CHAPTER 8:INTRODUCTION TO TRIGNOMETRY

The notes for trigonometry class 10 Maths is provided here. Get the complete concept on trigonometry which is covered in Class 10 Maths. Also, get the various trigonometric ratios for specific angles, the relationship between trigonometric functions, trigonometry tables, various identities are given below.

Trigonometric Ratios

Opposite & Adjacent Sides in a Right Angled Triangle

In the $\Delta ABC$ right-angled at B, BC is the side opposite to $\angle A,$ AC is the hypotenuse and AB is the side adjacent to $\angle A.$

Trigonometric Ratios

For the right $\Delta ABC$, right-angled at $\angle B$, the trigonometric ratios of the $\angle A$ are as follows:

• sin A=opposite side/hypotenuse=BC/AC
• cos A=adjacent side/hypotenuse=AB/AC
• tan A=opposite side/adjacent side=BC/AB
• cosec A=hypotenuse/opposite side=AC/BC
• sec A=hypotenuse/adjacent side=AC/AB
• cot A=adjacent side/opposite side=AB/BC

Visualization of Trigonometric Ratios Using a Unit Circle

Draw a circle of unit radius with the origin as the centre. Consider a line segment OP joining a point P on the circle to the centre which makes an angle $\theta$ with the x-axis. Draw a perpendicular from P to the x-axis to cut it at Q.

• sinθ=PQ/OP=PQ/1=PQ
• cosθ=OQ/OP=OQ/1=OQ
• tanθ=PQ/OQ=sinθ/cosθ
• cosecθ=OP/PQ=1/PQ
• secθ=OP/OQ=1/OQ
• cotθ=OQ/PQ=cosθ/sinθ

Visualisation of Trigonometric Ratios Using a Unit Circle

Relation between Trigonometric Ratios

• cosec θ =1/sin θ
• sec θ = 1/cos θ
• tan θ = sin θ/cos θ
• cot θ = cos θ/sin θ=1/tan θ

Trigonometric Ratios of Specific Angles

Range of Trigonometric Ratios from 0 to 90 degrees

For $0^{\circ }\le \theta \le {90}^{\circ },$

• 0≤sinθ≤1
• 0≤cosθ≤1
• 0≤tanθ<∞
• 1≤secθ<∞
• 0≤cotθ<∞
• 1≤cosecθ<∞

$tan\theta$ and $sec\theta$ are not defined at  ${90}^{\circ }.$
$cot\theta$ and $cosec\theta$ are not defined at $0^{\circ }.$

Variation of trigonometric ratios from 0 to 90 degrees

As $\theta$ increases from $0^{\circ }$ to ${90}^{\circ }$

• $sin\theta$ increases from $0$ to $1$
• $cos\theta$ decreases from $1$ to $0$
• $tan\theta$ increases from $0$ to $\infty$
• $cosec\theta$ decreases from $\infty$ to $1$
• $sec\theta$ increases from $1$ to $\infty$
• $cot\theta$ decreases from $\infty$ to $0$

Standard values of Trigonometric ratios

∠A	0o	30o	45o	60o	90o
sin A	0	1/2	1/√2	√3/2	1
cos A	1	√3/2	1/√2	1/2	0
tan A	0	1/√3	1	√3	not defined
cosec A	not defined	2	√2	2/√3	1
sec A	1	2/√3	√2	2	not defined
cot A	not defined	√3	1	1/√3	0

Trigonometric Ratios of Complementary Angles

Complementary Trigonometric ratios

If $\theta$ is an acute angle, its complementary angle is ${90}^{\circ }-\theta .$ The following relations hold true for trigonometric ratios of complementary angles.

• $sin({90}^{\circ }-\theta )=cos\theta$
• $cos({90}^{\circ }-\theta )=sin\theta$
• $tan({90}^{\circ }-\theta )=cot\theta$
• $cot({90}^{\circ }-\theta )=tan\theta$
• $cosec({90}^{\circ }-\theta )=sec\theta$
• $sec({90}^{\circ }-\theta )=cosec\theta$

Trigonometric Identities

• $si{n}^2\theta +co{s}^2\theta =1$
• $1+co{t}^2\theta =coes{c}^2\theta$
• $1+ta{n}^2\theta =se{c}^2\theta$