Trigonometry Formulas
2026-02-28 08:47 Diff
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Last updated on August 11, 2025

In mathematics, trigonometry is the study of the relationships between the angles and sides of triangles. Trigonometric formulas are essential for solving problems involving angles and distances. In this topic, we will learn the key trigonometry formulas.

List of Trigonometry Formulas

Trigonometry involves various formulas related to the angles and sides of triangles. Let’s learn the essential trigonometry formulas.

Basic Trigonometric Ratios

The basic trigonometric ratios are sine, cosine, and tangent. They are defined as follows for a right-angled triangle: 

Sine (sin) of angle θ = Opposite side/Hypotenuse 

Cosine (cos) of angle θ = Adjacent side/Hypotenuse 

Tangent (tan) of angle θ = Opposite side/Adjacent side

Reciprocal Trigonometric Ratios

The reciprocal trigonometric ratios are cosecant, secant, and cotangent: 

Cosecant (csc) of angle θ = 1/sin(θ) = Hypotenuse/Opposite side 

Secant (sec) of angle θ = 1/cos(θ) = Hypotenuse/Adjacent side 

Cotangent (cot) of angle θ = 1/tan(θ) = Adjacent side/Opposite side

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Pythagorean Identities

The Pythagorean identities relate the squares of the basic trigonometric ratios: - sin²(θ) + cos²(θ) = 1 - 1 + tan²(θ) = sec²(θ) - 1 + cot²(θ) = csc²(θ)

Importance of Trigonometry Formulas

In math and real life, trigonometry formulas help solve problems related to angles and distances. Here are some key points: 

Trigonometry is used in various fields such as physics, engineering, and architecture. 

By learning these formulas, students can easily understand concepts like wave functions, rotations, and oscillations. 

Trigonometry helps in calculating heights and distances that are otherwise difficult to measure directly.

Tips and Tricks to Memorize Trigonometry Formulas

Students often find trigonometry formulas challenging. Here are some tips to master them: 

Use mnemonics like SOH-CAH-TOA to remember sine, cosine, and tangent ratios. 

Relate trigonometry to real-world examples like shadows, ramps, and ladders. 

Use flashcards to memorize the formulas and rewrite them for quick recall; create a formula chart for reference.

Common Mistakes and How to Avoid Them While Using Trigonometry Formulas

Students often make errors when using trigonometry formulas. Here are some common mistakes and how to avoid them:

Problem 1

Find the sine of angle θ in a triangle where the opposite side is 3 and the hypotenuse is 5.

Okay, lets begin

The sine of angle θ is 0.6.

Explanation

Using the sine formula: sin(θ) = Opposite/Hypotenuse = 3/5 = 0.6

Well explained 👍

Problem 2

Find the cosine of angle θ in a triangle where the adjacent side is 4 and the hypotenuse is 5.

Okay, lets begin

The cosine of angle θ is 0.8.

Explanation

Using the cosine formula: cos(θ) = Adjacent/Hypotenuse = 4/5 = 0.8

Well explained 👍

Problem 3

A triangle has an opposite side of 7 and an adjacent side of 24. Find the tangent of the angle.

Okay, lets begin

The tangent of the angle is 0.2917.

Explanation

Using the tangent formula: tan(θ) = Opposite/Adjacent = 7/24 ≈ 0.2917

Well explained 👍

Problem 4

Calculate the secant of angle θ where the adjacent side is 9 and the hypotenuse is 15.

Okay, lets begin

The secant of angle θ is 1.6667.

Explanation

Using the secant formula: sec(θ) = Hypotenuse/Adjacent = 15/9 ≈ 1.6667

Well explained 👍

Problem 5

In a right triangle, if the opposite side is 12 and the hypotenuse is 13, find the cosecant of the angle.

Okay, lets begin

The cosecant of the angle is 1.0833.

Explanation

Using the cosecant formula: csc(θ) = Hypotenuse/Opposite = 13/12 ≈ 1.0833

Well explained 👍

FAQs on Trigonometry Formulas

1.What is the sine formula?

The formula for sine is: sin(θ) = Opposite side/Hypotenuse

2.What is the formula for cosine?

The formula for cosine is: cos(θ) = Adjacent side/Hypotenuse

3.How to find the tangent?

To find the tangent of an angle, use the formula: tan(θ) = Opposite side/Adjacent side

4.What is the secant of an angle?

The secant of an angle is the reciprocal of cosine: sec(θ) = Hypotenuse/Adjacent side

5.What is the cosecant of an angle?

The cosecant of an angle is the reciprocal of sine: csc(θ) = Hypotenuse/Opposite side

Glossary for Trigonometry Formulas

  • Sine: A trigonometric ratio representing the opposite side over the hypotenuse in a right triangle.
  • Cosine: A trigonometric ratio representing the adjacent side over the hypotenuse in a right triangle.
  • Tangent: A trigonometric ratio representing the opposite side over the adjacent side in a right triangle.
  • Secant: The reciprocal of the cosine, representing the hypotenuse over the adjacent side.
  • Cosecant: The reciprocal of the sine, representing the hypotenuse over the opposite side.

Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.