CBSE Class 10 Answered

What are the trigonometric identities formula??

Asked by Man | 11 Sep, 2015, 10:37: PM

Expert Answer

Trigonometric ratios of acute angle A in right triangle ABC:

$1. space sin A equals fraction numerator s i d e space o p p o s i t e space t o space angle A over denominator h y p o t e n u s e end fraction equals p over h 2. space cos A equals fraction numerator s i d e space a d j a c e n t space t o space angle A over denominator h y p o t e n u s e end fraction equals b over h 3. space tan A equals fraction numerator s i d e space o p p o s i t e space t o space angle A over denominator s i d e space a d j a c e n t space t o space angle A end fraction equals p over b 4. space cos e c A equals fraction numerator h y p o t e n u s e over denominator s i d e space o p p o s i t e space t o space angle A end fraction equals h over p 5. space s e c A equals fraction numerator h y p o t e n u s e over denominator s i d e space a d j a c e n t space t o space angle A end fraction equals h over b 6. space c o t A equals fraction numerator s i d e space a d j a c e n t space t o space angle A over denominator s i d e space o p p o s i t e space t o space angle A end fraction equals b over p$

Each trigonometric ratio is a real number. It has no unit.

All the trigonometric symbols, cosine, sine, tangent, cotangent, secant and cosecant, have no literal meaning.

$open parentheses sin theta close parentheses to the power of n$ is generally written as $s i n to the power of n theta$ , n being a positive integer. Similarly, other trigonometric ratios can also be written.

The values of the trigonometric ratios of an angle do not vary with the length of the sides of the triangle, if the angles remain the same.

Pythagoras theorem: In a right triangle, square of the hypotenuse is equal to the sum of the square of the other two sides.

When any two sides of a right triangle are given, its third side can be obtained by using Pythagoras theorem.

Relation between trigonometric ratios:

$cos e c theta equals fraction numerator 1 over denominator sin theta end fraction s e c theta equals fraction numerator 1 over denominator cos theta end fraction c o t theta equals fraction numerator 1 over denominator tan theta end fraction equals fraction numerator cos theta over denominator sin theta end fraction tan theta equals fraction numerator sin theta over denominator cos theta end fraction$

Values of Trigonometric ratios of some specific angles:

ÐA	0°	30°	45°	60°	90°
sin A	0	$1 half$	$fraction numerator 1 over denominator square root of 2 end fraction$	$fraction numerator square root of 3 over denominator 2 end fraction$	1
cos A	1	$fraction numerator square root of 3 over denominator 2 end fraction$	$fraction numerator 1 over denominator square root of 2 end fraction$	$1 half$	0
tan A	0	$fraction numerator 1 over denominator square root of 3 end fraction$	1	$square root of 3$	Not defined
cosec A	Not defined	2	$square root of 2$	$fraction numerator 2 over denominator square root of 3 end fraction$	1
sec A	1	$fraction numerator 2 over denominator square root of 3 end fraction$	$square root of 2$	2	Not defined
cot A	Not defined	$square root of 3$	1	$fraction numerator 1 over denominator square root of 3 end fraction$	0

The value of sin A or cos A never exceeds 1, whereas the value of sec A or cosec A is always greater than 1 or equal to 1.

The value of sin $theta$ increases from 0 to 1 when $theta$ increases from 0⁰ to 90⁰.

The value of cos $theta$ decreases from 1 to 0 when $theta$ increases from 0⁰ to 90⁰.

Trigonometric ratios of complementary angles:

sin (90^o – A) = cos A
cos (90^o – A) = sin A
tan (90^o – A) = cot A
cot (90^o – A) = tan A
sec (90^o – A) = cosec A
cosec (90^o – A) = sec A

An equation involving trigonometric ratios of an angle, say, is termed as a trigonometric identity if it is satisfied by all values of $theta$ .

Basic trigonometric identities:

$1. space sin squared theta plus cos squared theta equals 1 2. space 1 plus tan squared theta equals s e c squared theta 3. space 1 plus c o t squared theta equals cos e c squared theta$