to the greater angle (Geom. Th. 9). 3. The sum of the angles must be exactly two right angles. In spherical triangles, the first two principles also apply (Spher. Geom., App. III. p. 2, and Prop. 7). 4. The sum of the three angles must not be less than two, nor greater than six right angles (Spher. Geom., Prop. 14). 5. The sum of the three sides must be less than a circumference (Spher. Geom., Prop. 8). 6. Each side must be less than a semicircumference (Spher. Geom., Prop. 8, Note). 7. Ench angle must be less than two right angles (Spher. Geom., Prop. 8, Note). EXAMINATION QUESTIONS IN TRIGONOMETRY. What is the object of Trigonometry? How many elements are there in a triangle, and what are they? How many elements must be given in order to determine the rest? In plane triangles what must one element always be? Why? What is the difference between Geometrical and Trigonometrical solutions? Are trigonometrical solutions perfectly accurate? Whence arises the very small inaccuracy? Ans. From the decimals neglected in calculating tables of logarithms. How is the circumference of a circle divided for the purposes of Trigonometry? What is the complement of an angle or arc? What is the supplement? What are complements of each other in a right angled triangle? What is the sine of an are? What is the cosine ? The tangent? Cotangent? Secant? Cosecant ? What trigonometrical line changes its sign with the sine? Ans. The cosecant. In which quadrant are they negative ? In passing through what values do quantities generally change their signs? Ans. Zero and infinity. What is the least value of the sine? Where is it 0 ? What is the greatest value of the sine? Where is it radius? How many times does it change its sign in going round the circumference? What is the least value of the tangent? Where is it 0? What is its greatest value? Where is it infinite? How many times does it change in going round the circumference! N. B. Let these questions be repeated for the secant, cosine, cotangent, and To what is the sine of a negative arc equal? The cosine of a negative arc? The Tangent? Secant? Cotangent? Cosecant? To what is the sine of the supplement of an arc equal? The Tangent? Secant? Cotangent? Cosecant ? The cosine of 900 plus an arc ? The Tangent? Secant? Cotangent? Cosecant ? What formula expresses the relation between the sine and cosine of an arc? Ans. Rsin+cos2. What is the expression for the tangent in terms of the sine and cosine? How are the tangents of two arcs to each other? To what is the tangent equal in terms of the cotangent? Ans. cal of the cotangent. Ans. Sin (a + b) = sin What is the formula for the sine of the sum of two arcs? a cos b+ sin b cos a or the sum of the rectangles of the alternate sines and cosines. The formula for the sine of the difference? Ans. Sin (a — b) = sin a cos b— sin b cos a. For the cosine of the sum? Ans. Cos (a+b) = cos a cos b difference of the rectangles of the cosines and sines. sin a sin b or the For the cosine of the difference? Ans. Cos (a - b) = cos a cos b+ sin a sin b. For the sine of an arc in terms of half the arc ? Ans. Sin a=2 sina cosa, or twice the sine of half the arc into the cosine of half the arc. From what is this formula deduced? The formula for the cosine in terms of half the arc? Ans. Cos a=cos2 a — sina. Whence derived? The formula for the sine of half an arc. Ans. Sina√} — § cosa. For the cosine of half an arc? Ans. Cosa =√} + § cos a. For the sum of the sines? Ans. Sin p+ sin q= 2 sin § ( p + q) cos § (p −q), or twice the sine of half the sum into the cosine of half the difference. The difference of the sines? Ans. Sin p-sin q=2 cos(p+q) sin What is the ratio of the sum of the sines to the difference of the sines ? Of the sine of the sum to the difference of the sines? Ans. sin (p + q) What is the formula for the sine in terms of the tangent? Ans. sin a= tan a √1+tan2 a What is the formula for the tangent of the sum of two arcs? Ans. tan (a+b) tan atan b tan atan b 1+tan a tan b From the tangent of the sum how is the tangent of twice an arc found? Of three times an arc ? For the tangent of the difference? Ans. tan (a — b) = RESOLUTION OF RIGHT ANGLED PLANE TRIANGLES. What are the three formulas for the solution of right angled triangles? = Ans. (1) Radius: the hypothenuse sine of one of the acute angles: the side opposite cosine: the side adjacent ; or radius being unity, hypothenuse X sine of either acute angle: side opposite and hyp. X cos of either angle: side adjacent. (2) R either of the perpendicular sides :: tangent of the angle adjacent or cotangent of the