Skip to main content

Introduction to Trigonometry Class 10 Mathematics Concept notes and Formula Sheet

 

Concept Notes

1. Introduction to Trigonometry

  • Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of right-angled triangles. It is widely used in various fields such as engineering, physics, architecture, and astronomy.

2. Basic Trigonometric Ratios

  • The basic trigonometric ratios are defined for an acute angle in a right-angled triangle. If ABC is a right-angled triangle with ∠B = 90° and θ is one of the acute angles, then:

    • Sine (sin θ):

      sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}

      (Opposite side: The side opposite to angle θ; Hypotenuse: The side opposite to the right angle)

    • Cosine (cos θ):

      cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}

      (Adjacent side: The side adjacent to angle θ)

    • Tangent (tan θ):

      tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

3. Reciprocals of Trigonometric Ratios

  • Cosecant (csc or cosec θ): csc⁡θ=1sin⁡θ=HypotenuseOpposite Side\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite Side}}
  • Secant (sec θ): sec⁡θ=1cos⁡θ=HypotenuseAdjacent Side\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent Side}}
  • Cotangent (cot θ): cot⁡θ=1tan⁡θ=Adjacent SideOpposite Side\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent Side}}{\text{Opposite Side}}

4. Trigonometric Ratios of Standard Angles

  • The values of trigonometric ratios for the angles 0°, 30°, 45°, 60°, and 90° are essential for solving problems.

    Angle (θ)0°30°45°60°90°
    sin θ01/21/√2√3/21
    cos θ1√3/21/√21/20
    tan θ01/√31√3∞
    csc θ∞2√22/√31
    sec θ12/√3√22∞
    cot θ∞√311/√30

5. Trigonometric Identities

  • Trigonometric identities are equations involving trigonometric ratios that hold true for all values of the variables involved.

    • Pythagorean Identities: sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

6. Complementary Angles

  • Two angles are called complementary if their sum is 90°.
  • The trigonometric ratios of complementary angles are related as follows: sin⁡(90∘−θ)=cos⁡θ,cos⁡(90∘−θ)=sin⁡θ\sin (90^\circ - \theta) = \cos \theta, \quad \cos (90^\circ - \theta) = \sin \theta tan⁡(90∘−θ)=cot⁡θ,cot⁡(90∘−θ)=tan⁡θ\tan (90^\circ - \theta) = \cot \theta, \quad \cot (90^\circ - \theta) = \tan \theta sec⁡(90∘−θ)=csc⁡θ,csc⁡(90∘−θ)=sec⁡θ\sec (90^\circ - \theta) = \csc \theta, \quad \csc (90^\circ - \theta) = \sec \theta

7. Applications of Trigonometry

  • Trigonometry is used to solve problems involving heights and distances, such as finding the height of a building, the distance between two objects, or the angle of elevation or depression.

Formula Sheet

  1. Basic Trigonometric Ratios:

    sin⁡θ=Opposite SideHypotenuse\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} cos⁡θ=Adjacent SideHypotenuse\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} tan⁡θ=Opposite SideAdjacent Side\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}

  2. Reciprocal Trigonometric Ratios:

    csc⁡θ=1sin⁡θ=HypotenuseOpposite Side\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite Side}} sec⁡θ=1cos⁡θ=HypotenuseAdjacent Side\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent Side}} cot⁡θ=1tan⁡θ=Adjacent SideOpposite Side\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent Side}}{\text{Opposite Side}}

  3. Trigonometric Ratios of Standard Angles:

    • sin⁡0∘=0\sin 0^\circ = 0, sin⁡30∘=12\sin 30^\circ = \frac{1}{2}, sin⁡45∘=12\sin 45^\circ = \frac{1}{\sqrt{2}}, sin⁡60∘=32\sin 60^\circ = \frac{\sqrt{3}}{2}, sin⁡90∘=1
    • cos⁡0∘=1\cos 0^\circ = 1, cos⁡30∘=32\cos 30^\circ = \frac{\sqrt{3}}{2}, cos⁡45∘=12\cos 45^\circ = \frac{1}{\sqrt{2}}, cos⁡60∘=12\cos 60^\circ = \frac{1}{2}, cos⁡90∘=0\cos 90^\circ = 0
    • tan⁡0∘=0\tan 0^\circ = 0, tan⁡30∘=13\tan 30^\circ = \frac{1}{\sqrt{3}}, tan⁡45∘=1, tan⁡60∘=3\tan 60^\circ = \sqrt{3}, tan⁡90∘=∞

  4. Pythagorean Identities:

    sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta

  5. Complementary Angle Relationships:

    sin⁡(90∘−θ)=cos⁡θ\sin (90^\circ - \theta) = \cos \theta cos⁡(90∘−θ)=sin⁡θ\cos (90^\circ - \theta) = \sin \theta tan⁡(90∘−θ)=cot⁡θ\tan (90^\circ - \theta) = \cot \theta cot⁡(90∘−θ)=tan⁡θ\cot (90^\circ - \theta) = \tan \theta sec⁡(90∘−θ)=csc⁡θ\sec (90^\circ - \theta) = \csc \theta csc⁡(90∘−θ)=sec⁡θ\csc (90^\circ - \theta) = \sec \theta

Examples for Practice

  1. Find the value of sin⁡30∘+cos⁡60∘\sin 30^\circ + \cos 60^\circ.
  2. If tan⁡θ=1\tan \theta = 1, find the value of θ\theta from the standard angles.
  3. Prove that 1+tan⁡245∘=sec⁡245∘1 + \tan^2 45^\circ = \sec^2 45^\circ.
Labels:

Comments

Post a Comment

Popular posts from this blog

Class 9 Science Notes | Matter in Our Surroundings Notes

 Here are detailed notes on the topic "Matter in Our Surroundings" from Class 9 Science: Introduction to Matter Definition of Matter: Anything that occupies space and has mass is called matter. States of Matter: Matter exists in three main states - solid, liquid, and gas - based on the arrangement and movement of particles. Characteristics of States of Matter Solid: Shape: Definite shape and volume. Particle Arrangement: Particles are closely packed in a regular pattern. Movement: Vibrational motion around fixed positions. Examples: Wood, iron, ice. Liquid: Shape: Indefinite shape, definite volume. Particle Arrangement: Particles are close but can move past each other. Movement: Particles move with random motion. Examples: Water, milk, oil. Gas: Shape: Indefinite shape and volume. Particle Arrangement: Particles are far apart and move freely. Movement: Random and rapid motion. Examples: Air, helium, oxygen. Changes of State Melting: Solid to liquid (Endothermic...
Post a Comment
Read more

The Beggar Class 9 English Moments Summary

Chapter 10: "The Beggar" Summary: "The Beggar" by Anton Chekhov is a story about kindness, transformation, and redemption. The story begins with a beggar named Lushkoff, who is a drunkard and in a pitiable state. He meets a lawyer named Skvortsov, who, instead of giving him money, offers him work. Lushkoff is reluctant but agrees to do some wood chopping for Skvortsov. The work is difficult for Lushkoff, but he manages to do it. Skvortsov continues to give him small jobs, and over time, Lushkoff begins to change. He stops drinking and becomes more responsible. He eventually finds a stable job as a notary. Years later, Lushkoff meets Skvortsov and thanks him for saving his life. However, Lushkoff reveals that it was not the work itself that transformed him but the kindness of Skvortsov’s cook, Olga, who secretly did most of the work for him. Her compassion and silent support inspired Lushkoff to change his ways. The story conveys the message that sometimes, silent a...
Post a Comment
Read more