Trigonometry Definition, Formulas, Ratios, & Identities 2024

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Trigonometry Definition

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. The word “trigonometry” is derived from the Greek words “trigonon,” meaning triangle, and “metron,” meaning measure. This mathematical discipline has applications in various fields, including physics, engineering, computer science, and more.

Trigonometric Ratios

Trigonometry primarily involves six trigonometric ratios that relate the angles and sides of a right-angled triangle. These ratios are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

Sine (sinθ)

Definition: In a right-angled triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Formula-

\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}

Cosine (cosθ)

Definition: The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse.

Formula-

\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

Tangent (tanθ)

Definition: Tangent is the ratio of the length of the side opposite the angle to the length of the adjacent side in a right-angled triangle.

Formulas-

\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

Cosecant (cscθ)

Definition: Cosecant is the reciprocal of sine. It is the ratio of the length of the hypotenuse to the length of the side opposite the angle.

Formulas-

\csc(\theta) = \frac{1}{\sin(\theta)}

Secant (secθ)

Definition: Secant is the reciprocal of cosine. It is the ratio of the length of the hypotenuse to the length of the adjacent side.

Formulas-

\sec(\theta) = \frac{1}{\cos(\theta)}

Cotangent (cotθ)

Definition: Cotangent is the reciprocal of tangent. It is the ratio of the length of the adjacent side to the length of the side opposite the angle.

Formulas-

\cot(\theta) = \frac{1}{\tan(\theta)}

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions. They are useful for simplifying expressions and solving trigonometric equations.

Some fundamental trigonometric identities include.

Pythagorean Identities

$\sin^2(\theta) + \cos^2(\theta) = 1$

$1 + \tan^2(\theta) = \sec^2(\theta)$

$1 + \cot^2(\theta) = \csc^2(\theta)$

Reciprocal Identities

$\csc(\theta) = \frac{1}{\sin(\theta)}$

$\sec(\theta) = \frac{1}{\cos(\theta)}$

$\cot(\theta) = \frac{1}{\tan(\theta)}$

Quotient Identities

$\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$

$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$

Co-function Identities

$\sin\left(\frac{\pi}{2} – \theta\right) = \cos(\theta)$

$\cos\left(\frac{\pi}{2} – \theta\right) = \sin(\theta)$

$\tan\left(\frac{\pi}{2} – \theta\right) = \cot(\theta)$

Trigonometric Formulas

Beyond basic ratios and identities, trigonometry involves various formulas for solving problems and analyzing different scenarios.

Some key formulas include

Angle Sum and Difference Formulas

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle Formulas

$\sin(2\theta) = 2\sin\theta\cos\theta$

$\cos(2\theta) = \cos^2\theta – \sin^2\theta$

$\tan(2\theta) = \frac{2\tan\theta}{1 – \tan^2\theta}$

Half Angle Formulas

$\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 – \cos\theta}{2}}$

$\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos\theta}{2}}$

$\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 – \cos\theta}{1 + \cos\theta}}$

Law of Sines

For any triangle with sides a, b, and c, and angles A, B, and C

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Law of Cosines

For any triangle with sides a, b, and c, and angle C

c^2 = a^2 + b^2 – 2ab\cos C

Area of a Triangle

For a triangle with sides a, b, and included angle C

\text{Area} = \frac{1}{2}ab\sin C

Product to Sum and Sum to Product Formulas

Product to Sum

$\sin A \sin B = \frac{1}{2}[\cos(A – B) – \cos(A + B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A – B) + \cos(A + B)]$

$\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A – B)]$

Sum to Product

$\sin A + \sin B = 2\sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A – B}{2}\right)$

$\sin A – \sin B = 2\cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A – B}{2}\right)$

$\cos A + \cos B = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A – B}{2}\right)$

$\cos A – \cos B = -2\sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A – B}{2}\right)$

