Trigonometry

Trigonometric functions are a common type of angle-related functions in mathematics.

Common trigonometric functions include the sine function (sin), cosine function (cos), and tangent function (tan or tg or tang); in navigation, surveying, engineering, and other disciplines, other trigonometric functions such as the cotangent function (cot or ctg), secant function (sec), cosecant function (csc), versine function, and haversine function are also used.

1. Definitions of Trigonometric Functions

The definitions of trigonometric functions are the starting point for understanding this type of function. This chapter clarifies the geometric meanings of basic trigonometric functions (such as sine being the ratio of the opposite side to the hypotenuse) based on the relationships between sides in a right-angled triangle, and extends to their definitions in the coordinate system (e.g., sine as the ratio of the y-coordinate to the radius). It establishes a direct connection between angles and numerical values. These definitions lay the foundation for subsequent formula derivation, property analysis, and practical applications, helping us understand the essence of trigonometric functions from both “geometric” and “numerical” perspectives.

Trigonometry

In a right-angled triangle, when the connections of three points A, B, and C on a plane, namely AB, AC, and BC, form a right-angled triangle with ∠ACB being the right angle. For ∠BAC, let the opposite side a = BC, the hypotenuse c = AB, the adjacent side b = AC, and denote θ = ∠BAC, then the following relationships exist:

1.1 Sine Formula

The \enspace sine \enspace of \enspace {\displaystyle \theta }：\\ sin{\displaystyle \theta }=\frac{opposite}{hypotenuse}=\frac{a}{c}
sin{\displaystyle \theta }=\frac{y}{r}

1.2 Cosine Formula

The \enspace cosine  \enspace of \enspace {\displaystyle \theta }：\\ cos{\displaystyle \theta }=\frac{Adjacent}{hypotenuse}=\frac{b}{c}
cos{\displaystyle \theta }=\frac{x}{r}

1.3 Tangent Formula

The \enspace tangent \enspace of \enspace {\displaystyle \theta }：\\ tan{\displaystyle \theta }=\frac{opposite}{Adjacent}=\frac{a}{b}
tan{\displaystyle \theta }=\frac{y}{x}

1.4 Cotangent Formula

The \enspace cotangent \enspace of \enspace {\displaystyle \theta }：\\cot{\displaystyle \theta }=\frac{Adjacent}{Opposite}=\frac{b}{a}
cot{\displaystyle \theta }=\frac{x}{y}

1.5 Secant Formula

The \enspace secant  \enspace of \enspace {\displaystyle \theta }：\\ sec{\displaystyle \theta }=\frac{Hypotenuse}{Adjacent}=\frac{c}{b}
sec{\displaystyle \theta }=\frac{r}{x}

1.6 Cosecant Formula

The \enspace cosecant \enspace of \enspace {\displaystyle \theta }：\\ csc{\displaystyle \theta }=\frac{Hypotenuse}{Opposite}=\frac{c}{a}
csc{\displaystyle \theta }=\frac{r}{y}

2. Function Formulas

This chapter focuses on the basic relationships between trigonometric functions, including reciprocal relationships (e.g., tangent and cotangent are reciprocals), quotient relationships (e.g., tangent equals the ratio of sine to cosine), and Pythagorean relationships (e.g., the sum of the squares of sine and cosine is 1). These formulas form the “skeleton” of the trigonometric system. They not only reveal the intrinsic connections between different functions but also provide key tools for simplifying expressions, proving equalities, and transforming operational forms, serving as the fundamental basis for solving trigonometric problems.

2.1 Reciprocal Relationships

tan{\displaystyle \alpha}*cot{\displaystyle \alpha}=1
sin{\displaystyle \alpha}*csc{\displaystyle \alpha}=1
cos{\displaystyle \alpha}*sec{\displaystyle \alpha}=1

2.2 Quotient Relationships

tan{\displaystyle \alpha}=\frac{sin{\displaystyle \alpha}}{cos{\displaystyle \alpha}}
cot{\displaystyle \alpha}=\frac{cos{\displaystyle \alpha}}{sin{\displaystyle \alpha}}

2.3 Pythagorean Relationships

sin^2{\displaystyle \alpha}+cos^2{\displaystyle \alpha}=1
1+tan^2{\displaystyle \alpha}=sec^2{\displaystyle \alpha}
1+cot^2{\displaystyle \alpha}=csc^2{\displaystyle \alpha}

3. Reduction Formulas

Reduction formulas are core tools for handling trigonometric functions of arbitrary angles. This chapter, through a series of formulas (e.g., relationships between trigonometric functions of angles with the same terminal side, π±α, π/2±α, etc., and those of α), converts the trigonometric values of arbitrary angles into those of acute angles, significantly simplifying calculation difficulty. The mnemonic “Odd changes, even remains; sign depends on the quadrant” helps learners quickly grasp the pattern of the formulas, making it easy to handle angle transformation problems.

