Trigonometric Identities & Equations Solver

Trigonometric Identities Guide

What's the key thing to understand about Pythagorean Identities?

sin²A + cos²A = 1 (the fundamental Pythagorean identity). Derived: tan²A + 1 = sec²A. 1 + cot²A = cosec²A. Example: if sin A = 3/5, find cos A. cos²A = 1 − 9/25 = 16/25. cos A = ±4/5 (sign depends on quadrant). These identities convert between sin, cos, and tan in simplification or solving equations. Combined with the unit circle, they allow finding all trig values from any one given value (within ambiguity over quadrant). Most-used identity in calculus and physics.

What should I know about Double Angle Identities?

sin(2A) = 2 sin A cos A. cos(2A) = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1. tan(2A) = 2 tan A / (1 − tan²A). Example: sin(2 × 30°) = sin(60°) = √3/2 ≈ 0.866. Verify: 2 × sin(30°) × cos(30°) = 2 × 0.5 × (√3/2) = √3/2 ✓. Applications: integration (∫sin²x dx uses cos(2x) = 1 − 2sin²x → sin²x = (1−cos(2x))/2). Wave physics (intensity proportional to amplitude², leading to factor-of-2 doubling). Used in solving equations like sin(2x) = sin(x): 2 sin x cos x = sin x → sin x (2 cos x − 1) = 0 → x = 0°, 60°, 180°, 300° within one full rotation.

What do I need to know about Sum and Difference Identities?

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). Useful for: exact values (sin 75° = sin(45° + 30°) using known exact values for 30° and 45°). Reducing complicated expressions to known angles. Phasor addition in physics — adding two oscillations with different phases. AC circuit analysis uses these identities constantly.

What's the key thing to understand about Half-Angle Identities?

sin(A/2) = ±√((1 − cos A)/2). cos(A/2) = ±√((1 + cos A)/2). tan(A/2) = (1 − cos A)/sin A = sin A/(1 + cos A). Sign depends on which quadrant A/2 is in. Weierstrass substitution: t = tan(A/2). sin A = 2t/(1+t²). cos A = (1−t²)/(1+t²). Converts any rational function of trig to a rational function of t — making many integrals possible to solve analytically. Half-angle identities also useful in solving complex trig equations and in the derivation of geometric formulas for triangle areas.

What should I know about Proving Identities Using Substitution?

To verify a trig identity numerically, substitute a specific angle into both sides and confirm they match — a quick sanity check before or instead of a full algebraic proof. For cos(2θ) = 1 − 2sin²(θ) at θ=30°: left side = cos(60°) = 0.5; right side = 1 − 2×(0.5)² = 1−0.5 = 0.5 ✓. This doesn't prove the identity holds for all θ (a single matching value could be coincidence), but it's an excellent way to catch errors in a derived or memorised identity before relying on it further, and it's exactly how to check your own algebraic identity proofs for mistakes — if both sides don't match at an easy test angle like 30°, 45°, or 60°, there's an error somewhere in the derivation that needs fixing before moving on.

What do I need to know about Using Identities to Simplify Expressions?

Trig identities are most useful for collapsing a complicated expression into a single term, often as a step inside a larger integration or equation-solving problem. For example, sin²(x) + cos²(x) always simplifies to 1 regardless of x — so an expression like 3sin²(x) + 3cos²(x) − 1 instantly simplifies to 3(1) − 1 = 2. Similarly, (1−cos(2x))/2 is identical to sin²(x) via the double-angle identity rearranged — this particular substitution is a standard trick for integrating sin²(x), which has no simple direct antiderivative but becomes straightforward once rewritten using the double-angle form. Recognising these patterns — a squared sine or cosine, a product of sine and cosine, or an expression matching a known sum/difference form — is the core skill tested in A-Level trig identity questions, more so than memorising the identities themselves.

What should I know about Reciprocal and Quotient Identities?

Beyond sine, cosine, and tangent, three reciprocal functions complete the full set of trigonometric ratios: cosecant (csc θ = 1/sin θ), secant (sec θ = 1/cos θ), and cotangent (cot θ = 1/tan θ = cos θ/sin θ). The quotient identities connect the three primary functions directly: tan θ = sin θ/cos θ, and cot θ = cos θ/sin θ, which follow immediately from the SOH-CAH-TOA definitions of each ratio in a right triangle. These identities are essential for simplifying more complex trig expressions — an expression mixing sin, cos, csc, and cot can almost always be rewritten entirely in terms of sin and cos using these relationships, which is usually the first step recommended when a trig identity proof looks stuck, since it reduces the number of different functions involved down to just two.

