Trignometry Class -10th

Trigonometry: Class 10 Notes

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of a triangle, especially right-angled triangles. It is a fundamental concept used widely in geometry, physics, engineering, and various applied sciences.

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Key Terminology in Trigonometry:

1. Angle: An angle is formed when two rays meet at a common point (vertex). The unit of measurement for angles is degrees (°) or radians.

2. Right-Angled Triangle: A triangle in which one of the angles is 90°. In such a triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called base and perpendicular (also known as opposite).

3. Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

4. Base: One of the two sides that form the right angle in a right-angled triangle.

5. Perpendicular: The other side that forms the right angle in a right-angled triangle.

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Trigonometric Ratios:

In a right-angled triangle, given an angle (say $\theta$), we define the following trigonometric ratios:

1. Sine (sin) of an angle $\theta$:

$$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

This ratio relates the length of the side opposite the angle $\theta$ to the length of the hypotenuse.

2. Cosine (cos) of an angle $\theta$:

$$\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}$$

This ratio relates the length of the adjacent side (base) to the length of the hypotenuse.

3. Tangent (tan) of an angle $\theta$:

$$\tan \theta = \frac{\text{Opposite}}{\text{Base}}$$

This ratio relates the length of the side opposite the angle $\theta$ to the length of the adjacent side (base).

4. Cosecant (csc) of an angle $\theta$:

$$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}}$$

The reciprocal of sine.

5. Secant (sec) of an angle $\theta$:

$$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Base}}$$

The reciprocal of cosine.

6. Cotangent (cot) of an angle $\theta$:

$$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Base}}{\text{Opposite}}$$

The reciprocal of tangent.

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Trigonometric Ratios Table:

Here’s a quick reference for the six trigonometric ratios for any angle $\theta$ in a right-angled triangle:

Ratio	Formula	Reciprocal
Sine (sin)	$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$	Cosecant (csc)
Cosine (cos)	$\cos \theta = \frac{\text{Base}}{\text{Hypotenuse}}$	Secant (sec)
Tangent (tan)	$\tan \theta = \frac{\text{Opposite}}{\text{Base}}$	Cotangent (cot)

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Trigonometric Identities:

There are several important identities in trigonometry that help in simplifying and solving problems. These include:

1. Pythagorean Identity:

For any angle $\theta$ in a right-angled triangle:

$$\sin^2 \theta + \cos^2 \theta = 1$$

This identity forms the basis of several trigonometric derivations.

2. Reciprocal Identities:

• $\csc \theta = \frac{1}{\sin \theta}$

• $\sec \theta = \frac{1}{\cos \theta}$

• $\cot \theta = \frac{1}{\tan \theta}$

3. Quotient Identities:

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$

• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

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Trigonometric Values for Special Angles:

There are certain common angles whose trigonometric values are frequently used. These are:

Angle (θ)	0°	30°	45°	60°	90°
$\sin \theta$	0	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	1
$\cos \theta$	1	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	0
$\tan \theta$	0	$\frac{1}{\sqrt{3}}$	1	$\sqrt{3}$	Undefined

These values should be memorized for quick reference.

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Applications of Trigonometry:

Trigonometry has several practical applications:

1. Height and Distance Problems: Trigonometry helps calculate the height of objects when the distance to the object and the angle of elevation/depression are known.

2. Navigation: Trigonometry is used in navigation, for example, to determine positions and course directions.

3. Architecture and Engineering: Trigonometric concepts are used to measure structural angles, lengths, and heights.

4. Surveying: Trigonometry helps in land surveying and determining distances and areas of land.

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Example Problems:

Problem 1:

Find the height of a building if the angle of elevation to the top of the building from a point 50 meters away from the base is 30°.

Solution:

Let the height of the building be $h$.

Using the formula for tangent:

$$\tan(30^\circ) = \frac{h}{50}$$

From the table, $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.

So,

$$\frac{1}{\sqrt{3}} = \frac{h}{50}$$

Multiplying both sides by 50:

$$h = \frac{50}{\sqrt{3}} = \frac{50\sqrt{3}}{3} \approx 28.87 \text{ meters}.$$

Thus, the height of the building is approximately 28.87 meters.

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Conclusion:

Trigonometry is an essential part of geometry and has various real-life applications. Understanding the fundamental trigonometric ratios, identities, and their use in problem-solving is key to mastering this topic.

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These notes should help you get a solid foundation of trigonometry in Class 10. If you have specific questions or need more examples, feel free to ask!