Trigonometric Functions and Identities Class 11 Questions for AGNIVEER NAVY

Six Trigonometric Functions — Definitions and Domains

The six trig functions (sin, cos, tan, cot, sec, cosec) are ratios of sides in a right triangle. In a right triangle with angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. The reciprocals are cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ. Each function has a specific domain (values of θ for which it is defined) and range (output values). For example, sin and cos are defined for all real angles and have range [-1, 1]. Tan and cot are undefined at certain angles (where denominator = 0).

• sin θ = opposite/hypotenuse; range [-1, 1]; defined for all θ
• cos θ = adjacent/hypotenuse; range [-1, 1]; defined for all θ
• tan θ = sin θ/cos θ; undefined when cos θ = 0 (at 90°, 270°, ...)
• cosec θ = 1/sin θ; undefined when sin θ = 0 (at 0°, 180°, ...)
• sec θ = 1/cos θ; undefined when cos θ = 0
• cot θ = cos θ/sin θ; undefined when sin θ = 0

Pythagorean Identities — The Holy Trinity

These three identities are derived from the Pythagorean theorem and are the most frequently tested in Agniveer exams. They allow you to convert between sin and cos, and simplify complex expressions. The first identity (sin²θ + cos²θ = 1) is the foundation. From it, you can derive the other two by dividing by cos²θ or sin²θ respectively. These identities are used to simplify expressions, prove other identities, and solve trig equations.

• sin²θ + cos²θ = 1 — the master identity; always true for any angle θ
• 1 + tan²θ = sec²θ — derived by dividing first identity by cos²θ
• 1 + cot²θ = cosec²θ — derived by dividing first identity by sin²θ
• Use these to eliminate one function and express in terms of another
• Rearrange to find sin θ from cos θ or vice versa: sin²θ = 1 - cos²θ

Compound Angle Formulas — Addition and Subtraction

These formulas express sin(A ± B), cos(A ± B), and tan(A ± B) in terms of sin A, cos A, sin B, cos B. They are critical for solving problems where angles are sums or differences of standard angles (like 15°, 75°, 105°). The formulas are easy to confuse, so use the mnemonic SOCA to remember: sin(A+B) uses sin-cos-cos-sin with alternating signs, cos(A+B) uses cos-cos-sin-sin with minus sign in middle.

• sin(A + B) = sin A cos B + cos A sin B
• sin(A - B) = sin A cos B - cos A sin B
• cos(A + B) = cos A cos B - sin A sin B
• cos(A - B) = cos A cos B + sin A sin B
• tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
• tan(A - B) = (tan A - tan B) / (1 + tan A tan B)

Double Angle and Half Angle Formulas

Double angle formulas express sin 2θ, cos 2θ, tan 2θ in terms of sin θ and cos θ. These are special cases of compound angle formulas (where A = B = θ). Half angle formulas do the reverse: express sin(θ/2), cos(θ/2) in terms of cos θ. Double angle formulas appear frequently in Agniveer papers for simplifying and solving equations. Half angle formulas are less common but useful for specific problems.

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ (three forms; choose based on what's given)
• tan 2θ = 2 tan θ / (1 - tan²θ)
• sin²θ = (1 - cos 2θ) / 2 (useful for power reduction)
• cos²θ = (1 + cos 2θ) / 2 (useful for power reduction)

Trigonometric Equations — Solving for θ

A trig equation is an equation involving trig functions where you solve for the angle θ. The general solution includes all angles (in degrees or radians) that satisfy the equation. For sin θ = k, the general solution is θ = nπ + (-1)^n arcsin(k) in radians, or θ = n(180°) + (-1)^n sin⁻¹(k) in degrees. For cos θ = k, it is θ = 2nπ ± arccos(k). For tan θ = k, it is θ = nπ + arctan(k). Always check the domain and the given range for θ.

• sin θ = k has general solution θ = nπ + (-1)^n sin⁻¹(k) (n = integer)
• cos θ = k has general solution θ = 2nπ ± cos⁻¹(k)
• tan θ = k has general solution θ = nπ + tan⁻¹(k)
• Always check if the value of k is in the valid range (e.g., -1 ≤ k ≤ 1 for sin and cos)
• If a range is given (e.g., 0 ≤ θ ≤ 2π), find only solutions in that range

• Confusing sin(A + B) with sin A + sin B. Remember: sin(A + B) = sin A cos B + cos A sin B, NOT sin A + sin B. The exam loves this trap.
• Forgetting the domain restrictions. tan θ is undefined at 90°, 270°, etc. If a problem asks for tan θ and θ = 90°, the answer is undefined, not infinity.
• Using the wrong form of cos 2θ. There are three equivalent forms; students often pick the wrong one. If only sin θ is given, use cos 2θ = 1 - 2sin²θ. If only cos θ is given, use cos 2θ = 2cos²θ - 1.
• Forgetting the ± in general solutions. For cos θ = k, the general solution is θ = 2nπ ± cos⁻¹(k), not just θ = 2nπ + cos⁻¹(k). Missing the ± means missing half the solutions.
• Mixing radians and degrees. If the problem gives angles in degrees, work in degrees throughout. If in radians, stay in radians. Switching mid-solution causes errors.

• SOCA for compound angles: sin(A+B) = Sin-cOs-cOs-sin (alternating pattern), cos(A+B) = cOs-cOs-sin-sin (minus in middle). Helps remember which function pairs go together.
• All Students Take Calculus (ASTC): In quadrant I, All trig functions are positive. In quadrant II, Sine (and cosec) are positive. In quadrant III, Tangent (and cot) are positive. In quadrant IV, Cosine (and sec) are positive. Helps determine sign of trig values.
• SOH-CAH-TOA: sin = Opposite/Hypotenuse, cos = Adjacent/Hypotenuse, tan = Opposite/Adjacent. The classic mnemonic for right triangle definitions.
• Double angle: sin 2θ = 2 sin θ cos θ (product of two), cos 2θ = cos²θ - sin²θ (difference of squares). Helps distinguish from compound angle formulas.

• Agniveer Navy papers often include 1-2 questions on simplifying trig expressions using Pythagorean identities. These are quick marks if you memorize the three identities. Expect questions like 'Simplify: (sin²θ + cos²θ) / cos²θ' — answer is sec²θ.
• Compound angle formulas appear in 1-2 questions, usually asking you to find sin/cos/tan of non-standard angles like 15°, 75°, 105°. Practice breaking these into sum/difference of 45°, 30°, 60°. Time pressure is real; pre-memorize sin 30°, cos 30°, sin 45°, cos 45°, sin 60°, cos 60°.
• Trig equation solving is tested occasionally. The exam usually asks for solutions in a specific range (0 to 2π or 0° to 360°). Always list all solutions in the given range; missing one costs marks. Double-check by substituting back.
• Double angle formulas appear less frequently but are tested in harder questions. If you see sin²θ or cos²θ in an expression, think power reduction: sin²θ = (1 - cos 2θ)/2.
• Watch out for questions mixing multiple concepts: e.g., 'If sin θ = 3/5 and 0 < θ < π/2, find sin 2θ and cos 2θ.' You must find cos θ first (using Pythagorean identity), then apply double angle formulas. These multi-step problems test conceptual clarity.