Trigonometric Functions — Study Notes

1. Introduction

Trigonometry is the branch of mathematics that studies the relationships between angles and sides of triangles, and more generally, the periodic behaviour of angles. In Class 11, the subject takes a decisive step forward: the trigonometric functions are defined for all real numbers using the unit circle, measured in radians, and their graphs, identities, and applications are studied in depth.

The chapter begins by recalling the radian measure and establishing the relationship between degrees and radians. It then defines the six trigonometric ratios for any real angle using the unit circle, derives a large collection of trigonometric identities, studies the signs of ratios in different quadrants, and works with sums and differences of angles. The chapter concludes with the general solution of trigonometric equations and applications to practical problems.

These functions are not merely abstract. Trigonometric functions model waves, oscillations, sound, light, and planetary motion. The identities developed here are used throughout physics, engineering, and higher mathematics, including calculus of trigonometric functions in Class 12. A firm grasp of the unit circle approach and the standard identities is essential for success in this chapter and beyond.

2. Angles and Their Measures

An angle is formed by the rotation of a ray about its initial point. The starting ray is called the initial side and the final position is called the terminal side. A positive angle results from anticlockwise rotation, and a negative angle from clockwise rotation.

Radian Measure

One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. If the radius is r and an arc of length s subtends an angle theta at the centre, then:

theta = s / r

This gives the arc length formula s = r . theta. A complete revolution measures 360 degrees, which equals 2pi radians. Hence:

pi radians = 180 degrees

Conversion Between Degrees and Radians

1 degree = pi/180 radians and 1 radian = 180/pi degrees.

Example: 60 degrees = 60 x (pi/180) = pi/3 radians. Conversely, pi/4 radians = (pi/4) x (180/pi) = 45 degrees.

Standard Angle Table

30 deg = pi/6, 45 deg = pi/4, 60 deg = pi/3, 90 deg = pi/2, 180 deg = pi, 270 deg = 3pi/2, 360 deg = 2pi.

3. Trigonometric Functions via the Unit Circle

Consider a unit circle centred at the origin and a point P(x, y) on the circle such that the angle between OP and the positive x-axis is theta. Then the six trigonometric functions of theta are defined as:

sin theta = y, cos theta = x, tan theta = y/x (x not equal to 0), cot theta = x/y (y not equal to 0), sec theta = 1/x (x not equal to 0), cosec theta = 1/y (y not equal to 0)

For any angle theta, sin theta and cos theta always exist. The functions tan theta and sec theta are not defined when x = 0, i.e., theta = (2n + 1)pi/2. The functions cot theta and cosec theta are not defined when y = 0, i.e., theta = n pi.

Fundamental Identity

For all real values of theta:

sin^2 theta + cos^2 theta = 1

From this follow: 1 + tan^2 theta = sec^2 theta and 1 + cot^2 theta = cosec^2 theta.

4. Signs of Trigonometric Functions in Quadrants

The sign of each trigonometric function depends on the quadrant in which the terminal side of the angle lies. Since sin theta is the y-coordinate and cos theta is the x-coordinate of the point on the unit circle:

• In the first quadrant (0 to pi/2): all functions are positive.
• In the second quadrant (pi/2 to pi): only sin and cosec are positive.
• In the third quadrant (pi to 3pi/2): only tan and cot are positive.
• In the fourth quadrant (3pi/2 to 2pi): only cos and sec are positive.

A useful mnemonic is the phrase "All Silver Tea Cups", which lists the positive functions in quadrants 1, 2, 3, 4 respectively.

5. Domain, Range and Graphs

The six functions have the following domains and ranges:

• sin theta: domain R, range [-1, 1], period 2pi.
• cos theta: domain R, range [-1, 1], period 2pi.
• tan theta: domain R minus {(2n+1)pi/2}, range R, period pi.
• cot theta: domain R minus {n pi}, range R, period pi.
• sec theta: domain R minus {(2n+1)pi/2}, range (-infinity, -1] union [1, infinity), period 2pi.
• cosec theta: domain R minus {n pi}, range (-infinity, -1] union [1, infinity), period 2pi.

The graph of sin theta is a smooth wave oscillating between -1 and 1 with period 2pi. The graph of cos theta is the same wave shifted left by pi/2. The graph of tan theta has vertical asymptotes at odd multiples of pi/2 and increases from -infinity to infinity between consecutive asymptotes.

6. Trigonometric Identities

Compound Angle Formulas

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

Double Angle Formulas

sin 2x = 2 sin x cos x cos 2x = cos^2 x - sin^2 x = 2 cos^2 x - 1 = 1 - 2 sin^2 x tan 2x = 2 tan x/(1 - tan^2 x)

Triple Angle Formulas

sin 3x = 3 sin x - 4 sin^3 x cos 3x = 4 cos^3 x - 3 cos x

Half Angle and Other Useful Formulas

1 - cos 2x = 2 sin^2 x 1 + cos 2x = 2 cos^2 x cos^2 x = (1 + cos 2x)/2 sin^2 x = (1 - cos 2x)/2

