Trigonometric Functions

In this chapter: degree and radian measure, arc length, trigonometric functions through the unit circle, signs in the four quadrants, domain, range, periodicity and graphs, allied angles, sum and difference formulas, double and triple angle formulas, sum-to-product and product-to-sum formulas, general solutions.

Measuring angles

An angle is the amount of rotation of a ray about its initial point. Anticlockwise rotation gives a positive angle and clockwise rotation a negative one.

• Degree: one complete revolution is 360°; 1° = 60′ (minutes) and 1′ = 60″ (seconds).
• Radian: the angle subtended at the centre of a circle by an arc whose length equals the radius. One complete revolution is 2π radians.

To convert degrees to radians multiply by π/180; to go back multiply by 180/π. When no unit is written, the angle is in radians, so sin 1 means the sine of 1 radian, not 1°. For example, an arc of a circle of radius 21 cm subtending 60° at the centre has length 21 × π/3 = 7π = 22 cm (taking π = 22/7).

Trigonometric functions through the unit circle

Take the circle of radius 1 centred at the origin. Rotate the radius OA, starting from A(1, 0), through an angle x to reach a point P. The coordinates of P are defined to be (cos x, sin x). Since P is on the unit circle, for every real x:

After one full turn P comes back to the same place, so sin(2nπ + x) = sin x and cos(2nπ + x) = cos x for every integer n. Also sin x = 0 exactly when x = nπ, and cos x = 0 exactly when x = (2n + 1)π/2. The other four are defined from sine and cosine: tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x, cosec x = 1/sin x, each wherever its denominator is not zero.

Values you should know without thinking

Domain, range and period

Sine is an odd function and cosine is even: sin(−x) = −sin x and cos(−x) = cos x. Tangent, cotangent and cosecant are odd; secant is even.

Allied angles

For angles of the form nπ/2 ± x: if n is odd, sin changes to cos (and tan to cot); if n is even, the function stays the same. The sign in front is the sign of the original function in the quadrant where the angle falls, treating x as acute.

Sum and difference formulas

The tangent formulas need none of x, y, x ± y to be an odd multiple of π/2; the cotangent formulas need none of them to be a multiple of π.

Worked example: Find sin 15° and tan 15°.
Solution: sin 15° = sin(45° − 30°) = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2). For the tangent, tan 15° = (tan 45° − tan 30°)/(1 + tan 45° tan 30°) = (1 − 1/√3)/(1 + 1/√3) = (√3 − 1)/(√3 + 1). Multiplying top and bottom by √3 − 1 gives (4 − 2√3)/2 = 2 − √3.

Double and triple angle formulas

Rearranging cos 2x gives the power-reducing forms 2cos²x = 1 + cos 2x and 2sin²x = 1 − cos 2x, which you will use again and again in integration.

Worked example: sin x = 3/5 and x lies in the second quadrant. Find cos x, tan x and sin 2x.
Solution: cos²x = 1 − 9/25 = 16/25, so cos x = ±4/5. Cosine is negative in the second quadrant, so cos x = −4/5. Then tan x = (3/5)/(−4/5) = −3/4 and sin 2x = 2(3/5)(−4/5) = −24/25.

Sum-to-product and product-to-sum

A typical use: (sin 5x + sin 3x)/(cos 5x + cos 3x) = (2 sin 4x cos x)/(2 cos 4x cos x) = tan 4x. Whenever you see a sum of two sines or cosines in a fraction, try converting to products and look for a common factor.

One more result used constantly in JEE: a sin x + b cos x always lies between −√(a² + b²) and √(a² + b²), and both extremes are attained. So 3 sin x + 4 cos x has maximum 5 and minimum −5.

General solutions

The rationalised NCERT textbook has dropped the section on trigonometric equations, but general solutions are part of the MHT‑CET syllabus (through the Maharashtra board Class 12 chapter on trigonometric functions), are listed in the JEE Advanced syllabus, and keep turning up inside JEE problems. For n ∈ Z:

For example, sin x = √3/2 = sin(π/3) gives x = nπ + (−1)ⁿπ/3; the solutions in [0, 2π) are π/3 and 2π/3.

Common mistakes: (1) Taking the positive square root out of habit: after finding cos²x, fix the sign from the quadrant. (2) Treating sin 2 as sin 2°; without a degree sign the angle is in radians. (3) Writing tan(x + y) with a plus sign in the denominator; it is 1 − tan x tan y. (4) Changing sin to cos for π ± x; the swap happens only for odd multiples of π/2. (5) Forgetting the minus sign in cos x − cos y = −2 sin((x + y)/2) sin((x − y)/2). (6) Giving the range of sec x as [−1, 1], which is exactly the set it avoids (apart from ±1).

Exam focus

• Radian and degree conversion, and arc length l = rθ.
• Signs of the ratios in each quadrant and values at allied angles, including large angles such as 765° or −1710°.
• Sum, difference, double and triple angle formulas, used to evaluate expressions like tan 75° or cos²15° − sin²15°.
• Sum-to-product conversions to simplify and prove identities.
• Maximum and minimum of a sin x + b cos x; domain, range and period of the six functions.
• General solutions of simple equations such as sin x = k or cos 2x = cos x.

Practice questions