Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It plays a crucial role in various fields, including physics, engineering, and navigation. In this comprehensive guide, we will explore the essential trigonometry formulas for Class 10 students. By understanding these formulas, students can solve trigonometric problems with ease and excel in their exams.
What is Trigonometry Ratio?
Trigonometric ratios are the ratios of the sides of a right triangle for a given angle, denoted as theta (θ). There are six trigonometric ratios: sine, cosine, tangent, cotangent, secant, and cosecant. These ratios help establish the relationship between the angles and sides of a right triangle.
Trigonometric Ratio Formulas for Class 10
Trigonometric ratio formulas establish the relation between trigonometric ratios and the sides of a right triangle. Let's explore these formulas in detail:
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Sine (sin) Formula: The sine of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse.
- sin θ = Perpendicular / Hypotenuse
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Cosine (cos) Formula: The cosine of an angle is equal to the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.
- cos θ = Base / Hypotenuse
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Tangent (tan) Formula: The tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
- tan θ = Perpendicular / Base
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Cosecant (cosec) Formula: The cosecant of an angle is equal to the reciprocal of the sine of the angle.
- cosec θ = 1 / sin θ
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Secant (sec) Formula: The secant of an angle is equal to the reciprocal of the cosine of the angle.
- sec θ = 1 / cos θ
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Cotangent (cot) Formula: The cotangent of an angle is equal to the reciprocal of the tangent of the angle.
- cot θ = 1 / tan θ
Trigonometric Ratio Table for Class 10
The trigonometric ratio table provides the values of all six trigonometric ratios at some standard angles: 0°, 30°, 45°, 60°, and 90°.
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | Not defined |
| cosec θ | Not defined | 2 | √2 | 2/√3 | 1 |
| sec θ | 1 | 2/√3 | √2 | 2 | Not defined |
| cot θ | Not defined | √3 | 1 | 1/√3 | 0 |
Trigonometric Formulas for Class 10
In addition to the trigonometric ratio formulas, there are several other important trigonometric formulas that are essential for Class 10 students. Let's explore them:
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Trigonometric Reciprocal Identities:
- sin θ = 1 / cosec θ ⇒ cosec θ = 1 / sin θ
- cos θ = 1 / sec θ ⇒ sec θ = 1 / cos θ
- tan θ = 1 / cot θ ⇒ cot θ = 1 / tan θ
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Trigonometric Identities:
- sin^2θ + cos^2θ = 1
- 1 + tan^2θ = sec^2θ
- 1 + cot^2θ = cosec^2θ
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Complementary Angle Formulas:
- sin(90 - θ) = cos θ
- cos(90 - θ) = sin θ
- tan(90 - θ) = cot θ
- cot(90 - θ) = tan θ
- sec(90 - θ) = cosec θ
- cosec(90 - θ) = sec θ
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Supplementary Angle Formulas:
- sin(180 - θ) = sin θ
- cos(180 - θ) = -cos θ
- tan(180 - θ) = -tan θ
- cot(180 - θ) = -cot θ
- sec(180 - θ) = -sec θ
- cosec(180 - θ) = cosec θ
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Even and Odd Angle Formulas:
- sin(-θ) = -sin θ
- cos(-θ) = cos θ
- tan(-θ) = -tan θ
- cot(-θ) = -cot θ
- sec(-θ) = sec θ
- cosec(-θ) = -cosec θ
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Double Angle Formulas:
- sin 2θ = 2sin θ.cos θ
- cos 2θ = 1 - 2sin^2θ
- tan 2θ = 2tan θ / (1 - tan^2θ)
These formulas provide a solid foundation for solving trigonometric problems and understanding the relationships between angles and sides in a right triangle.
Solved Examples:
Let's solve a few examples using the trigonometry formulas:
Example 1: If sin A = 3/4, calculate cos A and tan A.
Solution: Using the formula cos^2A = 1 - sin^2A, we can find cos A:
- cos^2A = 1 - (3/4)^2
- cos^2A = 1 - 9/16
- cos^2A = 7/16
- cos A = √(7/16)
Using the formula tan A = sin A / cos A, we can find tan A:
- tan A = (3/4) / √(7/16)
Example 2: Given 15 cot A = 8, find sin A and sec A.
Solution: Using the formula cot A = 1 / tan A, we can find tan A:
- cot A = 1 / tan A
- 15 = 1 / tan A
- tan A = 1/15
Using the formula cosec^2A = 1 + cot^2A, we can find cosec A:
- cosec^2A = 1 + (1/15)^2
- cosec^2A = 1 + 1/225
- cosec^2A = 226/225
- cosec A = √(226/225)
Using the formula sin A = 1 / cosec A, we can find sin A:
- sin A = 1 / √(226/225)
These examples demonstrate how the trigonometry formulas can be applied to solve various problems.
Practice Questions:
Now, let's practice applying the trigonometry formulas with some additional questions:
- Prove that (cos θ / (1 - tan θ)) + (sin θ / (1 - cot θ)) = sin θ + cos θ.
- Prove that tan^2θ + cot^2θ + 2 = sec^2θ * cosec^2θ.
- If cosec θ + cot θ = 9, find cos θ.
- If cos θ = 4/5, find all other trigonometric ratios.
- Find the value of sin 60° cos 30° + cos 30° sin 60°.
- Find the value of tan 30° / cot 60°.
- Find the value of (sin 30° + cos 30°) - (sin 60° + cos 60°).
- If tan θ + cot θ = 1, find the value of tan 10θ + cot 10θ.
- Find the value of sin 45° + cos 45°.
- Find the value of cos 19° / sin 71°.
By practicing these questions, students can further enhance their understanding of trigonometry and improve their problem-solving skills.
In conclusion, trigonometry is a fascinating branch of mathematics that helps establish relationships between angles and sides in a right triangle. By mastering the trigonometry formulas and practicing their application, Class 10 students can excel in their exams and develop a strong foundation for future mathematical endeavors.