Chapter 6: 9th Class Math Notes

Chapter 6: 9th Class Math Notes are available for students with step-by-step instructions. This chapter is about trigonometry.

This chapter requires knowledge of real numbers, including trigonometric ratios, trigonometric identities, and some techniques to solve the variables.

Chapter 6: 9th Class Math Notes

Class 9 Math Notes – Chapter 6 Summary

This summary covers key concepts and solution methods from Chapter 6 of your 9th class mathematics notes:

Angles and Their Measurement

• Quadrants: Angles are divided into four quadrants based on their measure:
  • 1st Quadrant: 0^∘ to 90^∘.
  • 2nd Quadrant: 90^∘ to 180^∘.
  • 3rd Quadrant: 180^∘ to 270^∘.
  • 4th Quadrant: 270^∘ to 360^∘.
• Co-terminal Angles: Angles that have the same initial and terminal sides but differ by a multiple of 360^∘.
• Conversion Between Degrees and Radians:
  • To convert degrees to radians: Multiply by π/180​.
  • To convert radians to degrees: Multiply by 180/π​.

Trigonometric Ratios

• Basic Ratios:
  • sin⁡θ=Perpendicular/Hypotenuse
  • cos⁡θ=Base/Hypotenuse​
  • tan⁡θ=Perpendicular/Base
• Reciprocal Ratios:
  • csc⁡θ=1/sin⁡θ​
  • sec⁡θ=1/cos⁡θ
  • cot⁡θ=1/tan⁡θ
• Pythagorean Identities:
  • sin⁡²θ+cos⁡²θ=1
  • 1+tan⁡²θ=sec⁡²θ
  • 1+cot⁡²θ=csc⁡²θ

Trigonometric Identities

• Key Identities:
  • sin⁡(A±B)=sin⁡Acos⁡B±cos⁡Asin⁡B
  • cos⁡(A±B)=cos⁡Acos⁡B∓sin⁡Asin⁡B
  • tan⁡(A±B)=tan⁡A±tan⁡B/1∓tan⁡Atan⁡B​
• Double Angle Formulas:
  • sin⁡2θ=2sin⁡θcos⁡θ
  • cos⁡2θ=cos⁡²θ−sin⁡²θ
  • tan⁡2θ=2tan⁡θ/1−tan⁡²θ​

Solving Right-Angled Triangles

• Pythagorean Theorem: In a right-angled triangle, a²+b²=c², where c is the hypotenuse.
• Trigonometric Ratios in Right-Angled Triangles:
  • Use sin⁡, cos⁡, and ⁡tan to find unknown sides or angles.
  • Example: If θ=30^∘ and the opposite side is 4 cm, find the hypotenuse using sin⁡θ=Perpendicular/Hypotenuse​.

Applications of Trigonometry

• Finding Unknown Sides and Angles:
  • Use trigonometric ratios to solve for missing sides or angles in right-angled triangles.
  • Example: Given one angle and one side, find the other sides using trigonometric ratios.
• Real-Life Problems:
  • Calculate heights, distances, and angles in real-world scenarios using trigonometry.
  • Example: Find the height of a wall using the length of a ladder and the distance from the wall.

Key Trigonometric Values

• Standard Angles:
  • 0^∘, 30^∘, 45^∘, 60^∘, 90^∘.
  • Memorize the values of sin⁡, cos⁡, and tan for these angles.
• Example:
  • sin⁡30^∘=1/2​, cos⁡45^∘=1/√2, tan⁡60^∘=√3​.

9th Class Math Notes Full Book