cosecant functionOften examined

Principal cosecant function (8.1.16)

\csc x=\dfrac{1}{\sin x} is defined wherever \sin x\ne0, that is for x\ne n\pi. Like secant, its values satisfy |\csc x|\ge1, so \csc x=k has many solutions for each k\in\mathbb{R}-(-1,1) and none for -1<k<1.

Sine is one-to-one on \left[-\frac{\pi}{2},\frac{\pi}{2}\right], so its reciprocal is one-to-one there, except at x=0 where \sin x=0. The end points \pm\frac{\pi}{2} are kept because \csc\left(\pm\frac{\pi}{2}\right)=\pm1.

This is the principal cosecant function. The unique solution of \operatorname{Csc} x=k is written x=\operatorname{Csc}^{-1} k: 'x is the real number in \left[-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right] whose cosecant is k'.

Inverse cosecant function (8.1.17)

Domain and range of the inverse cosecant function (8.1.18)

8.1Finding principal values: summary and methodOften examined

Every question of the type 'find the principal value' is solved the same way. First learn the six ranges; they decide where the answer must lie.

Properties I to VIProperties of inverse trigonometric functionsOften examined

These six groups of properties follow directly from the definitions and ranges. They are used to simplify expressions, to change one inverse function into another, and to prove identities. In each property, notice the condition on x: it is part of the property.

Property I: inverse of a function (angle in, angle out)

These hold only when x is already in the principal interval. Outside it, the answer is the angle in the principal interval that has the same trigonometric value.

Property II: function of the inverse (number in, number out)

This always works for every x in the domain, because \operatorname{Cos}^{-1} x is, by definition, an angle whose cosine is x. For example \cos\left(\operatorname{Cos}^{-1} 0.3\right)=0.3, while \cos\left(\operatorname{Cos}^{-1} 2\right) has no meaning because 2 is outside the domain.

Property III: negative arguments

Property IV: reciprocal arguments

The first two hold for every x with |x|\ge1, because \operatorname{Sin} and \operatorname{Csc} use the same interval (without 0), and \operatorname{Cos} and \operatorname{Sec} use the same interval (without \frac{\pi}{2}). Tangent and cotangent use different intervals, which is why the third one splits into two cases.

Property V: complementary pairs

(The book prints the last one as \operatorname{Sec}^{-1}(-x)+\operatorname{Csc}^{-1}(-x)=\frac{\pi}{2}. Since -x runs over the same set |x|\ge1, it is the same statement.)

Property VI: changing one inverse function into another

To express \operatorname{Sin}^{-1} x through the other functions, draw a right triangle with an acute angle \theta, opposite side x and hypotenuse 1. Pythagoras gives the adjacent side \sqrt{1-x^2}, and all six ratios of \theta can be read off.

for 0<x<1 (triangle: adjacent x, hypotenuse 1)

for x>0 (triangle: opposite x, adjacent 1, hypotenuse \sqrt{1+x^2})

8.2Graphs of inverse trigonometric functions

A graph shows at a glance what the domain and range only list: where the function starts and ends, whether it rises or falls, its symmetry, and its asymptotes. Every graph in this section is the reflection of the principal trigonometric function in the line y=x.

1. Draw the axes. The x-axis carries the input (a real number); the y-axis carries the output angle in radians, marked in steps of \frac{\pi}{6} or \frac{\pi}{4}.
2. Make a table of values from the standard angles, for example \operatorname{Cos}^{-1}\frac12=\frac{\pi}{3}.
3. Plot the points, join them with a smooth curve, mark included end points with filled dots, and draw asymptotes as dashed lines.

Shifting, stretching and reflecting

• Adding a constant outside moves the graph up or down: y=\operatorname{Sin}^{-1} x+c. Adding inside moves it left or right: y=\operatorname{Sin}^{-1}(x-c) is shifted c units right.
• Multiplying outside stretches the graph vertically and multiplies the range: y=2\operatorname{Sin}^{-1} x has range [-\pi,\pi]. Multiplying x inside compresses it horizontally and divides the domain: y=\operatorname{Sin}^{-1}(2x) has domain \left[-\frac12,\frac12\right].
• A negative factor reflects the graph: y=-\operatorname{Sin}^{-1} x in the x-axis, y=\operatorname{Sin}^{-1}(-x) in the y-axis.

8.2.1Graph of y = Cos⁻¹xOften examined

The graph of y=\operatorname{Cos}^{-1} x shows the angle y whose cosine is x. The domain is x\in[-1,1] and the range is y\in[0,\pi], so every output is an angle from 0 to \pi radians.

