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Page 1 Page # 41 SOLUTION OF TRIANGLE 1. Sine Rule: C sin c B sin b A sin a ? ? . 2. Cosine Formula: (i) cos A = b c a b c 2 2 2 2 ? ? (ii) cos B = c a b ca 222 2 ? ? (iii) cos C = a b c a b 2 2 2 2 ? ? 3. Projection Formula: (i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA 4. Napier’s Analogy  tangent rule: (i) tan 2 C B ? = c b c b ? ? cot 2 A (ii) tan 2 A C ? = a c a c ? ? cot B 2 (iii) tan A B ? 2 = a b a b ? ? cot C 2 5. Trigonometric Functions of Half Angles: (i) sin A 2 = ( ) ( ) s b s c b c ? ? ; sin B 2 = ( ) ( ) s c s a ca ? ? ; sin C 2 = ( ) ( ) s a s b a b ? ? (ii) cos A 2 = s s a b c ( ) ? ; cos B 2 = s s b ca ( ) ? ; cos C 2 = s s c a b ( ) ? (iii) tan A 2 = ( ) ( ) ( ) s b s c s s a ? ? ? = ? s s a ( ) ? where s = a b c ? ? 2 is semi perimetre of triangle. (iv) sin A = ) c s)( b s)( a s ( s bc 2 ? ? ? = bc 2 ? Page 2 Page # 41 SOLUTION OF TRIANGLE 1. Sine Rule: C sin c B sin b A sin a ? ? . 2. Cosine Formula: (i) cos A = b c a b c 2 2 2 2 ? ? (ii) cos B = c a b ca 222 2 ? ? (iii) cos C = a b c a b 2 2 2 2 ? ? 3. Projection Formula: (i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA 4. Napier’s Analogy  tangent rule: (i) tan 2 C B ? = c b c b ? ? cot 2 A (ii) tan 2 A C ? = a c a c ? ? cot B 2 (iii) tan A B ? 2 = a b a b ? ? cot C 2 5. Trigonometric Functions of Half Angles: (i) sin A 2 = ( ) ( ) s b s c b c ? ? ; sin B 2 = ( ) ( ) s c s a ca ? ? ; sin C 2 = ( ) ( ) s a s b a b ? ? (ii) cos A 2 = s s a b c ( ) ? ; cos B 2 = s s b ca ( ) ? ; cos C 2 = s s c a b ( ) ? (iii) tan A 2 = ( ) ( ) ( ) s b s c s s a ? ? ? = ? s s a ( ) ? where s = a b c ? ? 2 is semi perimetre of triangle. (iv) sin A = ) c s)( b s)( a s ( s bc 2 ? ? ? = bc 2 ? Page # 42 6. Area of Triangle ( ?) : ? = 2 1 ab sin C = 2 1 bc sin A = 2 1 ca sin B = ss a sb sc ( )( )( ) ? ? ? 7. m  n Rule: If BD : DC = m : n, then (m + n) cot ? ? ? ? ? m n n B m C cot cot cot cot ? ? 8. Radius of Circumcirlce : R = C sin 2 c B sin 2 b A sin 2 a ? ? = ? 4 c b a 9. Radius of The Incircle : (i) r = ? s (ii) r = (s ? a) tan A 2 = (s ? b) tan B 2 = (s ? c) tan C 2 (iii) r = 4R sin A 2 sin B 2 sin C 2 10. Radius of The Ex Circles : (i) r 1 = ? s a ? ; r 2 = ? s b ? ; r 3 = ? s c ? (ii) r 1 = s tan A 2 ; r 2 = s tan B 2 ; r 3 = s tan C 2 (iii) r 1 = 4 R sin A 2 . cos B 2 . cos C 2 Page 3 Page # 41 SOLUTION OF TRIANGLE 1. Sine Rule: C sin c B sin b A sin a ? ? . 2. Cosine Formula: (i) cos A = b c a b c 2 2 2 2 ? ? (ii) cos B = c a b ca 222 2 ? ? (iii) cos C = a b c a b 2 2 2 2 ? ? 3. Projection Formula: (i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA 4. Napier’s Analogy  tangent rule: (i) tan 2 C B ? = c b c b ? ? cot 2 A (ii) tan 2 A C ? = a c a c ? ? cot B 2 (iii) tan A B ? 2 = a b a b ? ? cot C 2 5. Trigonometric Functions of Half Angles: (i) sin A 2 = ( ) ( ) s b s c b c ? ? ; sin B 2 = ( ) ( ) s c s a ca ? ? ; sin C 2 = ( ) ( ) s a s b a b ? ? (ii) cos A 2 = s s a b c ( ) ? ; cos B 2 = s s b ca ( ) ? ; cos C 2 = s s c a b ( ) ? (iii) tan A 2 = ( ) ( ) ( ) s b s c s s a ? ? ? = ? s s a ( ) ? where s = a b c ? ? 2 is semi perimetre of triangle. (iv) sin A = ) c s)( b s)( a s ( s bc 2 ? ? ? = bc 2 ? Page # 42 6. Area of Triangle ( ?) : ? = 2 1 ab sin C = 2 1 bc sin A = 2 1 ca sin B = ss a sb sc ( )( )( ) ? ? ? 7. m  n Rule: If BD : DC = m : n, then (m + n) cot ? ? ? ? ? m n n B m C cot cot cot cot ? ? 8. Radius of Circumcirlce : R = C sin 2 c B sin 2 b A sin 2 a ? ? = ? 