3.1 Formula 1

Let α be any angle; the values of the same trigonometric function for angles with the same terminal side are equal.

sin(2k{\displaystyle \pi}+{\displaystyle \alpha})=sin{\displaystyle \alpha}, k∈Z
cos(2k{\displaystyle \pi}+{\displaystyle \alpha})=cos{\displaystyle \alpha}, k∈Z
tan(2k{\displaystyle \pi}+{\displaystyle \alpha})=tan{\displaystyle \alpha}, k∈Z
cot(2k{\displaystyle \pi}+{\displaystyle \alpha})=cot{\displaystyle \alpha}, k∈Z

3.2 Formula 2

Let α be any angle; the relationship between the trigonometric function values of π + α and α.

sin({\displaystyle \pi}+{\displaystyle \alpha})=-sin{\displaystyle \alpha}
cos({\displaystyle \pi}+{\displaystyle \alpha})=-cos{\displaystyle \alpha}
tan({\displaystyle \pi}+{\displaystyle \alpha})=tan{\displaystyle \alpha}
cot({\displaystyle \pi}+{\displaystyle \alpha})=cot{\displaystyle \alpha}

3.3 Formula 3

The relationship between the trigonometric function values of any angle α and -α.

sin(-{\displaystyle \alpha})=-sin{\displaystyle \alpha}
cos(-{\displaystyle \alpha})=cos{\displaystyle \alpha}
tan(-{\displaystyle \alpha})=-tan{\displaystyle \alpha}
cot(-{\displaystyle \alpha})=-cot{\displaystyle \alpha}

3.4 Formula 4

The relationship between the trigonometric function values of π – α and α.

sin({\displaystyle \pi}-{\displaystyle \alpha})=sin{\displaystyle \alpha}
cos({\displaystyle \pi}-{\displaystyle \alpha})=-cos{\displaystyle \alpha}
tan({\displaystyle \pi}-{\displaystyle \alpha})=-tan{\displaystyle \alpha}
cot({\displaystyle \pi}-{\displaystyle \alpha})=-cot{\displaystyle \alpha}

3.5 Formula 5

The relationship between the trigonometric function values of 2π – α and α.

sin(2{\displaystyle \pi}-{\displaystyle \alpha})=-sin{\displaystyle \alpha}
cos(2{\displaystyle \pi}-{\displaystyle \alpha})=cos{\displaystyle \alpha}
tan(2{\displaystyle \pi}-{\displaystyle \alpha})=-tan{\displaystyle \alpha}
cot(2{\displaystyle \pi}-{\displaystyle \alpha})=-cot{\displaystyle \alpha}

3.6 Formula 6

The relationship between the trigonometric function values of π/2 ± α and α.

sin(\frac{{\displaystyle \pi}}{2}+{\displaystyle \alpha})=cos{\displaystyle \alpha}
cos(\frac{{\displaystyle \pi}}{2}+{\displaystyle \alpha})=-sin{\displaystyle \alpha}
tan(\frac{{\displaystyle \pi}}{2}+{\displaystyle \alpha})=-cot{\displaystyle \alpha}
cot(\frac{{\displaystyle \pi}}{2}+{\displaystyle \alpha})=-tan{\displaystyle \alpha}
sin(\frac{{\displaystyle \pi}}{2}-{\displaystyle \alpha})=cos{\displaystyle \alpha}
cos(\frac{{\displaystyle \pi}}{2}-{\displaystyle \alpha})=sin{\displaystyle \alpha}
tan(\frac{{\displaystyle \pi}}{2}-{\displaystyle \alpha})=cot{\displaystyle \alpha}
cot(\frac{{\displaystyle \pi}}{2}-{\displaystyle \alpha})=-tan{\displaystyle \alpha}

3.7 Formula 7

The relationship between the trigonometric function values of 3π/2 ± α and α.

