Trigonometric Functions

An angle is formed by the rotation of a ray from an initial side to a terminal side about a fixed point called the vertex. If the rotation is anticlockwise the angle is taken as positive, and if clockwise it is taken as negative. Angles are measured in two systems.

In the sexagesimal (degree) system, one complete rotation is divided into 360 equal parts, each called one degree (1°). Further, 1° = 60 minutes (60') and 1' = 60 seconds (60").

In the circular (radian) system, one radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. Since the circumference of a circle of radius r is 2πr, a complete rotation (360°) corresponds to an arc length of 2πr, and therefore equals 2π radians. This gives the fundamental relation connecting the two systems: 180° = π radians.

Using this single relation, degree measure is converted to radian measure by multiplying by π/180, and radian measure is converted to degree measure by multiplying by 180/π. For a circle of radius r, if an arc of length l subtends an angle θ (in radians) at the centre, then l = rθ; the area of the corresponding sector is A = ½ r²θ.

Example: Convert 240° into radian measure, and find the length of the arc it subtends in a circle of radius 21 cm (take π = 22/7).

1. Radian measure = 240 × (π/180) = 240π/180 = 4π/3 radians.
2. Here θ = 4π/3 rad and r = 21 cm.
3. Arc length l = rθ = 21 × 4π/3 = 28π cm.
4. Taking π = 22/7, l = 28 × 22/7 = 88 cm.

So 240° = 4π/3 radians, and the corresponding arc length is 88 cm.

Take a unit circle (radius 1) centred at the origin, and let a point P start at (1, 0) and move a directed distance x along the circle (anticlockwise if x > 0, clockwise if x < 0). If P has coordinates (a, b), define cos x = a and sin x = b. Since x can be any real number, sine and cosine are functions defined on the whole of R. The remaining four functions are defined from these two: tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x, cosec x = 1/sin x. Because these are quotients, tan x and sec x are undefined wherever cos x = 0, and cot x and cosec x are undefined wherever sin x = 0.

The quadrant rule ("All Sin Tan Cos", read anticlockwise from quadrant I) tells us which functions are positive in each quadrant: in quadrant I all six functions are positive; in quadrant II only sin and cosec are positive; in quadrant III only tan and cot are positive; in quadrant IV only cos and sec are positive.

From a² + b² = 1 for a point on the unit circle, we get the fundamental identity sin²x + cos²x = 1, valid for every real x. Dividing through by cos²x and by sin²x gives the other two identities, 1 + tan²x = sec²x and 1 + cot²x = cosec²x.

Domain and range follow directly: sin x and cos x are defined for all real x and always lie between −1 and 1; tan x and sec x are undefined at odd multiples of π/2; cot x and cosec x are undefined at integer multiples of π; and since sec²x ≥ 1 and cosec²x ≥ 1, both |sec x| ≥ 1 and |cosec x| ≥ 1 always, so their range excludes the open interval (−1, 1). sin x, cos x, sec x and cosec x repeat every 2π; tan x and cot x repeat every π.

Example: If cos x = −3/5 and x lies in the third quadrant, find the values of the other five trigonometric functions.

1. sin²x = 1 − cos²x = 1 − 9/25 = 16/25, so sin x = ±4/5.
2. In the third quadrant sine is negative, so sin x = −4/5.
3. tan x = sin x/cos x = (−4/5)/(−3/5) = 4/5.
4. cosec x = 1/sin x = −5/4, sec x = 1/cos x = −5/3, cot x = 1/tan x = 5/4.

Note that tan x and cot x came out positive, exactly as the quadrant rule predicts for quadrant III.

The compound (sum and difference) angle formulas express sin, cos and tan of x±y in terms of the trigonometric functions of x and y separately. They are the master identities of this chapter, since every multiple-angle, sub-multiple-angle and transformation formula is derived from them.

The two building-block results are cos(x−y) = cos x cos y + sin x sin y and cos(x+y) = cos x cos y − sin x sin y. Replacing x by (π/2 − x) in these gives the sine formulas: sin(x+y) = sin x cos y + cos x sin y and sin(x−y) = sin x cos y − cos x sin y. Dividing the sine formula by the cosine formula gives the tangent formulas: tan(x+y) = (tan x + tan y)/(1 − tan x tan y) and tan(x−y) = (tan x − tan y)/(1 + tan x tan y), valid whenever all the terms are defined.

