By The formulas of trigonometry: ত্রিকোণমিতির সূত্র।
The formulas of trigonometry: ত্রিকোণমিতির সূত্র।
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It has a rich set of formulas that are essential for solving a wide range of problems. Here are some of the fundamental trigonometric formulas:
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1. **Sine, Cosine, and Tangent**:
– **Sine (sin θ)**: In a right triangle, it’s the ratio of the length of the side opposite an angle to the length of the hypotenuse. The formula is: `sin(θ) = opposite / hypotenuse`.
– **Cosine (cos θ)**: It’s the ratio of the length of the side adjacent to an angle to the length of the hypotenuse. The formula is: `cos(θ) = adjacent / hypotenuse`.
– **Tangent (tan θ)**: It’s the ratio of the sine of an angle to the cosine of the angle. The formula is: `tan(θ) = sin(θ) / cos(θ)`.
2. **Reciprocal Functions**:
– **Cosecant (csc θ)**: The reciprocal of sine, `csc(θ) = 1 / sin(θ)`.
– **Secant (sec θ)**: The reciprocal of cosine, `sec(θ) = 1 / cos(θ)`.
– **Cotangent (cot θ)**: The reciprocal of tangent, `cot(θ) = 1 / tan(θ)`.
3. **Pythagorean Identities**:
– **Pythagorean Theorem**: In a right triangle, it states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. `a² + b² = c²`, where ‘a’ and ‘b’ are the legs, and ‘c’ is the hypotenuse.
– **Pythagorean Identity**: `sin²(θ) + cos²(θ) = 1`.
4. **Trigonometric Addition Formulas**:
– **Sum of Angles**: `sin(A + B) = sin(A)cos(B) + cos(A)sin(B)`, `cos(A + B) = cos(A)cos(B) – sin(A)sin(B)`.
– **Double Angle**: `sin(2θ) = 2sin(θ)cos(θ)`, `cos(2θ) = cos²(θ) – sin²(θ)`.
5. **Trigonometric Ratios for Common Angles**:
– For special angles (0°, 30°, 45°, 60°, 90°), trigonometric ratios have simple values.
6. **Inverse Trigonometric Functions**:
– Inverse Sine (arcsin or sin⁻¹): Gives the angle whose sine is a given value.
– Inverse Cosine (arccos or cos⁻¹): Gives the angle whose cosine is a given value.
– Inverse Tangent (arctan or tan⁻¹): Gives the angle whose tangent is a given value.
These are the fundamental trigonometric formulas. They are used to solve a variety of problems related to angles, distances, and periodic phenomena in fields like physics, engineering, and astronomy, among others.
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