sin(\frac{{3\displaystyle \pi}}{2}+{\displaystyle \alpha})=-cos{\displaystyle \alpha}
cos(\frac{{3\displaystyle \pi}}{2}+{\displaystyle \alpha})=sin{\displaystyle \alpha}
tan(\frac{3{\displaystyle \pi}}{2}+{\displaystyle \alpha})=-cot{\displaystyle \alpha}
cot(\frac{3{\displaystyle \pi}}{2}+{\displaystyle \alpha})=-tan{\displaystyle \alpha}
sin(\frac{3{\displaystyle \pi}}{2}-{\displaystyle \alpha})=-cos{\displaystyle \alpha}
cos(\frac{3{\displaystyle \pi}}{2}-{\displaystyle \alpha})=-sin{\displaystyle \alpha}
tan(\frac{3{\displaystyle \pi}}{2}-{\displaystyle \alpha})=cot{\displaystyle \alpha}
cot(\frac{3{\displaystyle \pi}}{2}-{\displaystyle \alpha})=tan{\displaystyle \alpha}

Mnemonic for memorization: Odd multiples of 90° change the function, even multiples do not; the sign is determined by the quadrant.

That is, for forms like (2k + 1)90° ± α, the function name changes to its co-function: sine becomes cosine, cosine becomes sine, tangent becomes cotangent, and cotangent becomes tangent. For forms like 2k × 90° ± α, the function name remains unchanged.

4. Basic Formulas

This chapter covers more complex core formulas in trigonometric operations, including sum and difference formulas (for calculating trigonometric functions of the sum or difference of two angles), sum-to-product and product-to-sum formulas (converting between sum/difference and product forms of trigonometric functions), double-angle and half-angle formulas (handling angle multiple or half relationships), auxiliary angle formulas (transforming linear combinations of sine and cosine into a single trigonometric function), universal formulas (expressing any trigonometric function using the tangent of the half-angle), and the law of cosines (relating sides and angles of a triangle). These formulas are “powerful tools” for solving practical problems such as geometric calculations and physical motion analysis, and are key to in-depth study of trigonometric function properties.

4.1 Sum and Difference Angle Formulas

Angle Sum and Difference Formulas

sin({\displaystyle \alpha}+{\displaystyle \beta})\\=sin{\displaystyle \alpha}*cos{\displaystyle \beta}+cos{\displaystyle \alpha}*sin{\displaystyle \beta}
sin({\displaystyle \alpha}-{\displaystyle \beta})\\=sin{\displaystyle \alpha}*cos{\displaystyle \beta}-cos{\displaystyle \alpha}*sin{\displaystyle \beta}
cos({\displaystyle \alpha}+{\displaystyle \beta})\\=cos{\displaystyle \alpha}*cos{\displaystyle \beta}-sin{\displaystyle \alpha}*sin{\displaystyle \beta}
cos({\displaystyle \alpha}-{\displaystyle \beta})\\=cos{\displaystyle \alpha}*cos{\displaystyle \beta}+sin{\displaystyle \alpha}*sin{\displaystyle \beta}
tan({\displaystyle \alpha}+{\displaystyle \beta})=\frac{tan{\displaystyle \alpha}+tan{\displaystyle \beta}}{1-tan{\displaystyle \alpha}*tan{\displaystyle \beta}}
tan({\displaystyle \alpha}-{\displaystyle \beta})=\frac{tan{\displaystyle \alpha}-tan{\displaystyle \beta}}{1+tan{\displaystyle \alpha}*tan{\displaystyle \beta}}
cot({\displaystyle \alpha}+{\displaystyle \beta})=\frac{cot{\displaystyle \alpha}*cot{\displaystyle \beta}-1}{cot{\displaystyle \alpha}+cot{\displaystyle \beta}}
cot({\displaystyle \alpha}-{\displaystyle \beta})=\frac{cot{\displaystyle \alpha}*cot{\displaystyle \beta}+1}{-cot{\displaystyle \alpha}+cot{\displaystyle \beta}}

