Chapter 3 — TRIGONOMETRIC FUNCTIONS

Concept Used

In trigonometry, the six trigonometric functions are interrelated through fundamental identities. If one trigonometric ratio and the quadrant of the angle are known, all remaining ratios can be determined.

The most important identity used is

\[ \sin^2 x + \cos^2 x = 1 \]

The sign of trigonometric functions depends on the quadrant of the angle.

• First Quadrant: all trigonometric functions are positive
• Second Quadrant: only sine is positive
• Third Quadrant: sine and cosine are negative, tangent is positive
• Fourth Quadrant: cosine is positive

Since the given angle lies in the third quadrant, both sine and cosine will be negative.

Solution Roadmap

1. Use the identity \( \sin^2x + \cos^2x = 1 \) to find \( \sin x \).
2. Use the quadrant information to determine the correct sign.
3. Find \( \tan x = \dfrac{\sin x}{\cos x} \).
4. Use reciprocal identities to find \( \cot x, \sec x, \mathrm{cosec}\ x \).

Geometric Interpretation

The third quadrant corresponds to angles between \(180^\circ\) and \(270^\circ\). Both x-coordinate and y-coordinate are negative on the unit circle.

Solution

Final Values

• \( \sin x = -\dfrac{\sqrt3}{2} \)
• \( \tan x = \sqrt3 \)
• \( \cot x = \dfrac{1}{\sqrt3} \)
• \( \sec x = -2 \)
• \( \mathrm{cosec}\ x = -\dfrac{2}{\sqrt3} \)

Significance for Board and Competitive Exams

Problems of this type frequently appear in CBSE board examinations and competitive entrance tests such as JEE, NEET and BITSAT. Students are expected to quickly determine trigonometric ratios using identities and quadrant rules.

Key skills developed through this question include:

• Understanding trigonometric identities
• Applying quadrant sign conventions
• Deriving remaining trigonometric ratios efficiently
• Strengthening unit circle intuition

Mastery of these techniques is essential for solving more advanced trigonometric equations, inverse trigonometric problems and calculus applications in higher classes.

Concept Used

When one trigonometric ratio and the quadrant of an angle are known, the remaining trigonometric functions can be determined using fundamental identities and quadrant sign conventions.

The key identity used in this problem is

\[ \sin^2 x + \cos^2 x = 1 \]

The sign of trigonometric ratios depends on the quadrant in which the angle lies.

• First Quadrant: all trigonometric ratios are positive
• Second Quadrant: sine is positive while cosine and tangent are negative
• Third Quadrant: sine and cosine are negative while tangent is positive
• Fourth Quadrant: cosine is positive while sine and tangent are negative

Since the angle lies in the second quadrant, the value of cosine must be negative while sine remains positive.

Solution Roadmap

1. Use the identity \( \sin^2x + \cos^2x = 1 \) to determine \( \cos x \).
2. Choose the correct sign of cosine using the quadrant rule.
3. Use the quotient identity \( \tan x = \dfrac{\sin x}{\cos x} \).
4. Use reciprocal identities to determine \( \cot x, \sec x, \mathrm{cosec\ x} \).

Geometric Interpretation

Angles in the second quadrant lie between \(90^\circ\) and \(180^\circ\). In this region of the unit circle, the y-coordinate is positive while the x-coordinate is negative. Hence sine is positive and cosine is negative.

Solution

Final Values

• \( \cos x = -\dfrac{4}{5} \)


• \( \tan x = -\dfrac{3}{4} \)


• \( \cot x = -\dfrac{4}{3} \)


• \( \sec x = -\dfrac{5}{4} \)


• \( \mathrm{cosec\ x} = \dfrac{5}{3} \)

Significance for Board and Competitive Exams

Questions of this form are extremely common in CBSE board examinations as well as competitive entrance tests such as JEE, NEET and BITSAT. They test the student’s understanding of trigonometric identities and quadrant sign conventions.