4 c b a 9. Radius of The Incircle : (i) r = ? s (ii) r = (s ? a) tan A 2 = (s ? b) tan B 2 = (s ? c) tan C 2 (iii) r = 4R sin A 2 sin B 2 sin C 2 10. Radius of The Ex Circles : (i) r 1 = ? s a ? ; r 2 = ? s b ? ; r 3 = ? s c ? (ii) r 1 = s tan A 2 ; r 2 = s tan B 2 ; r 3 = s tan C 2 (iii) r 1 = 4 R sin A 2 . cos B 2 . cos C 2 Page # 43 11. Length of Angle Bisectors, Medians & Altitudes : (i) Length of an angle bisector from the angle A = ? a = 2 2 bc b c A cos ? ; (ii) Length of median from the angle A = m a = 1 2 2 2 2 2 2 b c a ? ? & (iii) Length of altitude from the angle A = A a = a 2 ? 12. Orthocentre and Pedal Triangle: The triangle KLM which is formed by joining the feet of the altitudes is called the Pedal Triangle. (i) Its angles are ? ? ? 2A, ? ? ? 2B and ? ? ? 2C. (ii) Its sides are a cosA = R sin 2A, b cosB = R sin 2B and c cosC = R sin 2C (iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal. Page 4 Page # 41 SOLUTION OF TRIANGLE 1. Sine Rule: C sin c B sin b A sin a ? ? . 2. Cosine Formula: (i) cos A = b c a b c 2 2 2 2 ? ? (ii) cos B = c a b ca 222 2 ? ? (iii) cos C = a b c a b 2 2 2 2 ? ? 3. Projection Formula: (i) a = b cosC + c cosB (ii) b = c cosA + a cosC (iii) c = a cosB + b cosA 4. Napier’s Analogy  tangent rule: (i) tan 2 C B ? = c b c b ? ? cot 2 A (ii) tan 2 A C ? = a c a c ? ? cot B 2 (iii) tan A B ? 2 = a b a b ? ? cot C 2 5. Trigonometric Functions of Half Angles: (i) sin A 2 = ( ) ( ) s b s c b c ? ? ; sin B 2 = ( ) ( ) s c s a ca ? ? ; sin C 2 = ( ) ( ) s a s b a b ? ? (ii) cos A 2 = s s a b c ( ) ? ; cos B 2 = s s b ca ( ) ? ; cos C 2 = s s c a b ( ) ? (iii) tan A 2 = ( ) ( ) ( ) s b s c s s a ? ? ? = ? s s a ( ) ? where s = a b c ? ? 2 is semi perimetre of triangle. (iv) sin A = ) c s)( b s)( a s ( s bc 2 ? ? ? = bc 2 ? Page # 42 6. Area of Triangle ( ?) : ? = 2 1 ab sin C = 2 1 bc sin A = 2 1 ca sin B = ss a sb sc ( )( )( ) ? ? ? 7. m  n Rule: If BD : DC = m : n, then (m + n) cot ? ? ? ? ? m n n B m C cot cot cot cot ? ? 8. Radius of Circumcirlce : R = C sin 2 c B sin 2 b A sin 2 a ? ? = ? 4 c b a 9. Radius of The Incircle : (i) r = ? s (ii) r = (s ? a) tan A 2 = (s ? b) tan B 2 = (s ? c) tan C 2 (iii) r = 4R sin A 2 sin B 2 sin C 2 10. Radius of The Ex Circles : (i) r 1 = ? s a ? ; r 2 = ? s b ? ; r 3 = ? s c ? (ii) r 1 = s tan A 2 ; r 2 = s tan B 2 ; r 3 = s tan C 2 (iii) r 1 = 4 R sin A 2 . cos B 2 . cos C 2 Page # 43 11. Length of Angle Bisectors, Medians & Altitudes : (i) Length of an angle bisector from the angle A = ? a = 2 2 bc b c A cos ? ; (ii) Length of median from the angle A = m a = 1 2 2 2 2 2 2 b c a ? ? & (iii) Length of altitude from the angle A = A a = a 2 ? 12. Orthocentre and Pedal Triangle: The triangle KLM which is formed by joining the feet of the altitudes is called the Pedal Triangle. (i) Its angles are ? ? ? 2A, ? ? ? 2B and ? ? ? 2C. (ii) Its sides are a cosA = R sin 2A, b cosB = R sin 2B and c cosC = R sin 2C (iii) Circumradii of the triangles PBC, PCA, PAB and ABC are equal. Page # 44 13. The triangle formed by joining the three excentres ? 1 , ? 2 and ? 3 of ? ABC is called the excentral or excentric triangle. (i) ? ABC is the pedal triangle of the ? ? 1 ? 2 ? 3 . (ii) Its angles are ? 2 2 ? A , ? 2 2 ? B & ? 2 2 ? C . (iii) Its sides are 4 R cos A 2 , 4 R cos B 2 & 4 R cos C 2 . (iv) ? ? ? 1 = 4 R sin A 2 ; ? ? ? 2 = 4 R sin B 2 ; ? ? ? 3 = 4 R sin C 2 . (v) Incentre ? of ? ABC is the orthocentre of the excentral ? ? 1 ? 2 ? 3 . 14. Distance Between Special Points : (i) Distance between circumcentre and orthocentre OH 2 = R 2 (1 – 8 cosA cos B cos C) (ii) Distance between circumcentre and incentre O ? 2 = R 2 (1 – 8 sin 2 A sin 2 B sin 2 C ) = R 2 – 2Rr (iii) Distance between circumcentre and centroid OG 2 = R 2 – 9 1 (a 2 + b 2 + c 2 )Read More
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