Here, you will learn formulas for inverse trigonometric functions, equation and inequations involving inverse trigonometric function.
Let’s begin –
Simplified Inverse Trigonometric Functions
(a) y = f(x) = sin−1(2x1+x2) = {2tan−1x,if |x|≤1π–2tan−1x,if x>1−(π+2tan−1x),if x<−1
(b) y = f(x) = cos−1(1−x21+x2) = {2tan−1x,if |x|≥0–2tan−1x,if x<0
(c) y = f(x) = tan−1(2x1−x2) = {2tan−1x,if |x|<1π+2tan−1x,if x<−1−(π–2tan−1x),if x>1
(d) y = f(x) = sin−1(3x–4x3) = {−(π+3sin−1x),if −1≤x≤−123sin−1x,if −12≤x≤12π–3sin−1x,if 12≤x≤1
(e) y = f(x) = cos−1(4x3–3x) = {3cos−1x–2π,if −1≤x≤−122π–3cos−1x,if −12≤x≤123cos−1x,if 12≤x≤1
(f) y = f(x) = sin−1(2x√1−x2) = {−(π+2sin−1x),if −1≤x≤−1√22sin−1x,if −1√2≤x≤1√2π–2sin−1x,if 1√2≤x≤1
(g) y = f(x) = cos−1(2x2−1) = {2cos−1x,if 0≤x≤12π–2cos−1x,if −1≤x≤0
Example : Prove that : 2tan−112 + tan−117 = tan−13117
Solution : We have, 2tan−112 + tan−117
= 2tan−1(2×121−(12)2) + tan−117 [∵ 2tan−1x = tan−12x1−x2]
tan−143 + tan−117 = tan−1[43+171–43×17] = tan−13117
Equations involving Inverse trigonometric functions
Example : Prove that the equation 2cos−1x + sin−1x = 11π6 has no solution.
Solution : Given equation is 2cos−1x + sin−1x = 11π6
⟹ cos−1x + (cos−1x + sin−1x) = 11π6
⟹ cos−1x + π2 = 11π6
⟹ cos−1x = 4π3
which is not possible as cos−1x ∈ [0, π]. Hence no solution.
Inequations involving Inverse trigonometric functions
Example : Find the complete solution set of sin−1(sin5) > x2 – 4x.
Solution : sin−1(sin5) > x2 – 4x ⟹ sin−1[sin(5−2π)] > x2 – 4x
⟹ x2 – 4x < 5 – 2π ⟹ x2 – 4x + 2π – 5< 0
⟹ 2 – √9−2π < x < 2 + √9−2π ⟹ x ∈ (2 – √9−2π, 2 + √9−2π)
Hope you learnt formulas for inverse trigonometric functions, equation and inequations involving inverse trigonometric function, learn more concepts of inverse trigonometric functions and practice more questions to get ahead in competition. Good Luck!