Construction steps

1. Draw the axes. Mark the x-axis from -1 to 1 and the y-axis from 0 to \pi in steps of \frac{\pi}{6}, using a suitable scale.
2. Plot the points of the table.
3. Join them with a smooth curve and mark both end points with filled dots.

Properties

• Domain and range: domain [-1,1], range [0,\pi].
• Key points: (1,0), \left(0,\frac{\pi}{2}\right), (-1,\pi). The y-intercept is \frac{\pi}{2} and the x-intercept is 1.
• Appearance: a smooth curve that starts at (-1,\pi), passes through \left(0,\frac{\pi}{2}\right) and ends at (1,0).
• Monotonicity: strictly decreasing; as x increases, the angle decreases.
• Symmetry: neither even nor odd. Because \operatorname{Cos}^{-1}(-x)=\pi-\operatorname{Cos}^{-1} x, the point (-x,\pi-y) is on the graph whenever (x,y) is, so the graph is symmetric about the point \left(0,\frac{\pi}{2}\right).
• Reflection: it is the reflection of y=\operatorname{Cos} x (0\le x\le\pi) in the line y=x; the coordinates of every point are swapped.
• Inverse relations: \cos(\operatorname{Cos}^{-1} x)=x for x\in[-1,1] and \operatorname{Cos}^{-1}(\cos x)=x for x\in[0,\pi].

8.2.2Graph of y = Sin⁻¹xOften examined

The graph of y=\operatorname{Sin}^{-1} x shows the angle y whose sine is x. The domain is x\in[-1,1] and the range is y\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right].

Construction steps

1. Draw the axes. Mark the x-axis from -1 to 1 and the y-axis from -\frac{\pi}{2} to \frac{\pi}{2} in steps of \frac{\pi}{6}.
2. Plot the points of the table and join them with a smooth curve; mark the end points \left(\pm1,\pm\frac{\pi}{2}\right) with filled dots.

Properties

• Domain and range: domain [-1,1], range \left[-\frac{\pi}{2},\frac{\pi}{2}\right].
• Key points: \left(-1,-\frac{\pi}{2}\right), (0,0), \left(1,\frac{\pi}{2}\right). The only intercept is the origin.
• Monotonicity: strictly increasing.
• Odd function: \operatorname{Sin}^{-1}(-x)=-\operatorname{Sin}^{-1} x, so the graph is symmetric about the origin.
• Reflection: it is the reflection of y=\operatorname{Sin} x \left(-\frac{\pi}{2}\le x\le\frac{\pi}{2}\right) in the line y=x.
• Inverse relations: \sin(\operatorname{Sin}^{-1} x)=x for x\in[-1,1] and \operatorname{Sin}^{-1}(\sin x)=x for x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right].
• Identities: \operatorname{Sin}^{-1} x=\operatorname{Cos}^{-1}\sqrt{1-x^2} for 0\le x\le1, and \operatorname{Sin}^{-1} x=-\operatorname{Cos}^{-1}\sqrt{1-x^2} for -1\le x<0 (the minus sign is needed because \operatorname{Cos}^{-1} is never negative).

8.2.3Graph of y = Tan⁻¹x

The graph of y=\operatorname{Tan}^{-1} x shows the angle y whose tangent is x. The domain is all real numbers and the range is the open interval \left(-\frac{\pi}{2},\frac{\pi}{2}\right).

Construction steps

1. Draw the axes. The x-axis must be long, because the domain is all of \mathbb{R}; mark the y-axis from -\frac{\pi}{2} to \frac{\pi}{2}.
2. Draw the horizontal asymptotes y=\pm\frac{\pi}{2} as dashed lines.
3. Plot the points of the table and join them smoothly, making the curve flatten towards the asymptotes at both ends.

Properties

• Domain and range: domain (-\infty,\infty), range \left(-\frac{\pi}{2},\frac{\pi}{2}\right).
• Key points: \left(-1,-\frac{\pi}{4}\right), (0,0), \left(1,\frac{\pi}{4}\right). The book also lists '\left(\pm\infty,\pm\frac{\pi}{2}\right)': these are the directions the curve approaches, not points on it.
• Asymptotes: y=-\frac{\pi}{2} as x\to-\infty and y=\frac{\pi}{2} as x\to+\infty.
• Shape: an S-shaped curve through the origin, strictly increasing.
• Odd function: \operatorname{Tan}^{-1}(-x)=-\operatorname{Tan}^{-1} x; the graph is symmetric about the origin.
• Reflection: the reflection of y=\operatorname{Tan} x \left(-\frac{\pi}{2}<x<\frac{\pi}{2}\right) in y=x.
• Inverse relations: \tan(\operatorname{Tan}^{-1} x)=x for all x, and \operatorname{Tan}^{-1}(\tan x)=x for x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right).

