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Class 10 Maths Concepts for chapter Trignometry



What is Trigonometry

Trigomometry from Greek trigõnon, "triangle" and metron, "measure") is a branch of mathematics that studies relationships involving lengths and angles of triangles. The field emerged during the 3rd century BC from applications of geometry to astronomical studies.

Trigonometry is most simply associated with planar right angle triangles (each of which is a two-dimensional triangle with one angle equal to 90 degrees). The applicability to non-right-angle triangles exists, but, since any non-right-angle triangle (on a flat plane) can be bisected to create two right-angle triangles, most problems can be reduced to calculations on right-angle triangles. Thus the majority of applications relate to right-angle triangles

Right triangle trigonometry and Trigonometric ratios

In a right angle triangle ABC where B=90° ,

trigonometric ratios

We can define following term for angle A

Base :  Side adjacent to angle

Perpendicular: Side Opposite of angle

Hypotenuse: Side opposite to right angle

We can define the trigonometric ratios for angle A as

sin A= Perpendicular/Hypotenuse =BC/AC

cosec A= Hypotenuse/Perpendicular =AC/BC

cos A= Base/Hypotenuse =AB/AC

sec A= Hypotenuse/Base=AC/AB

tan A= Perpendicular/Base =BC/AB

cot A= Base/Perpendicular=AB/BC

Notice that each ratio in the right-hand column is the inverse, or the reciprocal, of the ratio in the left-hand column.

The reciprocal of sin A is cosec A ; and vice-versa.

The reciprocal of cos A is sec A

And the reciprocal of tan A is cot A

These are valid for acute angles.

We can define tan A = sin A/cos A

And Cot A =cos A/ Sin A

Important Note

Since the hypotenuse is the longest side in a right triangle, the value of

sin A or cos A is always less than 1 (or, in particular, equal to 1).

Similarly we can have define these for angle $C$

trigonometric ratios

We can define the trigonometric ratios for angle $C$  as

sin C= Perpendicular/Hypotenuse =AB/AC

cosec C= Hypotenuse/Perpendicular =AC/ABB

cos C= Base/Hypotenuse =BC/AC

sec C= Hypotenuse/Base=AC/BC

tan A= Perpendicular/Base =AB/BC

cot A= Base/Perpendicular=BC/AB

Trigonometric Ratios of Common angles

We can find the values of trigonometric ratio’s various angle

trig reference table

Trigonometric ratio’s of complimentary angles

We know that for Angle A, the complementary angle is 90 – A

In a right angle triangle ABC

A+B+C=180

Now B=90

So A +C =90

Or C=90-A

We have see in Previous section the value for  trigonometric ratios for angle $C$

sin C= Perpendicular/Hypotenuse =AB/AC

cosec C= Hypotenuse/Perpendicular =AC/ABB

cos C= Base/Hypotenuse =BC/AC

sec C= Hypotenuse/Base=AC/BC

tan C= Perpendicular/Base =AB/BC

cot C= Base/Perpendicular=BC/AB

This can be rewritten as

sin (90-A)  =AB/AC

cosec (90-A)  =AC/AB

cos (90-A) =BC/AC

sec (90-A)=AC/BC

tan (90-A)=AB/BC

cot( 90-A) =BC/AB

Also we know that

sin A=BC/AC

cosec A= AC/BC

cos A= AB/AC

sec A =AC/AB

tan A =BC/AB

cot A =AB/BC

From both of these, we can easily make it out

Sin (90-A) =cos(A)

Cos(90-A) = sin A

Tan(90-A) =cot A

Sec(90-A)= cosec A

Cosec (90-A) =sec A

Cot(90- A) =tan A

Trigonometric identities

Sin2 A + cos2 A  =0

1 + tan2 A =sec2 A

1 + cot2 A =cosec2 A


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