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