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75. DEFINITION. To define the three principal Trigonometrical Ratios of an angle.

E

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Let ROE be an angle.

M

R

In OE one of the lines containing the angle take any point P, and from P draw PM perpendicular to the other line OR, or, if necessary, to RO produced.

Then, in the right-angled triangle OPM, the side MP, which is opposite the angle under consideration, is called the perpendicular.

The side OP, which is opposite the right angle, is called the hypotenuse.

The third side OM (which is adjacent to the right angle and to the angle under consideration) is called the base. Then the ratio

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NOTE. The order of the letters, in MP, OM and OP, indicates direction and decides their algebraical signs. [Art. 132.]

76. If A stand for the angle ROE, these ratios are called sine A, cosine A and tangent A, and are usually abbreviated thus:

sin A,

cos A,

tan A.

77. There are three

other Trigonometrical Ratios, formed by inverting the sine, cosine and tangent respectively, which are called the cosecant, secant, and cotangent respectively.

78. To define the three other Trigonometrical Ratios of any angle.

The same construction and figure as in Art. 75 being made, then the ratio

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79.

Thus if A stand as before for the angle ROE, these

ratios are called cosecant A, secant A, and cotangent A. They are abbreviated thus,

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81. The above definitions apply to an angle of any magnitude. (We shall return to this subject in Chapter X.) For the present the student may confine his attention to angles which are each less than a right angle.

82. The powers of the Trigonometrical Ratios are expressed as follows:

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It is

The student must notice that 'sin A' is a single symbol. the name of a number, or fraction, belonging to the angle A; and if it be at any time convenient, we may denote sin ▲ by a single letter, such as s or x. Also sin2 A is an abbreviation for (sin A)2, that is for (sin A) × (sin A). Such abbreviations are used because they are convenient.

83. The Trigonometrical Ratios are always the same for the same angle.

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Take any angle ROE; let P be any point in OE one of the lines containing the angle, and let P', '" be any two points in OR the other line containing the angle. Draw PM perpendicular to OR, and P'M', P"M" perpendiculars to OE.

Then the three triangles OMP, OM'P, OM"P" each contain a right angle, and they have the angle at O common; therefore their third angles must be equal. Thus the three triangles are equiangular.

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to the angle at O; that is, they are each sin ROE.

Thus, sin ROE is the same whatever be the position of the point P on either of the lines containing the angle ROE. Therefore sin ROE is always the same.

84. A similar proof holds good for each of the other ratios.

85. Also if two angles are equal, it is clear that the numerical values of their Trigonometrical Ratios will be the

same.

We have already shown (Art. 74), that the values of these ratios are different for different angles.

Hence for each particular value of A, sin A, cos A, tan A, etc. have definite numerical values.

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86. In the following examples the student should notice

(i) the angle referred to,

(ii) that there is a right angle in the same triangle as the angle referred to,

(iii) the perpendicular, which is opposite the angle referred to, and is perpendicular to one of the lines containing the angle,

(iv) the hypotenuse, which is opposite the right angle, (v) the base, the third side of the triangle.

Example. In the second figure on the next page, in which BDA is a right angle, find sin DBA and cos DBA.

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(ii) BDA is a right angle in the same triangle as the angle DBA.

(iii) DA is the perpendicular, for it is opposite DBA and is perpendicular to BD.

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(1) Let ABC be any triangle and let AD be drawn perpendicular to BC. Write down the perpendicular, and the base when the following angles are referred to: (i) the angle ABD, (ii) the angle BAD, (iii) the angle ACD, (iv) the angle DAC.

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