Inverse Trigonometric Ratios

Trigonometric Ratio	Inverse Ratio
$x = sin θ$	$θ = sin^{- 1} x$
$x = cos θ$	$θ = cos^{- 1} x$
$x = tan θ$	$θ = tan^{- 1} x$
$x = cot θ$	$θ = cot^{- 1} x$
$x = sec θ$	$θ = sec^{- 1} x$
$x = csc θ$	$θ = csc^{- 1} x$

Trigonometric Ratio

Inverse Ratio

x = sin θ

θ = sin^{- 1} x

x = cos θ

θ = cos^{- 1} x

x = tan θ

θ = tan^{- 1} x

x = cot θ

θ = cot^{- 1} x

x = sec θ

θ = sec^{- 1} x

x = csc θ

θ = csc^{- 1} x

There are three different notations for each inverse trigonometric ratios. For example, inverse trigonometric sine of

x

can be written as follows.

sin^{- 1} x arcsin x A r c s i n x

In the first notation, it is important to not mistake

- 1

for the negative power of

1 .

For that reason, writing

arcsin x

A r c s i n x

is sometimes preferred as it leaves no space for misinterpretation. Note that there is a slight difference between these two notations.

	$arcsin x$	$A r c s i n x$
Algebraic Definition	${θ \in R : sin θ = x}$	${θ \in [- \frac{π}{2}, \frac{π}{2}] : sin θ = x}$
Meaning	All real angles $θ$ that satisfy $sin θ = x .$	One or more unique angles $θ$ in the interval $[- \frac{π}{2}, \frac{π}{2}]$ that satisfy $sin θ = x .$

In other words, $arcsin x$ represents all the angles whose sine equals $x,$ while $A r c s i n x$ represents only the principal angle in the main interval $[- \frac{π}{2}, \frac{π}{2}] .$ However, this notation is not always followed. In practice, in most cases the lower case notation is used to denote the principal value, and the upper case notation can be not used at all.

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Inverse Trigonometric Ratios

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