Trigonometrical Ratios – Summary Notes

INTRODUCTION The word ‘Trigonometry’ means measurement of triangles. In this unit, we shall be dealing with the relations of sides and angles of right-angled triangles only.

CONCEPT OF PERPENDICULAR, BASE AND HYPOTENUSE IN A RIGHT TRIANGLE. For any acute angle (which is also known as the angle of reference) in a right-angled triangle; the side opposite to the acute angle is called the perpendicular; the side adjacent to it is called the base and the side opposite to the right angle is called the hypotenuse. Examples:

NOTATION OF ANGLES To indicate an angle, any letter of the English alphabet can be used, but in trigonometry, in general, the following Greek letters are used: (i)   t h e t a (ii)   phi (iii)   alpha (iv)   b e t a (v)   gamma , etc.

TRIGONOMETRICAL RATIOS The ratio between the lengths of a pair of two sides of a right-angled triangle is called a trigonometrical ratio. The three sides of a right-angled triangle give six trigonometrical ratios; namely: sine, cosine, tangent, cotangent, secant and cosecant. In short, these ratios are written as: sin, cos, tan, cot, sec and cosec respectively.

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In a right-angled triangle ABC, for acute angle A: 1. sine (sin) is defined as the ratio between the lengths of perpendicular and hypotenuse. perpendicular BC sin A  hypotenuse AC 2. cosine (cos) is defined as the ratio between the lengths of base an d hypotenuse. base AB cos A   hypotenuse AC 3. tangent (tan) is defined as the ratio between the lengths of perpendicular and base. perpendicular BC tan A  base AB 4. cotangent (cot) is defined as the ratio between the lengths of base and perpendicular. base AB cot A  perpendicular BC 5. secant (sec) is defined as the ratio between the lengths of hypotenuse and base. hypotenuse AC sec A  base AB 6. cosecant (cosec) is defined as the ratio between the lengths of hypotenuse and perpendicular. hypotenuse AC cosec A  perpendcular BC Similarly, for acute angle C in the given right triangle, perpendcular AB (i) sinC  hypotenuse AC base BC (ii) cosC  hypotenuse AC perpendicular AB (iii) tan C  and so on. base BC Each trigonometrical ratio is a real number and has no unit.

RECIPROCAL RELATIONS perpendicular hypotenuse 1. Since, sin A  and cosec A  hypotenuse perpendicular sin Acosec A are reciprocal of each other. 1 1 sin A and cosec A  cosec A sin A

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1 1 Similarly, c o s A  and se c A  se c A c o s A 1 1 ta n A  and c o t A  c o t A ta n A perpendicular sin A hypotenuse perpendicular 2. Since,    tan A cos A base base hypotenuse sin A cos A  tan A  and cot A  . cos A sin A