Trigonometry plays a pivotal role in almost all physics problems especially those related to vector quantities where changes in angular values are key in solving the given problem. Here we look at 1) the basic and essential trignometric functions and, in the second post 2) their role with respect to some recent iit jee questions.
a. Definition : The branch of mathematics dealing with the study of the relationships between sides and angles of triangles and the behaviour of wave functions.
b.Basic Trigonometric functions : If A, B, and C are the measures of the angles of a right triangle (C=90°) and a, b, and c are the lengths of the respective sides opposite these angles, then the six functions are expressed for one of the acute angles, say A, as various ratios of the opposite side (a), the adjacent side (b), and the hypotenuse (c).
Eventhough , the actual lengths of the sides of a right triangle can have any values, the ratios of the lengths will be the same for all similar right triangles, and , these ratios depend only on the angles and not on the actual lengths. The functions occur in pairs-sine and cosine, tangent and cotangent, secant and cosecant-called cofunctions. In equations they are usually represented as sin, cos, tan, cot, sec, and csc. Since in ordinary (Euclidean) plane geometry the sum of the angles of a triangle is 180°, angles A and B must add up to 90° and therefore are complementary angles. The definitions of the functions denote that, sin B=cos A, cos B=sin A, tan B=cot A, and sec B=csc A; in general, the function of an angle is equal to the cofunction of its complement. Since the hypotenuse (c), is always the longest side of a right triangle, the values of the sine and cosine are always between zero and one, the values of the secant and cosecant are always equal to or greater than one, and the values of the tangent and cotangent are unbounded, increasing from zero without limit.
For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 :√3 : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°=√3/2, tan 30°=cot 60°=1/√3, cot 30°=tan 60°=√3, sec 30°=csc 60°=2/√3, and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.
A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a/sin A=b/sin B=c/sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a2=b2+c2−2bc cosA and the Law of Tangents holds that (a−b)/(a+b)=[tan 1/2(A−B)]/[tan 1/2(A+B)].
c. Cartesian co-ordinates and extension of Trignometric functions :
By defining the functions with respect to the cartesian co-ordinates , usage of trigonometric funstions can be extended beyond 90°. Let r be a line of unit length from the origin to the point P (x,y), and let θ be the angle r makes with the positive x-axis. The six functions become sin θ =y/r=y, cos θ=x/r=x, tan θ=y/x, cot θ=x/y, sec θ=r/x=1/x, and csc θ=r/y=1/y. As θ increases beyond 90°, the point P crosses the y-axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270° P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive.
As the positions of r for angles of 360° or more coincide with those already taken by r as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on.This periodic nature of trigonometric functions plays a key role in the repeating / periodic phenomena related to physics eg. electromagnetic waves.
Interestingly , all of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O.
- for a chord AB of the circle, where θ is half of the subtended angle, sin(θ) is AC (half of the chord).
- cos(θ) is the horizontal distance OC, and versin(θ) = 1 − cos(θ) is CD.
- tan(θ) is the length of the segment AE of the tangent line through A. cot(θ) is another tangent segment, AF.
- sec(θ) = OE and csc(θ) = OF are segments of secant lines (intersecting the circle at two points), and can also be viewed as projections of OA along the tangent at A to the horizontal and vertical axes, respectively.
- DE is exsec(θ) = sec(θ) − 1 (the portion of the secant outside, or ex, the circle).
The above constructions prove that , the secant and tangent functions diverge as θ approaches π/2 (90 degrees) and that the cosecant and cotangent diverge as θ approaches zero.
Law of Sines – Also called the ‘sine rule ‘ , for an arbitrary triangle states that –
where R is the radius of the circumscribed circle of the triangle:
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
Law of Cosines – Also called the ‘ cos’ rule , it is the extension of the Pythagorean theorem dealing with arbitrary triangles.
Law of Tangents – it states that –
Euler’s formula –
This formula states that eix = cosx + isinx, and produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i –
Trigonometric Functions –
The trigonometric functions are summarized in the table below . The angle θ is the angle between the hypotenuse and the adjacent line.
|Function||Abbreviation||Description||Identities (using radians)|
|Tangent||tan (or tg)|
|Cotangent||cot (or ctg or ctn)|
|Cosecant||csc (or cosec)|
( to continue………… )