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):

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

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

  • Cosine (cos θ):

    $$\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$$

    (Adjacent side: The side adjacent to angle θ)

  • Tangent (tan θ):

    $$\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}$$

3. Reciprocals of Trigonometric Ratios

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

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 θ	0	1/2	1/√2	√3/2	1
  cos θ	1	√3/2	1/√2	1/2	0
  tan θ	0	1/√3	1	√3	∞
  csc θ	∞	2	√2	2/√3	1
  sec θ	1	2/√3	√2	2	∞
  cot θ	∞	√3	1	1/√3	0

5. Trigonometric Identities

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

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

6. Complementary Angles

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

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:

   $$\sin \theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}}$$ $$\cos \theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}}$$ $$\tan \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}$$

2. Reciprocal Trigonometric Ratios:

   $$\csc \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite Side}}$$ $$\sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent Side}}$$ $$\cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent Side}}{\text{Opposite Side}}$$

3. Trigonometric Ratios of Standard Angles:

   • $\sin 0^\circ = 0$, $\sin 30^\circ = \frac{1}{2}$, $\sin 45^\circ = \frac{1}{\sqrt{2}}$, $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\sin 90^{\circ }=1$
   • $\cos 0^\circ = 1$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\cos 45^\circ = \frac{1}{\sqrt{2}}$, $\cos 60^\circ = \frac{1}{2}$, $\cos 90^\circ = 0$
     
   • $\tan 0^\circ = 0$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$, $\tan 45^{\circ }=\mathrm{1, }$$\tan 60^\circ = \sqrt{3}$, $\tan 90^{\circ }=\mathrm{\infty }$

4. Pythagorean Identities:

   $$\sin^2 \theta + \cos^2 \theta = 1$$ $$1 + \tan^2 \theta = \sec^2 \theta$$ $$1 + \cot^2 \theta = \csc^2 \theta$$

5. Complementary Angle Relationships:

   $$\sin (90^\circ - \theta) = \cos \theta$$ $$\cos (90^\circ - \theta) = \sin \theta$$ $$\tan (90^\circ - \theta) = \cot \theta$$ $$\cot (90^\circ - \theta) = \tan \theta$$ $$\sec (90^\circ - \theta) = \csc \theta$$ $$\csc (90^\circ - \theta) = \sec \theta$$

Examples for Practice

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