Triple Angle Sum Formula

sin({\displaystyle \alpha}+{\displaystyle \beta}+{\displaystyle \gamma})\\=sin{\displaystyle \alpha}*cos{\displaystyle \beta}*cos{\displaystyle \gamma}\\+cos{\displaystyle \alpha}*sin{\displaystyle \beta}*cos{\displaystyle \gamma}\\+cos{\displaystyle \alpha}*cos{\displaystyle \beta}*sin{\displaystyle \gamma}\\-sin{\displaystyle \alpha}*sin{\displaystyle \beta}*sin{\displaystyle \gamma}
cos({\displaystyle \alpha}+{\displaystyle \beta}+{\displaystyle \gamma})\\=-cos{\displaystyle \alpha}*cos{\displaystyle \beta}*cos{\displaystyle \gamma}\\-cos{\displaystyle \alpha}*sin{\displaystyle \beta}*sin{\displaystyle \gamma}\\-sin{\displaystyle \alpha}*cos{\displaystyle \beta}*sin{\displaystyle \gamma}\\-sin{\displaystyle \alpha}*sin{\displaystyle \beta}*sin{\displaystyle \gamma}

4.2 Sum-to-Product Formulas

sin({\displaystyle \alpha})+sin({\displaystyle \beta})\\=2sin(\frac{{\displaystyle \alpha}+{\displaystyle \beta}}{2})cos(\frac{{\displaystyle \alpha}-{\displaystyle \beta}}{2})
sin({\displaystyle \alpha})-sin({\displaystyle \beta})\\=2cos(\frac{{\displaystyle \alpha}+{\displaystyle \beta}}{2})sin(\frac{{\displaystyle \alpha}-{\displaystyle \beta}}{2})
cos({\displaystyle \alpha})+cos({\displaystyle \beta})\\=2cos(\frac{{\displaystyle \alpha}+{\displaystyle \beta}}{2})cos(\frac{{\displaystyle \alpha}-{\displaystyle \beta}}{2})
cos({\displaystyle \alpha})-cos({\displaystyle \beta})\\=-2sin(\frac{{\displaystyle \alpha}+{\displaystyle \beta}}{2})sin(\frac{{\displaystyle \alpha}-{\displaystyle \beta}}{2})
tan({\displaystyle \alpha})+tan({\displaystyle \beta})=\frac{sin({\displaystyle \alpha}+{\displaystyle \beta})}{{cos\displaystyle \alpha}*cos{\displaystyle \beta}}

Mnemonic:
Sine plus sine, sine comes first; cosine plus cosine, cosines stand side by side.
Sine minus sine, cosine comes first; cosine minus cosine, negative sine.

4.3 Product-to-Sum Formulas

cos{\displaystyle \alpha}*sin{\displaystyle \beta}\\=\frac{1}{2}[sin({\displaystyle \alpha}+{\displaystyle \beta})-sin({\displaystyle \alpha}-{\displaystyle \beta})]
sin{\displaystyle \alpha}*cos{\displaystyle \beta}\\=\frac{1}{2}[sin({\displaystyle \alpha}+{\displaystyle \beta})+sin({\displaystyle \alpha}-{\displaystyle \beta})]
cos{\displaystyle \alpha}*cos{\displaystyle \beta}\\=\frac{1}{2}[cos({\displaystyle \alpha}+{\displaystyle \beta})+cos({\displaystyle \alpha}-{\displaystyle \beta})]
sin{\displaystyle \alpha}*sin{\displaystyle \beta}\\=-\frac{1}{2}[cos({\displaystyle \alpha}+{\displaystyle \beta})-cos({\displaystyle \alpha}-{\displaystyle \beta})]

4.4 Multiple Angle Formulas

Double Angle Formulas

sin(2{\displaystyle \alpha})=2sin{\displaystyle \alpha}*cos{\displaystyle \alpha}
cos(2{\displaystyle \alpha})\\=cos^2{\displaystyle \alpha}-sin^2{\displaystyle \alpha}\\=2cos^2{\displaystyle \alpha}-1\\=1-2sin^2{\displaystyle \alpha}
tan{2\displaystyle \alpha}=\frac{2tan{\displaystyle \alpha}}{1-tan^2{\displaystyle \alpha}}

Triple Angle Formulas

sin(3{\displaystyle \alpha})=3sin{\displaystyle \alpha}-4sin^3{\displaystyle \alpha}
cos(3{\displaystyle \alpha})=4cos^3{\displaystyle \alpha}-3cos{\displaystyle \alpha}
tan(3{\displaystyle \alpha})\\=tan{\displaystyle \alpha}*tan(60^o-{\displaystyle \alpha})*tan(60^o+{\displaystyle \alpha})