8.2.4Graph of y = Cot⁻¹x

The graph of y=\operatorname{Cot}^{-1} x shows the angle y in (0,\pi) whose cotangent is x. The domain is all real numbers and the range is (0,\pi).

Construction steps

1. Draw the axes; mark the y-axis from 0 to \pi in steps of \frac{\pi}{6}.
2. Draw the asymptotes y=0 (the x-axis) and y=\pi.
3. Plot the points of the table and join them smoothly.

Properties

• Domain and range: domain (-\infty,\infty), range (0,\pi).
• Key points: \left(-1,\frac{3\pi}{4}\right), \left(0,\frac{\pi}{2}\right), \left(1,\frac{\pi}{4}\right). There is no x-intercept, because \operatorname{Cot}^{-1} x is never 0.
• Asymptotes: y=\pi as x\to-\infty and y=0 as x\to+\infty.
• Monotonicity: strictly decreasing (an S-shaped curve going down).
• Symmetry: neither even nor odd. \operatorname{Cot}^{-1}(-x)=\pi-\operatorname{Cot}^{-1} x means the graph is symmetric about the point \left(0,\frac{\pi}{2}\right).
• Reflection: the reflection of y=\operatorname{Cot} x (0<x<\pi) in y=x.
• Inverse relations: \cot(\operatorname{Cot}^{-1} x)=x for all x, and \operatorname{Cot}^{-1}(\cot x)=x for x\in(0,\pi).
• Identity: \operatorname{Cot}^{-1} x=\operatorname{Tan}^{-1}\frac1x for x>0, and \operatorname{Cot}^{-1} x=\pi+\operatorname{Tan}^{-1}\frac1x for x<0.

8.2.5Graph of y = Sec⁻¹x

The graph of y=\operatorname{Sec}^{-1} x shows the angle y in [0,\pi], y\ne\frac{\pi}{2}, whose secant is x. The domain is (-\infty,-1]\cup[1,\infty), so the graph has two separate branches, and the range is \left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right].

Construction steps

1. Draw the axes; mark the y-axis from 0 to \pi and draw the asymptote y=\frac{\pi}{2} as a dashed line.
2. Leave the strip -1<x<1 empty: no point of the graph lies there.
3. Plot the points of the table; join each branch smoothly, with filled dots at (1,0) and (-1,\pi).

Properties

• Domain and range: domain (-\infty,-1]\cup[1,\infty), range \left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right].
• Key points: (-2,\frac{2\pi}{3}), (-1,\pi), (1,0), (2,\frac{\pi}{3}). The only x-intercept is (1,0); there is no y-intercept because 0 is not in the domain.
• Asymptote: the horizontal line y=\frac{\pi}{2}, approached as x\to\pm\infty.
• Monotonicity: increasing on each branch: from 0 up towards \frac{\pi}{2} on [1,\infty), and from about \frac{\pi}{2} up to \pi on (-\infty,-1]. It is not increasing across the gap, because \operatorname{Sec}^{-1}(-1)=\pi is larger than \operatorname{Sec}^{-1}(1)=0.
• Symmetry: neither even nor odd; \operatorname{Sec}^{-1}(-x)=\pi-\operatorname{Sec}^{-1} x gives symmetry about the point \left(0,\frac{\pi}{2}\right).
• Reflection: the reflection of y=\operatorname{Sec} x (0\le x\le\pi, x\ne\frac{\pi}{2}) in y=x.
• Inverse relations: \sec(\operatorname{Sec}^{-1} x)=x for |x|\ge1, and \operatorname{Sec}^{-1}(\sec x)=x for x\in\left[0,\frac{\pi}{2}\right)\cup\left(\frac{\pi}{2},\pi\right].
• Identity: \operatorname{Sec}^{-1} x=\operatorname{Cos}^{-1}\frac1x for |x|\ge1.

8.2.6Graph of y = Csc⁻¹x

The graph of y=\operatorname{Csc}^{-1} x shows the angle y in \left[-\frac{\pi}{2},\frac{\pi}{2}\right], y\ne0, whose cosecant is x. The domain is (-\infty,-1]\cup[1,\infty) and the range is \left[-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right], so the outputs are angles between -\frac{\pi}{2} and \frac{\pi}{2}.

