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Class 10 Trigonometry Formula

Trigonometry, a branch of mathematics that studies relationships involving lengths and angles of triangles, plays a crucial role in Class 10 mathematics curriculum. It is fundamental not only for advanced studies in mathematics but also for various applications in science and engineering. This document provides a detailed overview of Class 10 Trigonometry, covering essential concepts, formulas, and applications.

त्रिकोणमिति गणित की एक शाखा है जो त्रिकोणों के कोणों और भुजाओं के बीच के संबंधों का अध्ययन करती है। यह कक्षा 10 के गणित के पाठ्यक्रम में एक महत्वपूर्ण विषय है और आगे की गणितीय पढ़ाई के लिए आधारशिला का कार्य करती है। यहां त्रिकोणमिति के कुछ प्रमुख सिद्धांत और शब्दावली का हिंदी में वर्णन किया गया है।

1. Introduction to Trigonometry

Trigonometry originates from the Greek words “trigonon” (triangle) and “metron” (measure). It primarily deals with right-angled triangles, focusing on the relationships between their sides and angles.

2. Trigonometric Ratios

The foundation of trigonometry lies in the trigonometric ratios, which are defined for a right-angled triangle. These ratios are:

• Sine (sin θ): Opposite side / Hypotenuse
• Cosine (cos θ): Adjacent side / Hypotenuse
• Tangent (tan θ): Opposite side / Adjacent side
• Cosecant (csc θ): Hypotenuse / Opposite side
• Secant (sec θ): Hypotenuse / Adjacent side
• Cotangent (cot θ): Adjacent side / Opposite side

3. Trigonometric Identities

Several fundamental identities in trigonometry simplify the process of solving equations and proving relationships:

Pythagorean Identities:

$$\sin^2 θ + \cos^2 θ = 1$$

sin2θ+cos2θ=1

$$1 + \tan^2 θ = \sec^2 θ$$

1+tan2θ=sec2θ

$$1+{\cot }^2\theta ={\mathrm{cosec}}^2\theta$$

 

$$1 + \cot^2 θ = \csc^2 θ$$

1+cot2θ=csc2θ

Reciprocal Identities:

$$\mathrm{cosec}\theta =\frac{1}{\sin \theta },\sec \theta =\frac{1}{\cos \theta },\cot \theta =\frac{1}{\tan \theta }$$

 

 

4. Trigonometric Ratios of Specific Angles

Certain angles, namely 0°, 30°, 45°, 60°, and 90°, have trigonometric ratios that are frequently used:

•  

  ​    $\cos 30° = \frac{\sqrt{3}}{2}$ 

  ​​    $\tan 30° = \frac{1}{\sqrt{3}}$ 

  ​

• 
  $\sin 45° = \cos 45° = \frac{1}{\sqrt{2}}$ 

      $\tan 45° = 1$ 

• 
  $\sin 60° = \frac{\sqrt{3}}{2}$ 

      $\cos 60° = \frac{1}{2}$ 

      $\tan 60° = \sqrt{3}$ 

  ​

• 
  $\sin 90° = 1$ 

    $\cos 90° = 0$ 

      $\tan 90\mathrm{°}=\mathrm{\infty }$ 

  $\tan 90° = \infty$ 

5. Complementary Angles

For any angle θ:

• 
  $\sin (90° – θ) = \cos θ$ 

  $\sin (90°-\theta )=\cos \theta$

• 
  $\cos (90° – θ) = \sin θ$ 

  $\cos (90°-\theta )=\sin \theta$

• 
  $\tan (90° – θ) = \cot θ$ 

  $\tan (90°-\theta )=\cot \theta$

• 
  $\cot (90° – θ) = \tan θ$ 

  $\cot (90°-\theta )=\tan \theta$

• 
  $\sec (90° – θ) = \csc θ$ 

  $\sec (90°-\theta )=\mathrm{cosec}\theta$

• 
  $\csc (90° – θ) = \sec θ$ 

  $\mathrm{cosec}(90°-\theta )=\sec \theta$

6. Trigonometric Applications

Trigonometry has numerous applications, especially in calculating heights and distances. Using trigonometric ratios, one can solve real-world problems involving right-angled triangles, such as finding the height of a building using the length of its shadow and the angle of elevation.

Practical Problems and Solutions

Example 1: Height and Distance

Problem: A ladder 10 meters long leans against a wall making an angle of 60° with the ground. Find the height of the point where the ladder touches the wall.

Solution: Using the sine ratio,

 

$$\sin 60\mathrm{°}=\frac{\text{Opposite side}}{\text{Hypotenuse}}=\frac{h}{10}$$

 

​

$$\frac{\sqrt{3}}{2}=\frac{h}{10}\Rightarrow h=10\times \frac{\sqrt{3}}{2}=5\sqrt{3}\approx 8.66\text{ meters}$$

 

 

Conclusion

Trigonometry in Class 10 sets the groundwork for more advanced mathematical concepts. Mastery of trigonometric ratios, identities, and their applications provides students with the tools to solve a wide array of problems both in academics and real-life scenarios. Consistent practice and application of these principles are essential for achieving proficiency in trigonometry.