Trigonometry Formulas

 Trigonometry Formulas: Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It’s a fundamental topic in mathematics and finds applications in various fields, from physics and engineering to astronomy and navigation. To master trigonometry, one needs to be well-versed in its numerous formulas and identities. In this comprehensive guide, we will explore a wide range of trigonometry formulas, making them accessible and understandable. Whether you’re a student looking to ace your trigonometry exams or a professional seeking to apply trigonometric principles in real-life scenarios, this article has got you covered.

Trigonometry Formulas List:

Let’s kick things off by presenting a list of the fundamental trigonometry formulas that we will delve into in more detail:

1. Basic Trigonometric Function Formulas
2. Reciprocal Identities
3. Trigonometry Table
4. Periodicity Identities (in Radians)
5. Cofunction Identities (in Degrees)
6. Sum & Difference Identities
7. Double Angle Identities
8. Triple Angle Identities
9. Half Angle Identities
10. Product Identities
11. Sum to Product Identities
12. Inverse Trigonometry Formulas
13. Trigonometry Formulas from Class 10 to Class 12
14. Trigonometry Formulas in Major Systems

Let’s explore these trigonometric concepts one by one, providing explanations, examples, and insights to make these formulas truly comprehensible.

Basic Trigonometric Function Formulas

• Basic trigonometric function formulas:
  • Sine (sin): Opposite side over the hypotenuse
  • Cosine (cos): Adjacent side over the hypotenuse
  • Tangent (tan): Opposite side over adjacent side
• Reciprocal identities:
  • cosecant (csc): 1 / sin
  • secant (sec): 1 / cos
  • cotangent (cot): 1 / tan
• Pythagorean theorem:
  • a^2 + b^2 = c^2
• 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)
• Double angle identities:
  • sin(2a) = 2sin(a)cos(a)
  • cos(2a) = cos^2(a) – sin^2(a)
  • tan(2a) = 2tan(a) / (1 – tan^2(a))
• Triple angle identities:
  • sin(3a) = 3sin(a) – 4sin^3(a)
  • cos(3a) = 4cos^3(a) – 3cos(a)
  • tan(3a) = (3tan(a) – tan^3(a)) / (1 – 3tan^2(a))
• Half angle identities:
  • sin(a/2) = sqrt((1 – cos(a)) / 2)
  • cos(a/2) = sqrt((1 + cos(a)) / 2)
  • tan(a/2) = sqrt((1 – cos(a)) / (1 + cos(a)))
• Product identities:
  • sin(a)sin(b) = 1/2(cos(a – b) – cos(a + b))
  • cos(a)cos(b) = 1/2(cos(a – b) + cos(a + b))
• Sum to product identities:
  • sin(a) + sin(b) = 2sin((a + b)/2)cos((a – b)/2)
  • sin(a) – sin(b) = 2cos((a + b)/2)sin((a – b)/2)
  • cos(a) + cos(b) = 2cos((a + b)/2)cos((a – b)/2)
  • cos(a) – cos(b) = -2sin((a + b)/2)sin((a – b)/2)

These basic trigonometric functions are the foundation of trigonometry and are essential for solving a wide range of problems involving angles and triangles.

Trigonometry Table

A trigonometry table, also known as a unit circle, provides the values of trigonometric functions at common angles. It’s a crucial tool for quick reference in trigonometric calculations. Here are some key values on the unit circle:

• 0 degrees (0 radians):
  • $\sin(0) = 0$
  • $\cos(0) = 1$
  • $\tan(0) = 0$
• 30 degrees ($\frac{\pi}{6}$ radians):
  • $\sin(30) = \frac{1}{2}$
  • $\cos(30) = \frac{\sqrt{3}}{2}$
  • $\tan(30) = \frac{1}{\sqrt{3}}$
• 45 degrees ($\frac{\pi}{4}$ radians):
  • $\sin(45) = \frac{1}{\sqrt{2}}$
  • $\cos(45) = \frac{1}{\sqrt{2}}$
  • $\tan(45) = 1$
• 60 degrees ($\frac{\pi}{3}$ radians):
  • $\sin(60) = \frac{\sqrt{3}}{2}$
  • $\cos(60) = \frac{1}{2}$
  • $\tan(60) = \sqrt{3}$
• 90 degrees ($\frac{\pi}{2}$ radians):
  • $\sin(90) = 1$
  • $\cos(90) = 0$
  • $\tan(90) = \infty$

These values are essential for solving trigonometric problems quickly and efficiently.

Periodicity Identities (in Radians)

Periodicity in trigonometry refers to the repetition of function values after a certain interval. In radians, the periodicity of trigonometric functions is as follows:

• Sine and Cosecant:
  • Period: $2\pi$ (i.e., $\sin(\theta) = \sin(\theta + 2\pi n)$)
• Cosine and Secant:
  • Period: $2\pi$ (i.e., $\cos(\theta) = \cos(\theta + 2\pi n)$)
• Tangent and Cotangent:
  • Period: $\pi$ (i.e., $\tan(\theta) = \tan(\theta + \pi n)$)

Here, $n$ represents an integer.

These identities are crucial in understanding the cyclic nature of trigonometric functions.

Cofunction Identities (in Degrees)

Cofunction identities express the relationship between a trigonometric function and its cofunction (complementary function). In degrees, the cofunction identities are as follows:

• Sine and Cosine:
  • $\sin(90^\circ – \theta) = \cos(\theta)$
  • $\cos(90^\circ – \theta) = \sin(\theta)$
• Tangent and Cotangent:
  • $\tan(90^\circ – \theta) = \cot(\theta)$
  • $\cot(90^\circ – \theta) = \tan(\theta)$

These identities are particularly useful when dealing with complementary angles.

