Trigonometric Functions

You already know the basics — sine, cosine, tangent, and how they relate to a right-angled triangle. But in Std 12, trigonometry goes from being about triangles to being about functions. And these functions are everywhere — they describe sound waves, light waves, seasonal temperature changes, the motion of a pendulum, the alternating current that powers your home, and even the signals that carry your WiFi data. This chapter is where you stop thinking of sine and cosine as ratios and start thinking of them as functions that repeat infinitely. The first big topic is compound angles — formulas for sin(A + B), cos(A + B), and tan(A + B). The single most important thing to remember: sin(A + B) is NOT sin A + sin B. That's the biggest mistake students make. The real formulas involve mixing sines and cosines. From these, we derive double angle formulas (sin 2A = 2 sin A cos A), triple angle formulas (sin 3A = 3 sin A − 4 sin³ A), and half-angle formulas. Cos 2A has three different forms — learn all three because each one is useful in different situations. Then come the factorization formulas that let you convert sums into products and vice versa. Why bother? Because sometimes a product is easier to integrate, differentiate, or solve. The transformation formulas — expressing a sin θ + b cos θ as R sin(θ + α) — are especially useful for finding maximum and minimum values. Next up: trigonometric equations. Unlike algebraic equations where solutions are finite, trigonometric equations typically have INFINITE solutions because trig functions are periodic. When you solve sin θ = 1/2, the answer isn't just 30° — it's 30° + 360°n and 150° + 360°n for integer n. General solutions follow patterns: for sin θ = sin α, θ = nπ + (−1)ⁿα; for cos θ = cos α, θ = 2nπ ± α; for tan θ = tan α, θ = nπ + α. These are the formulas you'll use repeatedly. Then there are inverse trigonometric functions — the 'undo' operations. If sin θ = x, then θ = sin⁻¹ x (also written as arcsin x). But since trigonometric functions are not one-to-one (multiple angles give the same value), we restrict the domain to get principal values. sin⁻¹ gives values in [−π/2, π/2], cos⁻¹ gives [0, π], and tan⁻¹ gives (−π/2, π/2). You'll need to learn the properties of inverse functions — like sin⁻¹(sin θ) = θ only when θ is in the principal range, and sin⁻¹ x + cos⁻¹ x = π/2. The chapter also covers solutions of triangles — a set of formulas that let you solve any triangle given some of its sides and angles. These include the sine rule (a/sin A = b/sin B = c/sin C = 2R), the cosine rule (a² = b² + c² − 2bc cos A), projection formulas, and area formulas. The sine rule is great when you know two angles and one side. The cosine rule is perfect when you know two sides and the included angle, or all three sides. You'll also learn about the Napier analogies (tangent rules) and the Mollweide formula. All of these help you find unknown parts of a triangle — whether it's a right triangle or an oblique one. This is used in navigation, surveying, astronomy, and engineering. By the end of this chapter, you'll be able to simplify complex trigonometric expressions, solve any trig equation, work comfortably with inverse functions, and solve any triangle using the right combination of rules.