xSkill: notes, lectures and practice for CBSE and CSE

1.1 Domain and principal value of every inverse function Each trigonometric function repeats values, so its inverse is defined only after we cut the graph down to one rising or falling piece. That piece fixes the principal value branch. Learn these six lines exactly; most one mark questions on this chapter are one of them in disguise. Function Domain Principal value branch sin inverse x [-1, 1] [-pi/2, pi/2] cos inverse x [-1, 1] [0, pi] tan inverse x R (-pi/2, pi/2) cot inverse x R (0, pi) sec inverse x R - (-1, 1) [0, pi] minus {pi/2} cosec inverse x R - (-1, 1) [-pi/2, pi/2] minus {0} The ranges are the part students lose marks on. tan inverse never reaches plus or minus pi/2, and sec inverse skips pi/2.

sin^{- 1} : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}], cos^{- 1} : [- 1, 1] \to [0, π]

Principal branches of sine inverse and cosine inverse

tan^{- 1} : R \to (- \frac{π}{2}, \frac{π}{2}), cot^{- 1} : R \to (0, π)

Principal branches of tangent inverse and cotangent inverse

1.2 Complementary and negative angle results Two inverse functions with complementary ranges add to a quarter turn. The same pattern appears three times, with different domains.

sin^{- 1} x + cos^{- 1} x = \frac{π}{2}, x \in [- 1, 1]

Sine inverse and cosine inverse add to pi by 2

tan^{- 1} x + cot^{- 1} x = \frac{π}{2}, x \in R

Tangent inverse and cotangent inverse add to pi by 2

sec^{- 1} x + cosec^{- 1} x = \frac{π}{2}, ∣ x ∣ \geq 1

Secant inverse and cosecant inverse add to pi by 2

1.3 Odd and even behaviour Sine, tangent and cosecant inverses are odd, so a minus sign can be pulled outside. Cosine, cotangent and secant inverses give a supplement instead, so the outer minus sign becomes pi minus the value.

sin^{- 1} (- x) = - sin^{- 1} x, tan^{- 1} (- x) = - tan^{- 1} x

Odd inverse functions

cos^{- 1} (- x) = π - cos^{- 1} x, cot^{- 1} (- x) = π - cot^{- 1} x, sec^{- 1} (- x) = π - sec^{- 1} x

Even-style inverse functions reflect as a supplement

1.4 Sum, difference and double angle formulas The tangent addition formula is the workhorse for equations of the type tan inverse a plus tan inverse b. Use it only inside the range where the combined angle still lies in the principal branch.

tan^{- 1} x + tan^{- 1} y = tan^{- 1} \frac{x + y}{1 - x y}, x y < 1

Sum of two tangent inverses when the product is below 1

tan^{- 1} x - tan^{- 1} y = tan^{- 1} \frac{x - y}{1 + x y}, x y > - 1

Difference of two tangent inverses

2 tan^{- 1} x = tan^{- 1} \frac{2 x}{1 - x ^{2}}, ∣ x ∣ < 1

Double angle form of tangent inverse

1.5 Conversions between trigonometric and inverse forms When an inverse function sits inside a trigonometric function, draw a right triangle. For sin inverse x, label the opposite side x and the hypotenuse 1, and the remaining side follows from Pythagoras.

sin (cos^{- 1} x) = 1 - x^{2}, cos (sin^{- 1} x) = 1 - x^{2}

Square root forms for a sine inside a cosine inverse or the reverse

tan (sin^{- 1} x) = \frac{x}{1 - x ^{2}}, sin (tan^{- 1} x) = \frac{x}{1 + x ^{2}}

Triangle ratios that appear in simplification questions

Exam

State the principal branch whenever a value is asked. A bare number such as pi by 4 is not enough if the question says find the principal value; name the branch or show the interval the answer sits in.