Trigonometric functions of any angle, their graphs and identities
In Class 11, angles are measured in radians as well as degrees: one radian is the angle at the centre of a circle subtended by an arc equal in length to the radius, and π radians equal 180°. Using the unit circle, sin θ and cos θ are defined for any angle as the y- and x-coordinates of a point on the circle, so they take positive or negative values depending on the quadrant. Sine and cosine have period 2π, tan has period π, and the values of sine and cosine always lie between −1 and 1.
Compound-angle formulas such as sin(A + B) = sin A cos B + cos A sin B and cos(A + B) = cos A cos B − sin A sin B lead to double-angle results, including sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A. Sum-to-product formulas turn sums of sines or cosines into products. A trigonometric equation has infinitely many solutions because the functions repeat, so its general solution is written using an integer n: if sin θ = sin α, then θ = nπ + (−1)ⁿα.
Why it matters
Because they repeat, trigonometric functions model anything periodic: sound and light waves, alternating current, tides, the swing of a pendulum and the seasons. Engineers rely on them in signal processing and electronics, and physics needs them to resolve forces and describe oscillations. In Class 12 they are differentiated and integrated, and their inverses form a separate chapter.
Key terms
- Radian
- The angle subtended at the centre of a circle by an arc equal in length to the radius.
- Period
- The smallest interval after which a function's values repeat.
- Unit circle
- A circle of radius one centred at the origin, used to define trigonometric functions for any angle.
- General solution
- A formula, using an integer n, that lists every solution of a trigonometric equation.