Maharashtra Board Class 12 Maths Part 1 Chapter 3 Trigonometric Functions PDF Download

Trigonometric Functions

Let's Study

3.1 Trigonometric Equations and their solutions

3.2 Solutions of triangle

3.2.1 Polar co-ordinates

3.2.2 Relation between the polar co-ordinates and the Cartesian co-ordinates

3.2.3 Solving a Triangle

3.2.4 The Sine rule

3.2.5 The Cosine rule

3.2.6 The Projection rule

3.2.7 Applications of the Sine rule, the Cosine rule and the Projection rule

3.3 Inverse Trigonometric Functions

Properties, Principal values of inverse trigonometric functions

Introduction

We are familiar with algebraic equations. In this chapter we will learn how to solve trigonometric equations, their principal and general solutions, their properties. Trigonometric functions play an important role in integral calculus.

Let's learn

3.1 Trigonometric Equations and their solutions

Trigonometric equation

An equation involving trigonometric function (or functions) is called a trigonometric equation.

For example: \(\sin\theta = \frac{1}{2}\), \(\tan\theta = 2\), \(\cos 3\theta = \cos 5\theta\) are all trigonometric equations. \(x = a + \sin(\omega t + \alpha)\) is also a trigonometric equation.

Solution of Trigonometric equation

A value of a variable in a trigonometric equation which satisfies the equation is called a solution of the trigonometric equation.

Teacher's Note

Trigonometric equations are very important in physics and engineering. For example, when a pendulum swings, we use trigonometric equations to find when it reaches a certain height.

Exam Trick

Remember: A trigonometric equation like \(\sin\theta = k\) has many solutions. You need to find solutions in \([0, 2\pi)\) which are called principal solutions.

Points to Remember

A trigonometric equation can have more than one solution.
Because of periodicity, a trigonometric equation may have infinite solutions.
We are interested in finding solutions in the interval \([0, 2\pi)\).
A principal solution is between 0 and \(2\pi\).
General solutions use the periodicity of trigonometric functions.

Principal Solutions

A solution \(\alpha\) of a trigonometric equation is called a principal solution if \(0 \leq \alpha < 2\pi\).

For example, \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\) are the principal solutions of the trigonometric equation \(\sin\theta = \frac{1}{2}\).

Note that \(\frac{13\pi}{6}\) is a solution but not a principal solution of \(\sin\theta = \frac{1}{2}\) because \(\frac{13\pi}{6} \notin [0, 2\pi)\).

0 is the principal solution of equation \(\sin\theta = 0\) but \(2\pi\) is not a principal solution.

Trigonometric equation \(\cos\theta = -1\) has only one principal solution. \(\theta = \pi\) is the only principal solution of this equation.

Teacher's Note

Principal solutions are the basic answers we find first. Think of them as the simplest answers in one complete cycle of the trigonometric function.

Exam Trick

To find principal solutions: First find one solution. Then use allied angle formulas like \(\sin(\pi - \theta) = \sin\theta\) to find other solutions in \([0, 2\pi)\).

Points to Remember

Principal solutions must be in the range \([0, 2\pi)\).
For \(\sin\theta = k\), there are usually two principal solutions.
For \(\cos\theta = k\), there are usually two principal solutions.
For \(\tan\theta = k\), there is usually one principal solution.
Use the unit circle to identify which quadrants contain your solutions.

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