Sum & Difference Identities

Sum and difference identities allow us to express trigonometric functions of the sum or difference of two angles in terms of the trigonometric functions of the individual angles. Here are the formulas:

1. Sine of Sum and Difference:
   • $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
2. Cosine of Sum and Difference:
   • $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
3. Tangent of Sum and Difference:
   • $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

These identities are instrumental in solving trigonometric equations involving sums and differences of angles.

Double Angle Identities

Double angle identities express trigonometric functions of double angles in terms of the trigonometric functions of the original angles:

1. Sine of Double Angle:
   • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
2. Cosine of Double Angle:
   • $\cos(2\theta) = \cos^2(\theta) – \sin^2(\theta) = 2\cos^2(\theta) – 1 = 1 – 2\sin^2(\theta)$
3. Tangent of Double Angle:
   • $\tan(2\theta) = \frac{2\tan(\theta)}{1 – \tan^2(\theta)}$

Triple Angle Identities

Triple angle identities express the trigonometric functions of triple angles in terms of the trigonometric functions of the original angles:

1. Sine of Triple Angle:
   • $\sin(3\theta) = 3\sin(\theta) – 4\sin^3(\theta)$
2. Cosine of Triple Angle:
   • $\cos(3\theta) = 4\cos^3(\theta) – 3\cos(\theta)$
3. Tangent of Triple Angle:
   • $\tan(3\theta) = \frac{3\tan(\theta) – \tan^3(\theta)}{1 – 3\tan^2(\theta)}$

Half Angle Identities

Half-angle identities express the trigonometric functions of half-angles in terms of the trigonometric functions of the original angles:

1. Sine of Half Angle:
   • $\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 – \cos(\theta)}{2}}$
2. Cosine of Half Angle:
   • $\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}$
3. Tangent of Half Angle:
   • $\tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 – \cos(\theta)}{1 + \cos(\theta)}}$

The plus or minus sign depends on the quadrant in which the half angle falls.

Half-angle identities are particularly useful for simplifying expressions and making calculations more manageable.

Sum to Product Identities

Sum-to-product identities allow us to express the sum or difference of trigonometric functions in terms of products of trigonometric functions:

1. Sine of Sum to Product:
   • $\sin(\alpha) + \sin(\beta) = 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha – \beta}{2}\right)$
2. Sine of Difference to Product:
   • $\sin(\alpha) – \sin(\beta) = 2\cos\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha – \beta}{2}\right)$
3. Cosine of Sum to Product:
   • $\cos(\alpha) + \cos(\beta) = 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha – \beta}{2}\right)$
4. Cosine of Difference to Product:
   • $\cos(\alpha) – \cos(\beta) = -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha – \beta}{2}\right)$

Sum-to-product identities provide an elegant way to simplify trigonometric expressions.

Inverse Trigonometry Formulas

Inverse trigonometric functions allow us to find angles given the values of trigonometric ratios. Here are the key inverse trigonometry formulas:

1. Arcsine (sin⁻¹):
   • $\sin^{-1}(x) = \theta \implies x = \sin(\theta)$ and $-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$
2. Arccosine (cos⁻¹):
   • $\cos^{-1}(x) = \theta \implies x = \cos(\theta)$ and $0 \leq \theta \leq \pi$
3. Arctangent (tan⁻¹):
   • $\tan^{-1}(x) = \theta \implies x = \tan(\theta)$ and $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$

Inverse trigonometry formulas are indispensable for solving problems where you need to find angles.

Trigonometry Formulas from Class 10 to Class 12

Trigonometry is a part of the curriculum in mathematics classes ranging from 10th grade to 12th grade. Here’s a brief overview of the key topics covered during these classes:

NCERT Class 10:

• Introduction to Trigonometry: Basics of trigonometry, right-angled triangles, and trigonometric ratios.
• Trigonometric Ratios: Sine, cosine, tangent, cosecant, secant, and cotangent.
• Applications: Solving real-life problems involving heights and distances.
• Introduction to Trigonometric Identities: Basic identities like $\sin^2(\theta) + \cos^2(\theta) = 1$.

Class 11:

• Trigonometric Functions: Extending trigonometric ratios to all angles.
• Trigonometric Identities: Verifying and applying trigonometric identities.
• Trigonometric Equations: Solving equations involving trigonometric functions.

Class 12:

• Inverse Trigonometric Functions: Understanding inverse trigonometric functions and their properties.
• Heights and Distances: Advanced applications of trigonometry in real-world scenarios.
• Additional Identities: Sum and difference identities, double angle identities, half angle identities, and more.

Trigonometry Formulas in Major Systems

Trigonometry is not only a staple in school mathematics but also plays a crucial role in various fields, and different systems are used to measure angles:

• Degree System: The most commonly used system for measuring angles, especially in everyday life. A full circle contains 360 degrees, with 90 degrees in each quadrant.
• Radian System: Widely used in mathematics and science, a radian is the angle subtended when the length of an arc is equal to the radius of the circle. A full circle contains $2\pi$ radians.
• Gradian System: Less common but used in some engineering applications. A full circle contains 400 gradians, with 100 gradians in each quadrant.
• Decimal Degrees: This system divides a degree into 100 parts, creating a decimal representation. It simplifies angle calculations in some surveying and geospatial applications.

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