| --- |
| source: ncert |
| topic: trigonometry |
| class_level: class_12 |
| chapter: ch2 |
| difficulty: advanced |
| --- |
| |
| Class 12 | Ch2: Inverse Trigonometric Functions |
| DOMAINS AND RANGES (Principal Value Branch): |
| sin^(-1): domain [-1,1], range [-pi/2, pi/2]. |
| cos^(-1): domain [-1,1], range [0, pi]. |
| tan^(-1): domain R, range (-pi/2, pi/2). |
| cosec^(-1): domain R-(-1,1), range [-pi/2,pi/2]-{0}. |
| sec^(-1): domain R-(-1,1), range [0,pi]-{pi/2}. |
| cot^(-1): domain R, range (0,pi). |
| KEY PROPERTIES: |
| sin^(-1)(x)+cos^(-1)(x)=pi/2 for x in [-1,1]. |
| tan^(-1)(x)+cot^(-1)(x)=pi/2 for all x. |
| sec^(-1)(x)+cosec^(-1)(x)=pi/2 for |x|>=1. |
| sin^(-1)(-x)=-sin^(-1)(x). cos^(-1)(-x)=pi-cos^(-1)(x). |
| tan^(-1)(-x)=-tan^(-1)(x). |
| sin^(-1)(1/x)=cosec^(-1)(x) for |x|>=1. |
| cos^(-1)(1/x)=sec^(-1)(x) for |x|>=1. |
| tan^(-1)(1/x)=cot^(-1)(x) for x>0. |
| ADDITION FORMULAS: |
| tan^(-1)(x)+tan^(-1)(y)=tan^(-1)((x+y)/(1-xy)) if xy<1. |
| tan^(-1)(x)+tan^(-1)(y)=pi+tan^(-1)((x+y)/(1-xy)) if xy>1 and x>0. |
| 2 tan^(-1)(x)=sin^(-1)(2x/(1+x^2))=cos^(-1)((1-x^2)/(1+x^2))=tan^(-1)(2x/(1-x^2)). |
| SOLVED EXAMPLES: |
| Example 1: Find principal value of sin^(-1)(-1/2). |
| sin^(-1)(1/2)=pi/6. sin^(-1)(-1/2)=-pi/6. (In range [-pi/2,pi/2]). |
| Example 2: Find principal value of cos^(-1)(-sqrt3/2). |
| cos^(-1)(sqrt3/2)=pi/6. cos^(-1)(-sqrt3/2)=pi-pi/6=5pi/6. (In range [0,pi]). |
| Example 3: Simplify tan^(-1)(cosx-sinx)/(cosx+sinx). |
| Divide by cosx: tan^(-1)((1-tanx)/(1+tanx))=tan^(-1)(tan(pi/4-x))=pi/4-x. |
| (Valid when pi/4-x is in range (-pi/2,pi/2).) |
| Example 4: Prove sin^(-1)x+cos^(-1)x=pi/2. |
| Let sin^(-1)x=theta. Then x=sin(theta). cos^(-1)x=cos^(-1)(sin theta)=cos^(-1)(cos(pi/2-theta))=pi/2-theta. |
| Sum=theta+pi/2-theta=pi/2. Proved. |
| COMMON MISTAKES: |
| sin^(-1)(x) is NOT 1/sin(x). It is the inverse FUNCTION. |
| Principal value of cos^(-1): range is [0,pi], NOT [-pi/2,pi/2]. |
| cos^(-1)(-x)=pi-cos^(-1)(x) (NOT -cos^(-1)(x) like sine). |
