Introduction
Trigonometric identities are equations involving trigonometric functions (sin, cos, tan, etc.) that are true for all values of the angle (where both sides are defined). These identities are essential in algebra, calculus, physics, and engineering because they help simplify expressions and solve trigonometric equations efficiently.
In this complete guide, you will learn all major trigonometric identities, their classifications, explanations, and solved examples.
What Are Trigonometric Identities?
Trigonometric identities are formulas that relate trigonometric functions to each other. They are derived from the unit circle, right triangle definitions, or Euler’s formula.
They help in: Simplifying complex trigonometric expressions, Solving equations, Proving mathematical results, Calculus integration and differentiation
1. Basic Trigonometric Ratios
For a right triangle, there are six trigonometric ratios.
\( \begin{aligned}
& 1. \sin \theta = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{p}{h} \\
& 2. \cos \theta = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{b}{h} \\
& 3. \tan \theta = \frac{\text{Perpendicular}}{\text{Base}} = \frac{p}{b} \\
\end{aligned}\)
\( \begin{aligned}
& 4. \text{ cosec } \theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{h}{p} \\
& 5. \sec \theta = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{h}{b} \\
& 6. \cot \theta = \frac{\text{Base}}{\text{Perpendicular}} = \frac{b}{p} \\
\end{aligned}\)
2. Reciprocal Identities
These identities express trigonometric functions as reciprocals:
\( \begin{aligned}
& 1. \sin \theta = \frac{1}{\csc \theta} \\
& 2. \cos \theta = \frac{1}{\sec \theta} \\
& 3. \tan \theta = \frac{1}{\cot \theta}
\end{aligned}\)
\( \begin{aligned}
& 4. \text{ cosec } \theta = \frac{1}{\sin \theta} \\
& 5. \sec \theta = \frac{1}{\cos \theta} \\
& 6. \cot \theta = \frac{1}{\tan \theta}
\end{aligned}\)
3. Quotient Identities
These relate tangent and cotangent with sine and cosine:
\( \begin{aligned}
& 1. \tan \theta = \frac{\sin \theta}{\cos \theta} \\
& 2. \cot \theta = \frac{\cos \theta}{\sin \theta}
\end{aligned}\)
4. Pythagorean Identities
These are the most important trigonometric identities derived from the Pythagorean theorem. These identities are frequently used to convert between trigonometric functions.
\(\begin{aligned}
& a. sin^{2}\theta + cos^{2}\theta = 1 \\
& b. 1 + tan^{2}\theta = sec^{2}\theta \\
& c. 1 + cot^{2}\theta = cosec^{2}\theta
\end{aligned}\)
5. Co-function Identities
These identities relate complementary angles:
\(\begin{aligned}
& A. (90^\circ - \theta) \\
& 1. \sin(90^\circ - \theta) = \cos\theta \\
& 2. \cos(90^\circ - \theta) = \sin\theta \\
& 3. \tan(90^\circ - \theta) = \cot\theta \\
& 4. \csc(90^\circ - \theta) = \sec\theta \\
& 5. \sec(90^\circ - \theta) = \csc\theta \\
& 6. \cot(90^\circ - \theta) = \tan\theta \\ \\
\end{aligned}\)
6. Even–Odd Identities
These identities depend on whether functions are even or odd.
Even Functions
\(\cos(-\theta) = \cos \theta, \quad \sec(-\theta) = \sec \theta\)
Odd Functions
\(\sin(-\theta) = -\sin \theta, \quad \tan(-\theta) = -\tan \theta\)
\(\cot(-\theta) = -\cot \theta, \quad \csc(-\theta) = -\csc \theta\)
7. Sum and Difference Identities
These identities are used to find trigonometric values of compound angles.
\(\begin{aligned}
& 1. sinC + sinD = 2sin(\frac{C+D}{2})cos(\frac{C-D}{2}) \\
& 2. sinC - sinD = 2cos(\frac{C+D}{2})sin(\frac{C-D}{2}) \\
& 3. cosC + cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2}) \\
& 4. cosC - cosD = 2sin(\frac{C+D}{2})sin(\frac{C-D}{2}) \\
\end{aligned}\)
8. Double Angle Identities
Used when angle is multiplied by 2.
\(\begin{aligned}
& 1. sin2A = 2sinAcosA, \frac{2tanA}{1 + tan^{2}A} \\
& 2. cos2A = cos^{2}A – sin^{2}A, 2cos^{2}A – 1, 1 – 2sin^{2}A \\
& 3. tan2A = \frac{2tanA}{1 – tan^{2}A} \\
& 4. sin3A = 3sinA – 4sin^{3}A \\
& 5. cos3A = 4cos^{3}A – 3cosA \\
& 6. tan3A = \frac{3tanA – tan^{3}A}{1-3tan^{2}A}
\end{aligned}\)
9. Half-Angle Identities
Used when the angle is halved.
\(\begin{aligned}
& 1. sinA = 2sin\frac{A}{2}cos\frac{A}{2} \\
& 2. cosA = cos^{2}\frac{A}{2} – sin^{2}\frac{A}{2}, 2cos^{2}\frac{A}{2} – 1, 1 – 2sin^{2}\frac{A}{2} \\
& 3. tanA = \frac{2tan\frac{A}{2}}{1 – tan^{2}\frac{A}{2}} \\
& 4. sinA = 3sin\frac{A}{3} – 4sin^{3}\frac{A}{3} \\
& 5. cosA = 4cos^{3}\frac{A}{3} – 3cos\frac{A}{3} \\
& 6. tanA = \frac{3tan\frac{A}{3} – tan^{3}\frac{A}{3}}{1-3tan^{2}\frac{A}{3}}
\end{aligned}\)
10. Product-to-Sum Identities
Used to convert products into sums.
\(\begin{aligned}
& 1. 2sinA.cosB = sin(A+B) + sin(A-B) \\
& 2. 2cosA.sinB = sin(A+B) - sin(A-B) \\
& 3. 2cosA.cosB = cos(A+B) + cos(A-B) \\
& 4. 2sinA.sinB = cos(A-B) - sin(A+B) \\
\end{aligned}\)
11. Sum-to-Product Identities
Reverse of product-to-sum:
\(\begin{aligned}
& 1. sinC + sinD = 2sin(\frac{C+D}{2})cos(\frac{C-D}{2}) \\
& 2. sinC - sinD = 2cos(\frac{C+D}{2})sin(\frac{C-D}{2}) \\
& 3. cosC + cosD = 2cos(\frac{C+D}{2})cos(\frac{C-D}{2}) \\
& 4. cosC - cosD = 2sin(\frac{C+D}{2})sin(\frac{C-D}{2}) \\
\end{aligned}\)
Importance of Trigonometric Identities
Trigonometric identities are widely used in:
- Geometry and coordinate geometry
- Calculus (integration & differentiation)
- Physics waves and oscillations
- Engineering signal processing
- Astronomy and navigation
Conclusion
Trigonometric identities form the backbone of trigonometry. Mastering reciprocal, quotient, Pythagorean, angle sum/difference, double angle, and half-angle identities enables you to simplify complex expressions and solve advanced mathematical problems efficiently.
