Trigonometry - Trigonometry

[Source: NCERT]

Trigonometric functions are the functions that are real and define the functions of a right-angled triangle. It relates the angles of a triangle with its sides. The six trigonometric functions are:

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• Secant (sec)
• Cosecant (cosec)
• Cotangent (cot)

Sine: The ratio of the side opposite to the angle with the hypotenuse of a right-angled triangle.

If x is the angle, then sin x = opposite side/hypotenuse.

Cosine: The side's ratio adjacent to the angle with the hypotenuse of a triangle.

cos x = adjacent side/hypotenuse

Tangent: The opposite side's ratio to the adjacent side of an angle in a right-angled triangle.

tan x = opposite side/adjacent side

Secant: The inverse of the cos function.

sec x = 1/cos x

Cosecant: The inverse of sin function.

cosec x = 1/sin x

Cotangent: The inverse of the cot function.  

cot x = 1/tan x

Pythagoras Theorem gives us the following identities:

sin2θ + cos2θ = 1

sec2θ = 1 + tan2θ

cosec2θ = 1 + cot2θ

The sum and difference identities are given below:

sin(x+y) = sin(x).cos(y)+cos(x).sin(y)

sin(x–y) = sin(x).cos(y)–cos(x).sin(y)

tan(x+y) = [tan(x)+tan(y)]/[1-tan(x)tan(y)]

tan(x-y) = [tan(x)-tan(y)]/[1+tan(x)tan(y)]

cos(x+y) = cosx.cosy–sinx.siny

cos(x–y) = cosx.cosy+sinx.siny

Only sec and cos are even functions in trigonometry. The remaining are odd functions.

π and 2π are periodic cycles. 2π is the periodic cycle for sin, cos, cosec, and sec functions, whereas π is the periodic cycle for tan and cot functions.

You will be introduced to trigonometry concepts in Class X where it will carry a maximum weightage of 12 marks.

Class XI will give you a more in-depth insight into this topic. Together with trigonometric equations, this topic carries a weightage of 23 marks in Class XI.

You will learn about inverse trigonometric functions in Class XII, and it will carry a weightage of 8 marks.

• Calculate cos 105.

cos 105 can be written as cos (60 + 45)

We have, cos(x+y) = cosx.cosy–sinx.siny

cos (60 + 45) = cos 60.cos 45 - sin 60.sin45 = ½ x 1/√2 - √3/2 x 1/√2 = 1 - √3/2/√2

• If sinx = 3, find tanx.

sinx =3 means that sin2x = 9

sin2x + cos2x = 1

cos2x = 1 - 9

cosx = square root of (-8) = 2.83 approx.

tan x = sinx/cosx = 3/2.83 = 1.06.

• What is cos60 as per the trigonometric table?

As per the trigonometric table, cos60 = ½.

Q: Why are trigonometric functions used?

Q: Name the basic trigonometric functions.

Q: What are the values of sin30 and tan30?

Q: Who established the Pythagoras Theorem?

Q: What can trigonometric functions be used for?