angle opposite: the other side, or R being 1, one perp. side X tan of adjacent angle = side opp. = (3) Square of the hypothenuse sum of the squares of the other two sides. Square of either perp. side rectangle of sum and dif. of the other two sides. In a right angled triangle how many elements must be given? Ans. Two. Why should each required element be found in terms of the two given ? In finding each unknown element how many logarithms will be employed? When the logarithms are added what must be rejected from their sum? When one is subtracted from the other what must be added to the latter? When the hypothenuse is given or required with an angle, which formula is employed? When the hypothenuse is neither given nor required? When two sides are given to find the third ? RESOLUTION OF PLANE TRIANGLES IN GENERAL. Two sides and the included angle of a triangle being given, how are the other elements determined? Ans. a+b: a − b : : tan} (A + B) : tan § (A — B) A+B 180° - C = } (A + B) + } (A — B) — A, } (A + B) — } (A — B) = B Two angles and the interjacent side being given? Ans. 180° — (A + B) = C. sin Cc sin A: a:: sin B: 6 Two of the three given parts being a side and its opposite angle? Ans. By the sine proportion, or sines of the angles are as the opposite sides. What is the formula for the cosine of an angle in terms of the three sides of a plane b2 + c2 — a2 triangle? Ans. Cos A 2bc or the sum of the squares of the sides which contain it, minus the square of the opposite side, divided by twice the rectangle of the containing sides. Which is the fundamental formula in Spherical Trigonometry? SOLUTION OF RIGHT ANGLED SPHERICAL TRIANGLES. Upon what are Napier's rules founded? Ans. Upon the formulas in right angled spherical trigonometry. How many parts are considered for the application of his rules? What are they? Ans. The base, perpendicular, the complement of the hypothenuse, and the complements of the two oblique angles. What are the rules? Ans. Sin of the middle part = product of the cosines of the opposite parts product of the tangents of the adjacent parts. = N. B. Radius must be introduced homogeneously. When two parts are given how are the rules applied? SOLUTION OF SPHERICAL TRIANGLES IN GENERAL. Three sides of a spherical triangle being given how are the three angles found? * All the formulas for the solution of spherical triangles may be derived from this; for applied to the three angles it gives three equations containing the six elements of the triangle from which any two elements being eliminated, an equation results containing the other four elements. Why is not the formula for the cosine of an angle in terms of the three sides suitable for the application of logarithms ? The three angles being given how are the three sides found? Ans. Sina R cos Two sides and the included Ans. By Napier's analogies. (A+B), sin (a + b) sin Cos angle being given how are the other parts found? (a + b): cos (a - b):: cotC tan (a - b): cot C: tan(A-B). And then ↓ (A + B) + ǹ (A — B) = A and § (A + B) — § (A — B) = B, and finally the sine proportion sin A: sin B:: sin C: sin c. set. Two angles and the interjacent side being given? Ans. Napier's Analogies, 2d Cos (A+B): cos § (A — B) :: tan c: tan } (a + b) Sin (A+B): sin } (A — B) :: tanc: tan† (a — b) § (a + b) + 1 (a — b) = a and } (a + b) — § (a — b) = b, sin a : sin A :: sin c : sin C. Two of the three given parts being a side and its opposite angle? Ans. From the vertex of that unknown angle which is opposite the unknown side, let fall a perpen. dicular upon this side and apply Napier's rules to the two right angled triangles thus formed. The parts of the given triangle are by this means found either directly, or by adding the parts of the two right angled triangles together. QUESTIONS ON LOGARITHMS. What is a logarithm? What is the constant number which is raised to a power called? To what is the logarithm of the base equal? To what is the logarithm of unity equal? What is the base of the common system? In the common system what is the logarithm of 100? Of 1000 ? Of all numbers between 100 and 1000? Of all numbers between 1000 and 10,000? What is the entire part of a logarithm called? How does it compare with the number of digits in the number to which the logarithm belongs? How is the logarithm of a number consisting of three figures found from the tables? Of one of four? Of one of more than four? By the tables of Callet? How is the number corresponding to any given logarithm found from the tables? What is the rule for multiplication by logarithms? For division? |