Putting y = ±π/2, ±π etc. into these formulas produces the standard allied-angle results used to reduce any angle to an acute reference angle, such as sin(π/2 − x) = cos x, cos(π/2 − x) = sin x, sin(π − x) = sin x, cos(π − x) = −cos x, sin(π + x) = −sin x, and cos(π + x) = −cos x. Together with sin(−x) = −sin x and cos(−x) = cos x, these allow every trigonometric value to be reduced to one for an angle between 0 and π/2.

Example: Find the values of sin 75° and cos 105° using the compound angle formulas.

1. Write 75° = 45° + 30°. Then sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
2. Write 105° = 60° + 45°. Then cos 105° = cos 60° cos 45° − sin 60° sin 45° = (1/2)(√2/2) − (√3/2)(√2/2) = (√2 − √6)/4.

So sin 75° = (√6+√2)/4 ≈ 0.9659 and cos 105° = (√2−√6)/4 ≈ −0.2588, consistent with 105° being obtuse (cosine negative).

Putting y = x in the compound angle formulas gives the double-angle formulas: sin 2x = 2 sin x cos x, and cos 2x = cos²x − sin²x, which (using sin²x+cos²x=1) can also be written as cos 2x = 2cos²x − 1 = 1 − 2sin²x. Dividing sin 2x by cos 2x gives tan 2x = 2 tan x/(1 − tan²x). Writing 3x as 2x + x and expanding similarly gives the triple-angle formulas sin 3x = 3 sin x − 4 sin³x and cos 3x = 4 cos³x − 3 cos x.

A separate pair of results, called transformation formulas, convert a sum or difference of two sines/cosines into a product, and vice versa. Setting x + y = A and x − y = B in the compound angle formulas and adding/subtracting in pairs gives: sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2), sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2), cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2), and cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2). The reverse (product-to-sum) forms are 2 sin x cos y = sin(x+y) + sin(x−y), 2 cos x sin y = sin(x+y) − sin(x−y), 2 cos x cos y = cos(x+y) + cos(x−y), and 2 sin x sin y = cos(x−y) − cos(x+y).

Example: Prove that (sin 5x − sin 3x)/(cos 5x + cos 3x) = tan x.

1. Numerator: sin 5x − sin 3x = 2 cos((5x+3x)/2) sin((5x−3x)/2) = 2 cos 4x sin x.
2. Denominator: cos 5x + cos 3x = 2 cos((5x+3x)/2) cos((5x−3x)/2) = 2 cos 4x cos x.
3. Ratio = (2 cos 4x sin x)/(2 cos 4x cos x) = sin x/cos x = tan x, provided cos 4x ≠ 0.

Hence the identity is proved.

A trigonometric equation is an equation involving trigonometric functions of an unknown angle. Solutions lying in [0, 2π) are called principal solutions, while the complete set of solutions, expressed with an arbitrary integer n, is called the general solution.

Three master results generate the general solution of every basic trigonometric equation: if sin x = sin a then x = nπ + (−1)ⁿ a; if cos x = cos a then x = 2nπ ± a; and if tan x = tan a then x = nπ + a, in every case n ∈ Z and a is the principal value satisfying the equation. Two special cases follow at once: sin x = 0 gives x = nπ, and cos x = 0 gives x = (2n+1)π/2. Also, if sin²x = sin²a (or the analogous statement for cos² or tan²), the general solution is simply x = nπ ± a.

The general method is: (i) use identities to reduce the equation to one of these standard forms (often after factorising or solving a quadratic in sin x, cos x or tan x), (ii) find the principal value a, and (iii) write the corresponding general solution, discarding any values outside the natural range of the function (for example, sin x or cos x can never exceed 1 in magnitude).

Example: Solve tan²x = 3.

1. tan²x = 3 = (√3)² = tan²(π/3), since tan(π/3) = √3.
2. Using the result for tan²x = tan²a, the general solution is x = nπ ± π/3, n ∈ Z.
3. Check: at x = π/3, tan x = √3 so tan²x = 3 ✓; at x = −π/3, tan x = −√3 so tan²x = 3 ✓.

So the complete solution set is x = nπ ± π/3, n ∈ Z.