Construction steps

1. Draw the axes; mark the y-axis from -\frac{\pi}{2} to \frac{\pi}{2} in steps of \frac{\pi}{6}.
2. Leave the strip -1<x<1 empty. The x-axis (y=0) is the asymptote.
3. Plot the points of the table and join each branch smoothly, with filled dots at \left(\pm1,\pm\frac{\pi}{2}\right).

Properties

• Domain and range: domain (-\infty,-1]\cup[1,\infty), range \left[-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right].
• Key points: \left(-2,-\frac{\pi}{6}\right), \left(-1,-\frac{\pi}{2}\right), \left(1,\frac{\pi}{2}\right), \left(2,\frac{\pi}{6}\right). There are no intercepts.
• Asymptote: the horizontal line y=0, approached as x\to\pm\infty.
• Monotonicity: decreasing on each branch: from \frac{\pi}{2} down towards 0 on [1,\infty), and from just below 0 down to -\frac{\pi}{2} on (-\infty,-1].
• Odd function: \operatorname{Csc}^{-1}(-x)=-\operatorname{Csc}^{-1} x; the graph is symmetric about the origin.
• Reflection: the reflection of y=\operatorname{Csc} x \left(-\frac{\pi}{2}\le x\le\frac{\pi}{2},\ x\ne0\right) in y=x.
• Inverse relations: \csc(\operatorname{Csc}^{-1} x)=x for |x|\ge1, and \operatorname{Csc}^{-1}(\csc x)=x for x\in\left[-\frac{\pi}{2},0\right)\cup\left(0,\frac{\pi}{2}\right].
• Identity: \operatorname{Csc}^{-1} x=\operatorname{Sin}^{-1}\frac1x for |x|\ge1.

Addition and subtraction formulaeOften examined

Two inverse trigonometric functions can often be combined into one. The idea is simple: name the two angles, use the compound-angle formula for \tan(\alpha\pm\beta), \sin(\alpha\pm\beta) or \cos(\alpha\pm\beta), and then check that the combined angle lies in the principal range.

Inverse tangent

Replace B by -B in the addition formula and use \operatorname{Tan}^{-1}(-B)=-\operatorname{Tan}^{-1} B.

Put B=A in the addition formula (the condition AB<1 becomes A^2<1).

Inverse sine and inverse cosine

Worked examples

Applications: angles of elevation, depression and slopesOften examined

In practical problems we usually know lengths and want an angle. The inverse trigonometric functions give that angle directly from a ratio of two sides of a right triangle:

For an acute angle the ratio is positive, and the principal value is exactly the acute angle we want. This is what the \sin^{-1}, \cos^{-1} and \tan^{-1} keys of a calculator give.

Line of sight

The straight line from the observer's eye to the object being looked at.

Angle of elevation

The angle between the horizontal and the line of sight when the object is above the observer.

Angle of depression

The angle between the horizontal and the line of sight when the object is below the observer.

Angle of inclination (of a ramp or road)

The angle a slope makes with the horizontal: \theta=\operatorname{Tan}^{-1}\left(\dfrac{\text{rise}}{\text{run}}\right).

Gradient

The ratio \dfrac{\text{rise}}{\text{run}}=\tan\theta, often written as '1 in 12' and so on.

Formula sheet

Key terms

Inverse trigonometric function

A function that gives the angle whose trigonometric ratio is a given number, for example \operatorname{Sin}^{-1} x is the angle whose sine is x.

One-to-one function

A function in which different inputs always give different outputs. Only one-to-one functions have inverses.

Horizontal line test

A function is one-to-one if no horizontal line cuts its graph more than once. The full trigonometric functions fail this test.

Restricted (principal) domain

The interval on which a trigonometric function is kept so that it becomes one-to-one: [0,\pi] for cosine, \left[-\frac{\pi}{2},\frac{\pi}{2}\right] for sine, and so on.

Principal function

A trigonometric function restricted to its principal domain, written with a capital letter such as \operatorname{Cos} x or \operatorname{Tan} x.

Principal value

The single value of an inverse trigonometric function, lying in its fixed range; for example the principal value of \operatorname{Cos}^{-1}\left(-\frac12\right) is \frac{2\pi}{3}.

Principal cosine function

\operatorname{Cos} x=\cos x for 0\le x\le\pi; its inverse is \operatorname{Cos}^{-1} x.

Principal sine function

\operatorname{Sin} x=\sin x for -\frac{\pi}{2}\le x\le\frac{\pi}{2}; its inverse is \operatorname{Sin}^{-1} x.

Principal tangent function

\operatorname{Tan} x=\tan x for -\frac{\pi}{2}<x<\frac{\pi}{2}; its inverse is \operatorname{Tan}^{-1} x.