Product-to-Sum and Sum-to-Product Formulas

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) 2 sin x sin y = cos (x - y) - cos (x + y)

sin x + sin y = 2 sin ((x + y)/2) cos ((x - y)/2) sin x - sin y = 2 cos ((x + y)/2) sin ((x - y)/2) cos x + cos y = 2 cos ((x + y)/2) cos ((x - y)/2) cos x - cos y = -2 sin ((x + y)/2) sin ((x - y)/2)

7. Values of Trigonometric Functions of Specific Angles

Some standard values to memorise:

sin 0 = 0, sin 30 = 1/2, sin 45 = 1/sqrt 2, sin 60 = sqrt 3/2, sin 90 = 1, sin 180 = 0, sin 270 = -1. cos 0 = 1, cos 30 = sqrt 3/2, cos 45 = 1/sqrt 2, cos 60 = 1/2, cos 90 = 0, cos 180 = -1, cos 270 = 0. tan 0 = 0, tan 30 = 1/sqrt 3, tan 45 = 1, tan 60 = sqrt 3, tan 90 is undefined.

For angles in other quadrants, use the reference angle and the quadrant sign rules. For example, sin 150 = sin (180 - 30) = sin 30 = 1/2.

8. Trigonometric Equations and Their General Solutions

An equation involving trigonometric functions is a trigonometric equation. Solutions are often required for all real values of the variable.

General Solutions

• sin x = 0 gives x = n pi.
• cos x = 0 gives x = (2n + 1)pi/2.
• sin x = sin y gives x = n pi + (-1)^n y.
• cos x = cos y gives x = 2n pi +/- y.
• tan x = tan y gives x = n pi + y.
• sin^2 x = sin^2 y, cos^2 x = cos^2 y, tan^2 x = tan^2 y each give x = n pi +/- y.

Here n is any integer. These general solutions cover all solutions of the equation over the real line.

9. Applications: Solution of Triangles and Heights

Using the trigonometric ratios, problems about heights and distances can be solved. In a right triangle with angle theta, the opposite side, adjacent side, and hypotenuse satisfy:

sin theta = opposite/hypotenuse, cos theta = adjacent/hypotenuse, tan theta = opposite/adjacent

The law of sines and law of cosines extend these ideas to any triangle. For a triangle with sides a, b, c opposite angles A, B, C:

a/sin A = b/sin B = c/sin C (law of sines) c^2 = a^2 + b^2 - 2ab cos C (law of cosines)

Quick Revision Tables

Table 1: Signs of Trigonometric Functions by Quadrant

Table 2: Domain, Range and Period of Trigonometric Functions

Mind Map

Important Diagrams (SVG)

Diagram 1: Unit Circle with Trigonometric Functions

Diagram 2: Graphs of sin x and cos x

Common Mistakes

1. Using degrees in calculus-style problems where the formulas assume radians. Always convert angles to radians when required by the formula.
2. Forgetting that tan theta is undefined at odd multiples of pi/2 and cot theta is undefined at multiples of pi. Division by zero is the error.
3. Confusing the signs of functions in quadrants. For example, cos theta is negative in the second quadrant, not positive.
4. Applying compound angle formulas with wrong signs. For example, cos (x + y) = cos x cos y - sin x sin y, not cos x cos y + sin x sin y.
5. Using sin 2x = 2 sin x. This is wrong; sin 2x = 2 sin x cos x. Never cancel sin factors unless justified.
6. Forgetting the sign in cos x - cos y = -2 sin((x + y)/2) sin((x - y)/2). The leading negative sign is essential.
7. In the sum-to-product conversion, dividing by the wrong factor: sin x + sin y uses the midpoint (x + y)/2, not the difference.
8. Writing the general solution of sin x = sin y as x = 2n pi + y. The correct form uses (-1)^n: x = n pi + (-1)^n y.
9. Confusing sin 150 with sin 30 directly without the quadrant check; sin 150 = 1/2 (positive) because sine is positive in the second quadrant.

Exam Tips

1. Memorise the standard angle table: sin, cos, tan for 0, 30, 45, 60, 90, expressed in radians.
2. When finding values in other quadrants, first find the reference angle in the first quadrant and then apply the correct sign using All Silver Tea Cups.
3. For simplification problems, always look for the pattern of double angle or compound angle formulas before expanding fully.
4. To prove identities, generally start from the more complicated side and simplify towards the simpler side, writing every step.
5. For trigonometric equations, first reduce the equation to a single function, then apply the general solution formula, and specify the range if restricted.
6. Practice converting degrees to radians quickly: multiply by pi/180; conversely multiply by 180/pi.
7. When using the sum-to-product formulas, double-check whether the first sign is plus or minus, especially for cos x - cos y.

Conclusion

Trigonometric functions transform the study of angles into a powerful analytic tool. The unit circle definition extends trigonometry from right triangles to all real angles, and the rich collection of identities built in this chapter enables the simplification and solution of a huge range of problems. The signs in quadrants, general solutions of trigonometric equations, and the graphs of the functions are all central to physics, engineering, and Class 12 calculus. With careful attention to radian measure, quadrant signs, and identity signs, students can master this chapter and lay a solid foundation for differentiation and integration of trigonometric functions in the next class.