Inverse Trigonometric Functions - BSEB Inter Study Guide
Inverse Trigonometric Functions
This domain checks principal values, domain and range restrictions, and identity-based simplification. BSEB questions often ask for one exact angle or a proof using sin⁻¹x+cos⁻¹x=π/2.
Principal branches
- sin⁻¹: [−1,1]→[−π/2,π/2]
- cos⁻¹: [−1,1]→[0,π]
- tan⁻¹: R→(−π/2,π/2)
- cot⁻¹: R→(0,π)
- sec⁻¹: (−∞,−1]∪[1,∞)→[0,π] minus {π/2}
- cosec⁻¹: (−∞,−1]∪[1,∞)→[−π/2,π/2] minus {0}
Identities
sin⁻¹x+cos⁻¹x=π/2 and tan⁻¹x+cot⁻¹x=π/2. Also tan⁻¹x+tan⁻¹y=tan⁻¹((x+y)/(1−xy)) only when xy<1; if xy>1 and x,y>0, add π.
Worked example
Evaluate sin⁻¹(sin 5π/6). First sin 5π/6=1/2. Since 1/2 lies in [−1,1], sin⁻¹(1/2) is the angle in [−π/2,π/2] whose sine is 1/2, namely π/6. So the value is π/6, not 5π/6.
Traps
- Cancelling sin⁻¹(sin x)=x outside the principal range.
- Applying the tan⁻¹ addition formula without checking xy<1.
- Mixing radians and degrees in one answer.
- Forgetting that cos⁻¹ has range [0,π], not [−π/2,π/2].