CXIX.

SPHERICAL TRIGONOMETRY.

1. In a spherical triangle, having given a=70°, b=40°, c=38° 30', calculate numerically the angle A; assuming a suitable formula.

2. In a right-angled spherical triangle, if e denote the side opposite to the right angle, show that

cos2c=cos2 a cos2 b+sin2 a sin2 b.

3. Are the three angles of a spherical triangle sufficient data to enable us to determine the angular measure of the arcs which are the sides of the triangle? Are they sufficient to enable us to determine the lineal measure?

4. In any spherical triangle, a, b, c being the sides, A, B, C the angles, prove that

5. Find the area of a spherical triangle in terms of the spherical excess; and obtain an expression for the latter, in a form adapted to logarithmic calculation, in terms of the sides.

6. Prove the following formula of spherical trigonometry, and adapt it to logarithmic computation :

cos c=cos a cos b+sin a sin b cos C.

7. In any right-angled spherical triangle

9. If a and b are the sides, c the hypothenuse of a rightangled spherical triangle, prove that cos c=cos a

cos b. 10. In any spherical triangle show that the sines of the angles are proportional to the sines of the opposite sides. 11. Give Napier's rules for the solution of right-angled spherical triangles.

12. If a, b, c, be the sides of a spherical triangle, and if the arc be drawn from the angle A to bisect the side a,

Prove cos cos d=cos

b+c b-c

COS

13. In spherical trigonometry, state and prove the fundamental properties by which the primitive and polar triangles are related to each other.

14. Find generally the cosine of an angle of a spherical triangle in terms of the sides, and thence the cosine of a side in terms of the angles.

15. The angles of a spherical triangle of the surface of the earth are 42° 2′ 35′′, 67° 55′ 38′′, 70° 1′ 48′′. Find the area of the triangle in square miles, taking the earth's radius at 4000 miles.

16. Define the polar triangle of a spherical triangle, and establish the relation between the sides and angles of the two triangles.

17. Prove that in a triangle ABC

cos acos b cos c+sin b sin c cos A;

and, by help of the polar triangle, deduce a relation between two angles and a side of the triangle.

18. In a spherical triangle, prove the formula

(1) cot a sin b=cot A sin C+cos b cos C.

(2) cos a sin b=cos A sin C+cos b cos C sin a.

19. Enunciate Napier's rules for the solution of a rightangled triangle, and prove these rules for the case in which one of the angles is the middle part.

20. In a triangle ABC, right-angled at C, the length of the arc of a great circle from C, perpendicular to AB, is 8. Prove that

cot2 8=cot a+cot2 b.

21. If two sides and the included angle of a spherical triangle be given, show how to find the other parts of the triangle.

22. If the sides of a spherical triangle be small compared with the radius of a sphere, prove that each angle of the spherical triangle exceeds by one-third of the spherical excess the corresponding angle of the plane triangle, the sides of which are of the same length as the arcs of the spherical triangle. Explain the use of this theorem in the case of a spherical triangle in which two sides and the included angle are known.

23. If two sides of a spherical triangle be given, determine the relation between the small variations of any other pair of elements of the triangle.

24. Define a spherical triangle, and prove that the three angles of any spherical triangle are together greater than two right angles, and less than six right angles. Show also that any two sides are greater than the third side. 25. Prove the following relations in any spherical triangle :

26. Explain fully the ambiguous case in the solution of a right-angled spherical triangle.

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COORDINATE GEOMETRY.

CXX.

1. Find the equation to a straight line passing through a given point, and at right angles to a given straight line. 2. All the parallelograms circumscribing a given ellipse are equal to one another.

3. Find the coordinates of the points in which the line 3x+7y=1 meets the two rectangular axes.

y

4. If 2+2=1 be the equation to a straight line, what do

α

a and b represent? What is the condition that two straight lines shall be at right angles to each other?

5. Show that 2y2-3xy-2x2-7y-x+3=0 represents two straight lines at right angles to each other.

6. Find the length of the perpendicular let fall from the point (3, 5) upon the line 7x-3y=9.

7. What is the equation of a right line passing through the two given points (x', y'), (x", y'') ?

8. What is the length of the perpendicular drawn from the point (x', y') upon the line x cos a+y sin a—

axes being rectangular ?

a-p=0, the

9. What is the area of the triangle formed by joining the three points (x', y'), (x'', y''), (x''', y''') ?

10. Given the coordinates of the extremities of a right line, find the coordinates of its middle point.

11. Find the length of the perpendicular drawn from the point (3, 4) on the right line 3-2-1-0, the axes being rectangular.

4

12. Find the equation to a straight line (1) drawn from the origin and making an angle of 45° with the straight line y=ax+b, (2) to a straight line bisecting the angle between two given straight lines passing through the origin, and explain why the problem admits two solutions.

13. The equation 3y-8xy-3x2-9y+7x+6=0 represents two straight lines at right angles to each other. 14. Determine the geometrical signification of the equation

10x2-xy-21y2-9x-y+2=0.

15. Find the equation to the straight line passing through the origin and making an angle of 60° with the line x+y√3-1=0.

16. Draw the figure representing the position of the straight line whose equation referred to rectangular coordinates is 3y—5x+7=0.

17. Find the angle between the lines whose equations referred to rectangular coordinates are y-2x+3=0 and y+x-2=0.

18. Find the equation to the right line perpendicular to the line whose equation is y=mx, the axes being rectangular.

19. Determine the angle between the lines y=mx+n and y=m'x+n'.

20. Find the point of intersection of the two lines—

3x-2y+5=0, 4x+3y=0;

find also the distance of this point from the line 2y=x-1.

CXXI.

1. Find the general rectangular equation to a given circle: construct the curve x2+y2-6x-10y-15=0; and find the equation to the diameter passing through the origin.

2. Find the equation to the circle with radius r having its centre on the axis of x, and passing through the origin. 3. When does the equation

Ay2+Bxy+Cx2+ Dy + Ex+F=0 represent a circle, the axes being oblique ?

4. What is the equation to the circle circumscribing the