Trigonometry formulas

All 20 formulas in this topic

Sine Cosine Tangentsin⁡θ=opphyp,cos⁡θ=adjhyp,tan⁡θ=oppadj\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}Find missing sides or angles in right triangles using sine, cosine, and tangent ratios—essential for SAT and ACT geometry.Pythagorean Identitysin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1The Pythagorean Identity states that sine squared plus cosine squared always equals one, helping you simplify trigonometric expressions and solve equations.Law of Sinesasin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}Law of Sines connects side lengths to their opposite angles in a triangle, used to find unknown sides or angles when an angle-side pair is known.Law of Cosinesc2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos CFinds unknown sides or angles in any triangle when you know two sides and the angle between them, or all three sides.Unit Circle(cos⁡θ,sin⁡θ) on x2+y2=1(\cos\theta, \sin\theta) \text{ on } x^2 + y^2 = 1The unit circle is a fundamental tool that shows how angles map to sine and cosine coordinates: the point (cos θ, sin θ) always lies on x² + y² = 1.Radians to Degreesθdeg⁡=θrad⋅180π\theta_{\deg} = \theta_{\text{rad}} \cdot \frac{180}{\pi}Convert an angle from radians to degrees using this formula, essential for reading angle measures in different units.Reciprocal Identitiescsc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}Reciprocal identities convert trig functions to their reciprocals, helping you simplify expressions and solve equations with cosecant, secant, and cotangent.Sine Addition Formulasin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B\sin(A \pm B) = \sin A\cos B \pm \cos A\sin BThe Sine Addition Formula expands sine of a sum or difference into a product of sines and cosines, making it useful for exact angle calculations.Cosine Addition Formulacos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B\cos(A \pm B) = \cos A\cos B \mp \sin A\sin BFind the cosine of an angle sum or difference by using the cosine and sine values of the individual angles—essential for solving trigonometric equations.Tangent Addition Formulatan⁡(A±B)=tan⁡A±tan⁡B1∓tan⁡Atan⁡B\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A\tan B}Find the tangent of angle sums and differences using a single formula that combines the tangent values of each angle.Double Angle Formulassin⁡2θ=2sin⁡θcos⁡θ,cos⁡2θ=cos⁡2θ−sin⁡2θ\sin 2\theta = 2\sin\theta\cos\theta, \quad \cos 2\theta = \cos^2\theta - \sin^2\thetaDouble angle formulas express trig functions of 2θ in terms of single angles, letting you simplify complex expressions and solve trigonometric equations.Half Angle Formulassin⁡θ2=±1−cos⁡θ2\sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}Half angle formulas express trigonometric functions of half an angle in terms of the full angle, helping solve equations and simplify expressions.Sum to Product Formulassin⁡A+sin⁡B=2sin⁡A+B2cos⁡A−B2\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}Convert sums of sines or cosines into products, making trigonometric equations and expressions easier to simplify and solve.Area of a Triangle with SineA=12absin⁡CA = \frac{1}{2}ab\sin CFind the area of a triangle when you know two sides and the included angle—much faster than using just base and height.Inverse Trigonometric Functionsθ=arcsin⁡x  ⟺  sin⁡θ=x,;−π2≤θ≤π2\theta = \arcsin x \iff \sin\theta = x, ; -\tfrac{\pi}{2} \le \theta \le \tfrac{\pi}{2}Inverse trigonometric functions find angles when you know their sine, cosine, or tangent values; use them to solve equations where an angle is unknown.Amplitude and Periody=asin⁡(b(x−h))+k,period=2π∣b∣y = a\sin\big(b(x - h)\big) + k, \quad \text{period} = \frac{2\pi}{|b|}Amplitude and Period tell you how tall and wide a sine wave is, used when analyzing or graphing trigonometric functions and oscillating systems.Cofunction Identitiessin⁡(π2−θ)=cos⁡θ\sin\left(\tfrac{\pi}{2} - \theta\right) = \cos\thetaCofunction identities relate sine and cosine of complementary angles, letting you switch between them when angles sum to 90° or π/2 radians.Reference Angleθ′=acute angle to the x-axis\theta' = \text{acute angle to the } x\text{-axis}Find the acute angle between any angle's terminal side and the x-axis to determine trigonometric values using reference angle formula.Solving Trigonometric Equationssin⁡θ=c  ⟹  θ=arcsin⁡c+2πn,;π−arcsin⁡c+2πn\sin\theta = c \implies \theta = \arcsin c + 2\pi n, ; \pi - \arcsin c + 2\pi nFind all angles that produce a specific sine value by using inverse sine and the periodic nature of trigonometric functions.Polar Coordinatesx=rcos⁡θ,y=rsin⁡θ,r2=x2+y2x = r\cos\theta, \quad y = r\sin\theta, \quad r^2 = x^2 + y^2Polar coordinates describe positions using distance from the origin and angle, providing a natural framework for circles, rotations, and periodic patterns.

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