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**Facts that Matter**

**Trigonometric Ratios**

The certain ratios involving the sides of a right triangle are called trigonometric ratios. Look at the adjoining right triangle ABC, right angled at B. The Trigonometric Ratios of the angle A are:

Reciprocals of the above t-ratios are:

NOTE:I. â€˜sin Î¸ â€™ is a single symbol and â€˜sinâ€™ cannot be detached from â€˜Î¸ â€™. sin Î¸ â‰ sin Ã— Î¸. This remark is true for other t-ratios also.

II. The values of the trigonometric ratios of an angle depend only on the magnitude of the angle and not on the lengths of the sides of the triangle.

III. In a right angle, the hypotenuse is the longest side, therefore, the value of sin A or cos A is always less than 1 or at the most equal to 1.

**Trigonometric Ratios of Some Specific Angles**

The specific angles are 0Â°, 30Â°, 45Â°, 60Â° and 90Â°. Trigonometric ratios of these angles are given in the following table:

NOTE:I. The value of sin A increases from 0 to 1, as A increases from 0Â° to 90Â°.

II. The value of cos A decreases from 1 to 0, as A increases from 0Â° to 90Â°.

III. The value of tan A increases from 0 to âˆž , as A increases from 0Â° to 90Â°.

IV. âˆš2 = 1.414 and âˆš3 = 1.732.

**Trigonometric Identities**

Since an equation is called an identity when it is true for all the values of the variables involved.

So, an equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angles involved. Some of the useful trigonometric identities:

(i) cos^{2}A + sin^{2}A = 1

(ii) 1 + tan^{2}A = sec^{2}A

(iii) cot^{2}A + 1 = cosec^{2}A**Trigonometric Ratios of Complementary Angles**

Two angles are said to be complementary if their sum equals 90Â°.

In a right angled Î”, if âˆ B = 90Â°, then âˆ A + âˆ C = 90, i.e. âˆ A and âˆ C form a pair of complementary angles. Thus, âˆ A and (90Â°â€“ âˆ A) are complementary angles.

Let us remember that:

(i) sin (90Â° âˆ’ A) = cos A

(ii) cos (90Â° âˆ’ A) = sin A

(iii) tan (90Â° âˆ’ A) = cot A

(iv) cot (90Â° âˆ’ A) = tan A

(v) sec (90Â° âˆ’ A) = cosec A

(vi) cosec (90Â° âˆ’ A) = sec A

NOTE:I. tan 0Â° = cot 90Â° = 0

II. sec 0Â° = cosec 90Â° = 1

III. sec 90Â°, cosec 0Â°, cot 0Â° and tan 90Â° are not defined.

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