12th Mathematics (Arts & Science) Part 1 Chapter 3 Solution (Digest) Maharashtra state board

Chapter 3 Trigonometric Functions

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Trigonometric functions are fundamental in mathematics, particularly in the fields of geometry, calculus, and physics. These functions relate the angles of a triangle to the lengths of its sides and are essential for analyzing periodic phenomena, waves, and oscillations.

Basic Trigonometric Functions

1. Sine (sin):

   • Definition: For an angle $\theta$ in a right triangle, $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.
   • Unit Circle: The y-coordinate of the point on the unit circle corresponding to an angle $\theta$ measured from the positive x-axis.

2. Cosine (cos):

   • Definition: For an angle $\theta$ in a right triangle, $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.
   • Unit Circle: The x-coordinate of the point on the unit circle corresponding to an angle $\theta$.

3. Tangent (tan):

   • Definition: For an angle $\theta$ in a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin(\theta)}{\cos(\theta)}$.
   • Unit Circle: The slope of the line connecting the origin to the point on the unit circle.

4. Cosecant (csc):

   • Definition: The reciprocal of sine, $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}}$.

5. Secant (sec):

   • Definition: The reciprocal of cosine, $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}}$.

6. Cotangent (cot):

   • Definition: The reciprocal of tangent, $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}}$.

Properties and Identities

1. Pythagorean Identities:

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

2. Angle Sum and Difference Identities:

   • $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
   • $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
   • $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

3. Double Angle Identities:

   • $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
   • $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
   • $\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

4. Half-Angle Identities:

   • $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$
   • $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$
   • $\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} = \frac{\sin(\theta)}{1 + \cos(\theta)} = \frac{1 - \cos(\theta)}{\sin(\theta)}$

Applications

1. Geometry: Trigonometric functions are used to find missing sides and angles in triangles, particularly right triangles.
2. Calculus: They are essential in the study of integrals and derivatives, especially in problems involving periodic functions.
3. Physics: Trigonometric functions model wave motion, oscillations, and other periodic phenomena.
4. Engineering: They are used in signal processing, electrical engineering, and mechanical systems analysis.
5. Astronomy: Trigonometry is used to calculate distances and angles between celestial bodies.

Graphs of Trigonometric Functions

• Sine and Cosine: Both functions produce sinusoidal waves, with sine starting at (0,0) and cosine starting at (0,1). They have a period of $2\pi$ and an amplitude of 1.
• Tangent: The tangent function has asymptotes where $\cos(\theta) = 0$, with a period of $\pi$.
• Cosecant, Secant, and Cotangent: These are the reciprocals of sine, cosine, and tangent, respectively, and have corresponding graphs with asymptotes where their respective sine, cosine, and tangent functions are zero.

Understanding trigonometric functions is crucial for solving many mathematical problems and modeling real-world phenomena. They provide the tools to analyze and predict patterns, making them indispensable in various scientific and engineering disciplines.