What should I know about a Trigonometric Identity Quick Reference?

What should I know about a Worked Example: Solving a Trig Equation?

As a worked example combining the double-angle identity with equation-solving, solve cos(2x)+sin(x)=1 for 0°≤x≤360°. Rewrite cos(2x) using cos(2x)=1−2sin²(x): (1−2sin²x)+sin(x)=1. Simplifying: −2sin²x+sin(x)=0, so sin(x)(1−2sin(x))=0. This gives two cases: sin(x)=0, giving x=0°, 180°, 360°; or sin(x)=0.5, giving x=30° or 150°. The full solution set: x=0°, 30°, 150°, 180°, 360°. Checking x=30°: cos(60°)+sin(30°)=0.5+0.5=1 ✓. Checking x=150°: cos(300°)+sin(150°)=0.5+0.5=1 ✓. Using a double-angle identity to reduce an equation to a single trig function, then factoring, is the standard approach to solving most trig equations beyond the simplest cases.

What should I know about Triple Angle Identities?

Beyond the commonly used double-angle identities, triple-angle identities extend the same idea one step further: sin(3A) = 3sin(A) − 4sin³(A), and cos(3A) = 4cos³(A) − 3cos(A). These follow from repeatedly applying the sum identity (writing 3A as 2A+A and expanding), the same derivation technique used for double-angle identities, just applied one extra time. As a quick check at A=30°: sin(90°) should equal 1, and 3sin(30°)−4sin³(30°) = 3(0.5)−4(0.125) = 1.5−0.5 = 1 ✓. Triple-angle identities appear less frequently than double-angle ones in standard coursework, but they're essential in certain integration techniques and in deriving exact trigonometric values for angles like 15° and 20°.

What should I know about Product-to-Sum Identities?

Beyond the sum, difference, and double-angle identities, a further family converts a product of two trig functions into a sum (or vice versa) — useful in calculus and signal processing. The key identity: sinA cosB = ½[sin(A+B) + sin(A−B)]. Similarly, cosA cosB = ½[cos(A−B)+cos(A+B)], and sinA sinB = ½[cos(A−B)−cos(A+B)]. Worked example: express sin(50°)cos(20°) as a sum. Using the identity: sin(50°)cos(20°) = ½[sin(70°)+sin(30°)] = ½[0.9397+0.5] = 0.71985. Checking directly: sin(50°)×cos(20°) = 0.766×0.9397 ≈ 0.7198 ✓. These product-to-sum identities are the mathematical basis of amplitude modulation in radio broadcasting, where multiplying (mixing) two sine waves of different frequencies produces exactly the sum-and-difference frequency components these identities predict.

What should I know about Solving Trig Equations with Multiple Angles?

Equations like sin(3x) = 0.5 require extra care with the solution range, because the multiple angle (3x here) sweeps through a much wider range of values than x itself as x varies over its given interval. Solving for 0°≤x≤360° means 3x ranges over 0° to 1080° — three full rotations, not one — so every valid solution for 3x within that extended range must be found before dividing back down to get x. Since sin(θ)=0.5 gives θ=30°, 150°, 390°, 510°, 750°, 870° within 0° to 1080°, dividing each by 3 gives x=10°, 50°, 130°, 170°, 250°, 290° — six solutions in total, not just the two you'd get from solving sin(x)=0.5 directly. Forgetting to extend the range for the multiple angle before solving, and then dividing by the correct number of extra solutions found within that extended range, is one of the most common errors in trigonometric equation-solving at A-Level and beyond.

What should I know about Cofunction Identities?

Cofunction identities relate a trig function of an angle to its 'co' partner at the complementary angle: sin(90°−A)=cos(A), and cos(90°−A)=sin(A), following directly from the fact that the two non-right angles in a right triangle always sum to 90°, so swapping which angle you measure from swaps sine and cosine. For example, sin(60°)=cos(30°), both equal to √3/2≈0.866. This identity is why sine and cosine graphs are simply horizontal shifts of each other.