sin (A - C) (1) (3) 5. Prove that in every circle the angle subtended by an arc equal to the radius is an invariable angle. Express that angle in degrees and decimals of degrees, and find the circular measure of one minute. On a circle 10 feet in radius it was found that an angle of 22° 30′ was subtended by an arc 3 feet 11 inches in length; hence calculate, to four decimal places, the numerical ratio of the circumference of a circle to its diameter. 6. If a, b, c be the sides subtending respectively the angles A, B, C of a plane triangle, Prove : sin A sin (B - A) cos B cos A + (1) r= a 2√6c 4. 2 sin (C - B) cos C cos B + b (3) tan-1 +tan-1 a + c 7. If (r) be the radius of the circle inscribed in the triangle ABC, prove A с 2 2 2 area a+b+c =45° when C is a right angle. b. sin (2) r = =0. COS .sin B 2 8. In a right-angled triangle ABC, C being the right angle, find AB, if B=30° and BC= 100 feet. A'B'C' is also a right-angled triangle, C' the right angle and B' = 30°, find A'C' when the area of the triangle A'B'C' is three times the area of ABC. 9. When two sides and an included angle of a triangle are given, investigate a formula for determining the other two angles. Shew that for this determination it is not necessary to know the absolute lengths of the two sides, provided their ratio is given. log102 3010300, Log cot 35° 15' = 10*1507464, Log tan 19° 28′ 50′′ = 10*5486864. Ex. The included angle is 70° 30', the ratio of the containing sides is 53, find the other angles. Given II. Given that the number of combinations of n things taken 4 together is to the number of combinations of n-1 things taken 5 together as 5: 3, find n. 12. Assuming the form of the continued product of any number of binominal factors, deduce the truth of the binomial theorem for a positive integral value of the index. Shew that three consecutive terms of the expansion of (1+x)" can be in continued proportion only when n+1=0. 13. Expand (1 – 4x) to five terms; and shew that the general term may be thrown into the form 2r+ I (12)2 * *. IV. PLANE TRIGONOMETRY. (Including Solution of Triangles.) [N.B.-Great importance will be attached to accuracy in numerical results.] I. In plane trigonometry state how a right angle is usually divided by English and French mathematicians respectively. If the circumferences of the quadrants of two circles be divided similarly to the right angles they subtend, what would be the radius of a circle divided according to the French scale, in which the length of the arc of one grade would be equal to the length of the arc of one degree on a circle whose radius was 18 feet? Express in the French scale (1) the sum of the angles of a quindecagon, (2) one of the angles when the quindecagon is equiangular. 2. According to the ratio definition, point out which of the elementary trigonometrical functions are never less than unity, and which may be either less or greater than unity. Prove (sin A)2 + (cos A)2 = 1, and express the numerical values of sin 135° and tan 150° with their proper signs. 3. If (A+B) is an angle in the first quadrant, obtain by means of a geometrical figure the formula cos (A+B) = =cos A cos B- sin A sin B, and deduce from it the expression for cos (A – B). Find sec (A+B) in terms of sec A and sec B, and prove sec 105° - √√√2(1 + √√3). sin (A - C) (1) (3) 5. Prove that in every circle the angle subtended by an arc equal to the radius is an invariable angle. Express that angle in degrees and decimals of degrees, and find the circular measure of one minute. On a circle 10 feet in radius it was found that an angle of 22° 30' was subtended by an arc 3 feet 11 inches in length; hence calculate, to four decimal places, the numerical ratio of the circumference of a circle to its diameter. 6. If a, b, c be the sides subtending respectively the angles A, B, C of a plane triangle, Prove: sin A a b+c + (1)_r= = sin (B - A) cos B cos A a с 2 area a+b+c sin (C - B) cos C cos B + 2√bc b-c (2) a = =(b−c) sec 0, if tan 0= b (3) tan-1 +tan-1 a + c 7. If (r) be the radius of the circle inscribed in the triangle ABC, prove A с 2 2 sin (2) r = =45° when C is a right angle. =0. 8. In a right-angled triangle ABC, C being the right angle, find AB, if B=30° and BC= 100 feet. A'B'C' is also a right-angled triangle, C' the right angle and B' = 30°, find A'C' when the area of the triangle A'B'C' is three times the area of ABC. log102 3010300, Log cot 35° 15' = 10*1507464, Log tan 19° 28′ 50′′ = 10*5486864. 9. When two sides and an included angle of a triangle are given, investigate a formula for determining the other two angles. Shew that for this determination it is not necessary to know the absolute lengths of the two sides, provided their ratio is given. Ex. The included angle is 70° 30', the ratio of the containing sides is 5 3, find the other angles. Given II. Given that the number of combinations of n things taken 4 together is to the number of combinations of n-1 things taken 5 together as 5: 3, find n. 12. Assuming the form of the continued product of any number of binominal factors, deduce the truth of the binomial theorem for a positive integral value of the index. Shew that three consecutive terms of the expansion of (1+x)" can be in continued proportion only when n+1=0. (Including Solution of Triangles.) [N.B.-Great importance will be attached to accuracy in numerical results.] I. In plane trigonometry state how a right angle is usually divided by English and French mathematicians respectively. If the circumferences of the quadrants of two circles be divided similarly to the right angles they subtend, what would be the radius of a circle divided according to the French scale, in which the length of the arc of one grade would be equal to the length of the arc of one degree on a circle whose radius was 18 feet? Express in the French scale (1) the sum of the angles of a quindecagon, (2) one of the angles when the quindecagon is equiangular. 2. According to the ratio definition, point out which of the elementary trigonometrical functions are never less than unity, and which may be either less or greater than unity. Prove (sin A)2 + (cos A)2 = 1, and express the numerical values of sin 135° and tan 150° with their proper signs. 3. If (A+B) is an angle in the first quadrant, obtain by means of a geometrical figure the formula cos (A+B) = =cos A cos B – sin A sin B, and deduce from it the expression for cos (A – B). Find sec (A+B) in terms of sec A and sec B, and prove sec 105° - √√√2(1 + √3). 5. Prove that in every circle the angle subtended by an arc equal to the radius is an invariable angle. Express that angle in degrees and decimals of degrees, and find the circular measure of one minute. On a circle 10 feet in radius it was found that an angle of 22° 30′ was subtended by an arc 3 feet 11 inches in length; hence calculate, to four decimal places, the numerical ratio of the circumference of a circle to its diameter. 6. If a, b, c be the sides subtending respectively the angles A, B, C of a plane triangle, Prove : (2) a= + a b+c sin (B - A) cos B cos A = (1) r= a 2√bc 4. с sin (C- B) b (3) tan-1 +tan-1 a + c 7. If (r) be the radius of the circle inscribed in the triangle ABC, prove A с b. sin sin 2 2 2 area a+b+c =0. =45° when C is a right angle. 8. In a right-angled triangle ABC, C being the right angle, find AB, if B=30° and BC= 100 feet. A'B'C' is also a right-angled triangle, C' the right angle and B' = 30°, find A'C' when the area of the triangle A'B'C' is three times the area of ABC. 9. When two sides and an included angle of a triangle are given, investigate a formula for determining the other two angles. Shew that for this determination it is not necessary to know the absolute lengths of the two sides, provided their ratio is given. Ex. The included angle is 70° 30', the ratio of the containing sides is 53, find the other angles. Given log102 3010300, Log cot 35° 15' = 10*1507464, Log tan 19° 28′ 50′′ = 10*5486864. |
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