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Inverse Trigonometric Functions
Introduction
Basic Concepts
Practice: Principal Values of Inverse Trig Functions
Properties of Inverse Trigonometric Functions
Practice: Simplifying Inverse Trig Expressions
Miscellaneous Examples
Chapter Review: Mixed Practice
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Inverse Trigonometric FunctionsIntroduction

Reading time: ~10 min

It is Makar Sankranti, and every rooftop in Ahmedabad is full of kite flyers. Aman has let out all 100 m of his string, and his cousin Riya, on the next roof, measures that the kite is flying 50 m above his hand.

Aman wants just one number: the angle his string makes with the ground. The string is the hypotenuse of a right triangle and the height is the side opposite the angle, so the sine of the angle is 50 ÷ 100 = ½. For once the question runs backwards: we know the sine and want the angle.

Find the angle — all of them. The blue wave is y = \sin x, and the dashed line is the height ½. Drag the point along the wave. Every angle you find whose sine is exactly ½ gets a green dot.

Riya's calculator says 30°, and for a kite that is clearly the right answer. But the green dots on the wave have just shown you that \dfrac{\pi}{6}, \dfrac{5\pi}{6}, -\dfrac{7\pi}{6}, -\dfrac{11\pi}{6} and infinitely many other angles all have sine ½, because the sine function repeats itself. A function must give exactly one output for each input, so "the angle whose sine is ½" is not yet a function. In Chapter 1 you saw that only a one-one and onto function has an inverse, and the trigonometric functions are neither on their natural domains and ranges. This chapter shows how to restrict them so that their inverses exist, what the graphs of those inverses look like, and how they behave.

Turning a measured ratio back into an angle is everyday work for machines. A phone works out how far you have tilted it from the ratio its sensor reports; a robot arm turns the position of its gripper into the angles its joints must hold; a surveyor turns the height of a tower and its distance into an angle of elevation. Inverse trigonometric functions do that job, and in calculus they are also used to define many integrals.

From Shadows to Symbols

The study of trigonometry as we know it began in India. The siddhantas, Sanskrit works on astronomy, introduced the sine of an angle, and Aryabhata (476 C.E.), Brahmagupta (598 C.E.), Bhaskara I (600 C.E.) and Bhaskara II (1114 C.E.) found important results. From India the knowledge travelled to Arabia and from there to Europe. The Greeks had also studied the subject, but their methods were clumsy, and once the Indian approach became known it was adopted all over the world.

Bhaskara I gave formulae for the sines of angles greater than 90°. A sixteenth-century Malayalam work, the Yuktibhasa, contains a proof of the expansion of sin (A + B), and Bhaskara II gave exact expressions for the sines and cosines of 18°, 36°, 54° and 72°. The symbols \sin^{-1} x and \cos^{-1} x for "arc sin x" and "arc cos x" came much later: the astronomer Sir John F. W. Herschel suggested them in 1813.

The oldest inverse-trigonometry problem of all may be a shadow. Thales (about 600 B.C.E.) is said to have found the height H of a great pyramid by comparing its shadow S with the shadow s of a staff of known height h:

\dfrac{H}{S} = \dfrac{h}{s} = \tan(\text{sun's altitude})

Read the other way round, the sun's altitude is \tan^{-1}\dfrac{h}{s}. Try it: a staff 1 m tall casts a shadow \sqrt{3} m long, so the sun's altitude is \tan^{-1}\dfrac{1}{\sqrt{3}} = .

Thales is also said to have found the distance of a ship at sea using triangles, and height-and-distance problems solved with similar triangles appear in ancient Indian works too.

Sina