Trigonometry Formulas - Basic to Advanced Formulas

Ruhi Singh
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Trigonometry, a branch of mathematics, deals with the relationships between the sides and angles of triangles. It's a fundamental tool in various fields like physics, engineering, and navigation. This article will delve into the key trigonometric formulas that are essential for understanding and solving trigonometric problems.

Trigonometry Formulas - Basic to Advanced Formulas

Basic Trigonometric Ratios

The three primary trigonometric ratios are sine, cosine, and tangent. They are defined for a right-angled triangle with respect to an acute angle θ:

Sine (sin θ): Ratio of the length of the side opposite to angle θ to the length of the hypotenuse.

sin θ = Opposite side / Hypotenuse

Cosine (cos θ): Ratio of the length of the side adjacent to angle θ to the length of the hypotenuse.

cos θ = Adjacent side / Hypotenuse

Tangent (tan θ): Ratio of the length of the side opposite to angle θ to the length of the side adjacent to angle θ.

tan θ = Opposite side / Adjacent side

Reciprocal Trigonometric Ratios

The reciprocal ratios are:

Cosecant (csc θ): Reciprocal of sine θ.

csc θ = 1 / sin θ = Hypotenuse / Opposite side

Secant (sec θ): Reciprocal of cosine θ.

sec θ = 1 / cos θ = Hypotenuse / Adjacent side

Cotangent (cot θ): Reciprocal of tangent θ.

cot θ = 1 / tan θ = Adjacent side / Opposite side

Pythagorean Identities

These identities relate the squares of the trigonometric ratios:

sin²Î¸ + cos²Î¸ = 1

1 + tan²Î¸ = sec²Î¸

1 + cot²Î¸ = csc²Î¸

Trigonometric Ratios of Complementary Angles

If two angles are complementary, their sum is 90°. The following relationships hold:

* sin (90° - θ) = cos θ

* cos (90° - θ) = sin θ

* tan (90° - θ) = cot θ

* csc (90° - θ) = sec θ

* sec (90° - θ) = csc θ

* cot (90° - θ) = tan θ

Trigonometric Ratios of Sum and Difference of Angles

sin (A ± B) = sin A cos B ± cos A sin B

cos (A ± B) = cos A cos B ∓ sin A sin B

tan (A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Trigonometric Ratios of Multiple Angles

sin 2θ = 2 sin θ cos θ

cos 2θ = cos²Î¸ - sin²Î¸ = 2cos²Î¸ - 1 = 1 - 2sin²Î¸

tan 2θ = 2 tan θ / (1 - tan²Î¸)

Trigonometric Ratios of Half-Angles

sin (θ/2) = ±√[(1 - cos θ) / 2]

cos (θ/2) = ±√[(1 + cos θ) / 2]

tan (θ/2) = ±√[(1 - cos θ) / (1 + cos θ)] = sin θ / (1 + cos θ) = (1 - cos θ) / sin θ

Trigonometric Equations

Trigonometric equations involve trigonometric functions. To solve them, we often use trigonometric identities, algebraic techniques, and the unit circle.

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Applications of Trigonometry

Trigonometry has a wide range of applications, including:

Navigation: Determining distances and directions

Engineering: Designing structures and machines

Physics: Analyzing wave motion, optics, and acoustics

Astronomy: Calculating distances and orbits of celestial bodies

Surveying: Measuring land and creating maps


Trigonometry Formulas

Trigonometry Formulas

Trigonometry Formulas

Trigonometry Formulas

Trigonometry Formulas

Trigonometry Formulas

Trigonometry Formulas

Trigonometry Formulas


By understanding these fundamental trigonometric formulas and their applications, you can effectively solve a variety of mathematical and real-world problems.