Cofunction Identities

$\sin(\pi – \theta) = \sin \theta$

$\cos(\pi – \theta) = -\cos \theta$

$\tan(\pi – \theta) = -\tan \theta$

Power-Reducing Formulas

$\sin^2 \theta = \frac{1 – \cos(2\theta)}{2}$

$\cos^2 \theta = \frac{1 + \cos(2\theta)}{2}$

Triple Angle Formulas

$\sin(3\theta) = 3\sin\theta – 4\sin^3\theta$

$\cos(3\theta) = 4\cos^3\theta – 3\cos\theta$

$\tan(3\theta) = \frac{3\tan\theta – \tan^3\theta}{1 – 3\tan^2\theta}$

Sine and Cosine Addition Formulas

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B – \sin A \sin B$

Inverse Trigonometric Function Formulas

$\sin^{-1}(x) + \cos^{-1}(x) = \frac{\pi}{2}$

$\tan^{-1}(x) + \cot^{-1}(x) = \frac{\pi}{2}$

Half Angle Tangent Formula

\tan\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1 + \cos\theta}

Area of a Sector of a Circle

For a circle with radius r and central angle θ

$\text{Area} = \frac{\theta}{360^\circ} \pi r^2$

Law of Tangents

For a triangle with sides a,b,c and angles A, B, C

$\frac{a-b}{a+b} = \frac{\tan\left(\frac{A-B}{2}\right)}{\tan\left(\frac{A+B}{2}\right)}$

Euler’s Formula

e^{i\theta} = \cos \theta + i \sin \theta

This formula connects exponential functions, trigonometric functions, and imaginary numbers.

Sum of Cubes Formulas

$\sin^3 \theta = \frac{3}{4}\sin \theta – \frac{1}{4}\sin(3\theta)$

$\cos^3 \theta = \frac{3}{4}\cos \theta + \frac{1}{4}\cos(3\theta)$

Hyperbolic Trigonometric Functions

$\sinh x = \frac{e^x – e^{-x}}{2}$

$\cosh x = \frac{e^x + e^{-x}}{2}$

$\tanh x = \frac{\sinh x}{\cosh x}$

Inverse Hyperbolic Functions

$\sinh^{-1} x = \ln(x + \sqrt{x^2 + 1})$

$\cosh^{-1} x = \ln(x + \sqrt{x^2 – 1})$

$\tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)$

Sine and Cosine Integrals

$\text{Si}(x) = \int_0^x \frac{\sin t}{t} \,dt$

$\text{Ci}(x) = -\int_x^\infty \frac{\cos t}{t} \,dt$

Product of Sines and Cosines

$\sin A \sin B = \frac{1}{2}[\cos(A – B) – \cos(A + B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A – B) + \cos(A + B)]$

Trigonometric Substitution (Calculus)

Used in integration, such as

\int \sqrt{a^2 – x^2} \,dx

\int \sqrt{x^2 – a^2} \,dx

Periodicity Formulas

$\sin(\theta + 2\pi n) = \sin \theta$

$\cos(\theta + 2\pi n) = \cos \theta$

$\tan(\theta + \pi n) = \tan \theta$ , where $n$ is an integer.

General Angle Formulas

$\sin(A + n360^\circ) = \sin A$

$\cos(A + n360^\circ) = \cos A$

$\tan(A + n180^\circ) = \tan A$ , where $n$ is an integer.

Law of Sines for Non-Right Triangles

For any triangle with sides a, b, c and angles A, B, C

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Law of Cosines for Non-Right Triangles

For any triangle with sides a, b, c and angles A, B, C

$c^2 = a^2 + b^2 – 2ab\cos C$

These formulas and identities form the basis for solving a wide range of trigonometric problems in geometry, physics, engineering, and other disciplines. Understanding and applying these concepts is important for anyone dealing with angles, triangles, and periodic phenomena.

Conclusion

Trigonometry is a rich and versatile branch of mathematics that plays a fundamental role in various scientific and technological fields. The relationships and properties described by trigonometric ratios, identities, and formulas provide a powerful toolkit for solving problems related to angles and triangles, making it an essential subject for students and professionals.

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