limit x = 0

It only remains to find the value of k, and this can be obtained by dividing the last equation through by x and letting a approach 0 indefinitely, when we have

By the aid of these series the trigonometric functions of any angle are readily calculated.

In the computation it

must be remembered that x is the circular measure of the given angle.

EXERCISE XXVIII.

Verify by the series just obtained that

1. sin2x+cos2x = 1.

2. sin(x)=- sin x, cos(x)cos x.

3. sin 2x 2 sin x cos x.

4. cos 2x=1-2 sin2x.

5. Find the series for secx as far as the term containing the 6th power of x.

6. Find the series for x cot x, noting that x cot x =

sin x

COS X.

7. Calculate sin 10° and cos 10° to 6 places of decimals. 8. Calculate tan 15° to 6 places of decimals.

From the exponential values of sin x and cos x show that 9. cos 3x=4 cos3x

10. sin 3x3 sin x

3 cos x.

4 sin3x.

SPHERICAL TRIGONOMETRY.

CHAPTER VII.

THE RIGHT SPHERICAL TRIANGLE.

§ 48. INTRODUCTION.

THE object of Spherical Trigonometry is to show how spherical triangles are solved. To solve a spherical triangle is to compute any three of its parts when the other three parts are given.

The sides of a spherical triangle are arcs of great circles. They are measured in degrees, minutes, and seconds, and therefore by the plane angles formed by radii of the sphere drawn to the vertices of the triangle. Hence, their measures are independent of the length of the radius, which may be assumed to have any convenient numerical value; as, for example, unity.

The angles of the triangle are measured by the angles made by the planes of the sides. Each angle is also measured by the number of degrees in the arc of a great circle, described from the vertex of the angle as a pole, and included between its sides.

The sides may have any values from 0° to 360°; but in this work only sides that are less than 180° will be considered. The angles may have any values from 0° to 180°.

If any two parts of a spherical triangle are either both less than 90° or both greater than 90°, they are said to be alike in kind; but if one part is less than 90°, and the other part greater than 90°, they are said to be unlike in kind.

Spherical triangles are said to be isosceles, equilateral, equiangular, right, and oblique, under the same conditions as plane triangles. A right spherical triangle, however, may have one, two, or three right angles.

When a spherical triangle has one or more of its sides equal to a quadrant, it is called a quadrantal triangle.

It is shown in Solid Geometry, that in every spherical triangle

I. If two sides of a spherical triangle are unequal, the angles opposite them are unequal, and the greater angle is opposite the greater side; and conversely.

II. The sum of the sides is less than 360°.

III. The sum of the angles is greater than 180° and less than 540°.

IV. If, from the vertices as poles, arcs of great circles are described, another spherical triangle is formed so related to the first triangle that the sides of each triangle are supplements of the angles opposite them in the other triangle.

Two such triangles are called polar triangles, or supplemental triangles.

Let A, B, C (Fig. 37) denote the angles of one triangle;

EXERCISE XXIX.

1. The angles of a triangle are 70°, 80°, and 100°; find the sides of the polar triangle.

2. The sides of a triangle are 40°, 90°, and 125°; find the angles of the polar triangle.

3. Prove that the polar of a quadrantal triangle is a right triangle.

4. Prove that, if a triangle has three right angles, the sides of the triangle are quadrants.

5. Prove that, if a triangle has two right angles, the sides opposite these angles are quadrants, and the third angle is measured by the number of degrees in the opposite side.

6. How can the sides of a spherical triangle, given in degrees, be found in units of length, when the length of the radius of the sphere is known?

7. Find the lengths of the sides of the triangle in Example 2, if the radius of the sphere is 4 feet.

§ 49. FORMULAS RELATING to Right SPHERICAL TRIANGLES.

As is evident from § 48, Examples 4 and 5, the only kind of right spherical triangle requiring further investigation is that which contains only one right angle.

Let ABC (Fig. 38) be a right spherical triangle having

only one right angle; and let A, B, C denote the angles of the triangle; a, b, c, respectively, the opposite sides.

Let C be the right angle; and for the present suppose that each of the other parts is less than 90°, and that the radius of the sphere is 1.

Let planes be passed through the sides, intersecting in the radii OA, OB, and OC.

E

B

FIG. 38.

a

A