In Focus: Trigonometry

Trigonometry is one of the most important and basic chapters in the preparation of any competitive examination. It is taught in 10th or 11th standard because many of the topics in mathematics need a sound understanding of trigonometry.

Trigonometry ( from Greek trigonon, “triangle”  and metron, “measure”) was related to measuring the sides and angles of a triangle. This is called school trigonometry. Afterwards the definition got extended and measure of an angle was no longer bounded. Hence, the definitions of the trigonometric ratios were also adapted to the new definition. Trigonometry basically has 6 trigonometric ratios namely , “sine”, “cosine”, “tangent”, “cotangent”, “secant” and “cosecant”  or sin, cos, tan, cot, sec and cosec respectively.

Consider the unit circle as shown in the figure. Consider a ray OP making an angle θ in the anticlockwise positive convention. The definitions of the trigonometric ratios are given as follows

cosθ = x , sinθ = y ,

tanθ = sinθ/cosθ and secθ = 1/cosθ : if cosθ ≠ 0,

cotθ = cosθ/sinθ and cosecθ = 1/sinθ : if sinθ ≠ 0.

since P(x,y) lies on the unit circle, we have

sin²θ + cos²θ = 1,

1 + tan²θ = sec²θ   if cosθ ≠ 0,

1 + cot²θ = cosec²θ   if sinθ ≠ 0.

We also observe that θ and θ+2πn, n being an integer have the same terminal ray so the all trigonometric ratios for these two angles are the same. Thus the trigonometric ratios are periodic. The trigonometric ratios sinθ, cosθ, secθ and cosecθ are periodic with a fundamental period of 2π whereas tanθ and cotθ are periodic with a fundamental period of π.

The below table depicts the domain and range of the trigonometric functions and the quadrant in which they have a +ve sign.

The formulas for trigonometric ratios of sum or difference of two angles are given by

sin(A ± B) = sinAcosB ± cosAsinB,

cos(A ± B) = cosAcosB ∓ cosAcosB and

tan(A ± B) = (tanA ± tanB) /(1∓tanAtanB)

Trigonometry also has factorization and de-factorization formulae.

Find a complete list of the sum and difference formulae here!

Factorization Formulae:

cosA + cosB = 2cos((A+B)/2)*cos((A-B)/2)

cosA – cosB = -2sin((A+B)/2)*sin((A-B)/2)

sinA + sinB = 2sin((A+B)/2)*cos((A-B)/2)

sinA – sinB = 2cos((A+B)/2)*sin((A-B)/2)

De-factorization Formulae:

2cos A*cos B = cos ( A + B) + cos ( A – B)

–2sin A*sin B = cos ( A + B) – cos ( A – B)

2sin A*cos B = sin (A + B) + sin (A – B)

2cos A*sin B = sin (A + B) – sin (A – B).

You can check out an exhaustive sheet of formulae here.

Trigonometry is a very basic concept so it does not have a lot of pre-requisites. Basic arithmetic and triangles and their properties is necessary. Also Co-ordinate Geometry is necessary to understand the new definitions of the trigonometric ratios.

Trigonometry, as mentioned before, forms the basis of other chapters. Inverse trigonometric functions is completely based on trigonometry. Complex Numbers, Conics, Calculus, Vectors all use trigonometry in some or the other way. The basic polar form of a complex number is given by trigonometry. Parametrization of straight line, circle, ellipse and hyperbola makes the problem easier to solve. The topic of Solution of triangles, is completely based on trigonometry. Using the Sine law and Cosine law really simplifies stuff. Integrating using trigonometric substitutions is one of the most important methods of integration. Substituting a trigonometric ratio in place of a  variable simplifies the problem. Even in physics when it comes to resolution of vectors to its components, trigonometry comes in handy.

A free body diagram without using trigonometric ratios is almost impossible. To understand waves on a string knowledge about trigonometric ratios is imminent. Architects use trigonometry to calculate structural load, roof slopes, ground surfaces and many other aspects, including sun shading and light angles.

In conclusion, we see that trigonometry is one of the most basic and important chapters of mathematics. The students opting for maths must have a deep and thorough understanding of this chapter, especially the ones who opt for competitive examinations. Having a good grasp over this topic would really give you the edge required for scoring better than your fellow students.

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