Half Angle Formulas

sin\frac{{\displaystyle \alpha}}{2}=±\sqrt{\frac{1-cos\displaystyle \alpha}{2}}
cos\frac{{\displaystyle \alpha}}{2}=±\sqrt{\frac{1+cos\displaystyle \alpha}{2}}
tan\frac{{\displaystyle \alpha}}{2}\\=±\sqrt{\frac{1-cos\displaystyle \alpha}{1+cos\displaystyle \alpha}}\\=\frac{sin\displaystyle \alpha}{1+cos\displaystyle \alpha}\\=\frac{1-cos\displaystyle \alpha}{sin\displaystyle \alpha}
cot\frac{{\displaystyle \alpha}}{2}\\=±\sqrt{\frac{1+cos\displaystyle \alpha}{1-cos\displaystyle \alpha}}\\=\frac{sin\displaystyle \alpha}{1-cos\displaystyle \alpha}\\=\frac{1+cos\displaystyle \alpha}{sin\displaystyle \alpha}

The sign is determined by the quadrant in which α/2 lies.

Auxiliary Angle Formula

a*sin{\displaystyle \alpha}+b*cos{\displaystyle \alpha}\\=\sqrt{a^2+b^2}*sin(\displaystyle \alpha+\displaystyle \gamma), tan\displaystyle \gamma=\frac{b}{a}

4.5 Universal Formulas

sin\displaystyle \alpha=\frac{2tan\frac{\displaystyle \alpha}{2}}{1+tan^2\frac{\displaystyle \alpha}{2}}
cos\displaystyle \alpha=\frac{1-tan^2\frac{\displaystyle \alpha}{2}}{1+tan^2\frac{\displaystyle \alpha}{2}}
tan\displaystyle \alpha=\frac{2tan\frac{\displaystyle \alpha}{2}}{1-tan^2\frac{\displaystyle \alpha}{2}}

4.6 Law of Cosines

a^2=b^2+c^2-2bc*cos\displaystyle \alpha
b^2=c^2+a^2-2ca*cos\displaystyle \beta
c^2=a^2+b^2-2ab*cos\displaystyle \gamma

5. Area Calculation with Trigonometric Function Formulas

The triangle area formula introduced in this chapter skillfully combines trigonometric functions with geometric area calculation. When two sides of a triangle and their included angle are known, the area can be quickly obtained using the sine of the included angle without relying on the length of the height, simplifying the cumbersome step of “finding the height” in traditional area calculation. It is widely used in scenarios such as geometric proofs and engineering surveys.

S_{\displaystyle \Delta}\\=\frac{1}{2}ab*sinC\\=\frac{1}{2}bc*sinA\\=\frac{1}{2}ac*sinB

6. Video Tutorials on Trigonometric Functions

The video resources provided in this chapter visualize abstract trigonometric concepts (such as formula derivation and angle transformation) through dynamic explanations, example demonstrations, and step-by-step breakdowns. They are suitable for learners with different paces—whether beginners needing to understand from scratch or advanced learners needing to consolidate difficult points, the videos offer a more intuitive and accessible learning path.

7. Trigonometry table

Trigonometric tables are convenient tools for querying trigonometric values. This chapter distinguishes between basic standard tables (covering common angles such as 0°, 30°, 45°, etc.) and scientific tables (covering the full 0°-360° range, high-precision values, and special angles). They not only meet the needs of beginners for quick queries of basic angle values but also provide high-precision numerical support for fields such as physics, engineering, and astronomy. Meanwhile, the patterns in the table values help learners deepen their understanding of properties like periodicity and monotonicity of trigonometric functions.

Basic standard tables

The trigonometric table is simply a collection of the values of trigonometric ratios for various standard angles including 0°, 30°, 45°, 60°, 90°, sometimes with other angles like 180°, 270°, and 360° included, in a tabular format. Because of patterns existing within trigonometric ratios and even between angles, it is easy to both predict the values of the trigonometry table and use the table as a reference to calculate trigonometric values for various other angles. The trigonometric functions are namely the sine function, cosine function, tan function, cot function, sec function, and cosec function.

Trigonometry table
Trigonometry table

Scientific tables

Trigonometric tables

Standard Trigonometric tables of reference angles with aspects of each angle

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trigonometry table

Trigonometric Table for Scientific Purposes (Including Extended Angles and High-Precision Values)

In scientific fields such as physics, engineering, astronomy, and surveying, the demand for trigonometric function values goes beyond basic standard angles. It also requires coverage of key angles across the full range of 0°-360°, higher-precision values (usually retaining 4-8 decimal places), and exact values of some special angles (e.g., 15°, 75°).

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