Trigonometry for Class 10

Trigonometry for Class 10

Introduction

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. It is essential for understanding various concepts in geometry, physics, engineering, and even in daily life. Mastering trigonometry opens doors to further studies in mathematics and science.

"Trigonometry is the bridge that connects geometry to algebra."

Understanding Angles

An angle is formed by two rays with a common endpoint called the vertex. Angles can be measured in degrees (°) or radians. The most common types of angles include:

  • Acute Angle: Less than 90°
  • Right Angle: Exactly 90°
  • Obtuse Angle: Greater than 90° but less than 180°
  • Straight Angle: Exactly 180°

Trigonometric Ratios

In a right triangle, the trigonometric ratios relate the angles to the lengths of the sides. The primary ratios are:

  • Sine (sin): Ratio of the length of the opposite side to the hypotenuse.
  • Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse.
  • Tangent (tan): Ratio of the length of the opposite side to the adjacent side.

Defining Sine, Cosine, and Tangent

For a right triangle with angle θ \theta :

  • Sine: sin⁡(θ)=OppositeHypotenuse \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
  • Cosine: cos⁡(θ)=AdjacentHypotenuse \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
  • Tangent: tan⁡(θ)=OppositeAdjacent \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

Examples of Applying Trigonometric Functions

Example 1: Finding the Height of a Tree

A person standing 30 meters from a tree measures the angle of elevation to the top of the tree as 45°. To find the height of the tree, we use the tangent function:

tan⁡(45°)=Height30 \tan(45°) = \frac{\text{Height}}{30} \\ Since tan⁡(45°)=1 \tan(45°) = 1 , we have:

1=Height30⇒Height=30 meters 1 = \frac{\text{Height}}{30} \Rightarrow \text{Height} = 30 \text{ meters}

Example 2: Finding the Length of a Side

In a right triangle, if one angle is 30° and the hypotenuse is 10 meters:

Using the sine function:

sin⁡(30°)=Opposite10 \sin(30°) = \frac{\text{Opposite}}{10} \\ Since sin⁡(30°)=12 \sin(30°) = \frac{1}{2} , we get:

12=Opposite10⇒Opposite=5 meters \frac{1}{2} = \frac{\text{Opposite}}{10} \Rightarrow \text{Opposite} = 5 \text{ meters}

Conclusion

Trigonometry is a powerful tool in mathematics. Understanding the definitions of sine, cosine, and tangent, along with how to apply these functions in real-world problems, is essential for success in Class 10 and beyond. As you continue your studies, remember that practice is key to mastering trigonometric concepts.