Principal cotangent function

\operatorname{Cot} x=\cot x for 0<x<\pi; its inverse is \operatorname{Cot}^{-1} x.

Principal secant function

\operatorname{Sec} x=\sec x for 0\le x\le\pi, x\ne\frac{\pi}{2}; its inverse is \operatorname{Sec}^{-1} x.

Principal cosecant function

\operatorname{Csc} x=\csc x for -\frac{\pi}{2}\le x\le\frac{\pi}{2}, x\ne0; its inverse is \operatorname{Csc}^{-1} x.

Reference angle

The acute angle that has the same trigonometric ratio as |k|; used to place a principal value in the correct quadrant.

Reflection in y = x

The graph of an inverse function is the mirror image of the original graph in the line y=x: each point (a,b) becomes (b,a).

Horizontal asymptote

A horizontal line that a graph approaches but never reaches, for example y=\pm\frac{\pi}{2} for y=\operatorname{Tan}^{-1} x.

Odd function

A function with f(-x)=-f(x); its graph is symmetric about the origin. \operatorname{Sin}^{-1}, \operatorname{Tan}^{-1} and \operatorname{Csc}^{-1} are odd.

Angle of elevation

The angle between the horizontal and the line of sight to an object above the observer.

Angle of depression

The angle between the horizontal and the line of sight to an object below the observer.

Angle of inclination

The angle a slope (ramp, road, roof) makes with the horizontal, \theta=\operatorname{Tan}^{-1}\frac{\text{rise}}{\text{run}}.

FBISE 2026 questions

Questions from the FBISE HSSC-II 2026 (1st Annual) papers that belong to this unit, answered the way an examiner expects.

Exam focus

1. Find principal values such as \operatorname{Cos}^{-1}\left(-\frac12\right), \operatorname{Cot}^{-1}(-\sqrt3), \operatorname{Csc}^{-1}(-2), and sums of principal values. (short)
   The basic skill of the unit; Exercise 8.1 Q1 and Q2 are built on it and it appears in every MCQ on this topic.
   Go to answer
2. MCQ: evaluate \sin^{-1}\left(\sin\frac{5\pi}{6}\right) or \cos^{-1}\left(\cos\frac{4\pi}{3}\right). (mcq)
   Asked in both 2026 sets (Set B MCQ 16, Set D MCQ 17); the trap answer is the starting angle.
   Go to answer
3. Find the domain and range of \sin^{-1}(2x-1) or \operatorname{cosec}^{-1}(3x+2). (short)
   Section B, 4 marks, in both 2026 sets (Q.2 (x)).
   Go to answer
4. Draw the graph of y=\sin^{-1}x or y=\cos^{-1}x with a proper scale, labelling domain, range, intercepts and end points. (long)
   Section C, 8 marks, in both 2026 sets (Q.4 OR); also know why [-2\pi,2\pi] cannot be the domain.
   Go to answer
5. Height of a tower from two angles of elevation (30° and 60°, 40 m apart). (short)
   2026 Set B Q.2 (v) OR, 4 marks; a standard application problem.
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6. Angle of inclination and length of a ramp from its rise and run. (short)
   2026 Set D Q.2 (v) OR, 4 marks; uses the inverse tangent directly.
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7. Prove \operatorname{Sin}^{-1} x+\operatorname{Cos}^{-1} x=\frac{\pi}{2} and \operatorname{Cos}^{-1}(-x)=\pi-\operatorname{Cos}^{-1} x. (short)
   Short proofs that test the ranges; typical 3 to 4 mark questions.
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8. Exact values such as \sin\left(\operatorname{Tan}^{-1}\frac34\right) and \cos\left(\operatorname{Sin}^{-1}\frac34+\operatorname{Cos}^{-1}\frac{5}{13}\right). (short)
   Exercise 8.1 Q3; the right-triangle method and the sign of the root decide the marks.
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9. State and prove \operatorname{Tan}^{-1} A+\operatorname{Tan}^{-1} B=\operatorname{Tan}^{-1}\frac{A+B}{1-AB} and apply it, for example \operatorname{Tan}^{-1}\frac12+\operatorname{Tan}^{-1}\frac13=\frac{\pi}{4}. (long)
   Named in the unit's learning outcomes ('state, prove and apply'); the book pages for it are missing, so learn it from these notes.
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10. Domain, range, symmetry and asymptotes of all six inverse functions. (short)
    First learning outcome of the unit; the book's own table has several misprints, so use the corrected table.
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