Through problems like this, students develop the ability to:

• Use Pythagorean trigonometric identities efficiently
• Apply quadrant rules to determine correct signs
• Derive all trigonometric ratios from a single given ratio
• Build strong conceptual understanding of the unit circle

These skills form the foundation for advanced topics such as trigonometric equations, inverse trigonometric functions and calculus applications in higher mathematics.

Concept Used

When a trigonometric ratio such as \( \cot x \) is given along with the quadrant of the angle, the remaining trigonometric functions can be determined using trigonometric identities and sign conventions.

The quotient identity used here is

\[ \cot x = \dfrac{\cos x}{\sin x} \]

Another important identity used is

\[ \sec^2 x = 1 + \tan^2 x \]

The sign of trigonometric ratios depends on the quadrant in which the angle lies.

• First Quadrant: all trigonometric ratios are positive
• Second Quadrant: sine is positive
• Third Quadrant: sine and cosine are negative while tangent and cotangent are positive
• Fourth Quadrant: cosine is positive

Since the angle lies in the third quadrant, both sine and cosine are negative, while tangent and cotangent remain positive.

Solution Roadmap

1. Use the reciprocal relation \( \tan x = \dfrac{1}{\cot x} \).
2. Apply the identity \( \sec^2x = 1 + \tan^2x \) to determine \( \sec x \).
3. Find \( \cos x = \dfrac{1}{\sec x} \).
4. Use \( \sin^2x + \cos^2x = 1 \) to determine \( \sin x \).
5. Use reciprocal relations to determine \( \mathrm{cosec\ x} \).

Geometric Interpretation

In the third quadrant of the unit circle, both the x-coordinate and y-coordinate are negative. Therefore sine and cosine are negative while tangent and cotangent are positive.

Solution

Final Values

• \( \tan x = \dfrac{4}{3} \)


• \( \sec x = -\dfrac{5}{3} \)


• \( \cos x = -\dfrac{3}{5} \)


• \( \sin x = -\dfrac{4}{5} \)


• \( \mathrm{cosec\ x} = -\dfrac{5}{4} \)

Significance for Board and Competitive Exams

Questions based on determining all trigonometric ratios from a single given ratio frequently appear in CBSE board examinations and competitive tests such as JEE, NEET and BITSAT.

Such problems strengthen conceptual understanding of:

• Reciprocal and quotient identities
• Pythagorean trigonometric identities
• Quadrant-based sign conventions
• Geometric interpretation of trigonometric ratios

A strong grasp of these relationships is essential for solving trigonometric equations, simplifying expressions and handling calculus applications involving trigonometric functions in advanced mathematics.

Concept Used

When a reciprocal trigonometric ratio such as secant is given along with the quadrant of the angle, the remaining trigonometric ratios can be determined using reciprocal identities and Pythagorean identities.

The reciprocal relation used is

\[ \sec x = \dfrac{1}{\cos x} \]

The fundamental identity used is

\[ \sin^2 x + \cos^2 x = 1 \]

The sign of trigonometric ratios depends on the quadrant in which the angle lies.

• First Quadrant: all trigonometric ratios are positive
• Second Quadrant: sine is positive
• Third Quadrant: tangent is positive
• Fourth Quadrant: cosine and secant are positive while sine and tangent are negative

Since the angle lies in the fourth quadrant, cosine and secant are positive while sine and tangent are negative.

Solution Roadmap

1. Use the reciprocal identity \( \cos x = \dfrac{1}{\sec x} \).
2. Apply the identity \( \sin^2x + \cos^2x = 1 \) to determine \( \sin x \).
3. Use the quadrant rule to assign the correct sign of sine.
4. Determine \( \mathrm{cosec}\ x \), \( \tan x \), and \( \cot x \) using quotient and reciprocal identities.

Geometric Interpretation

Angles in the fourth quadrant lie between \(270^\circ\) and \(360^\circ\). In this region of the unit circle, the x-coordinate is positive while the y-coordinate is negative.

Solution

Final Values

• \( \cos x = \dfrac{5}{13} \)
• \( \sin x = -\dfrac{12}{13} \)
• \( \mathrm{cosec}\ x = -\dfrac{13}{12} \)
• \( \tan x = -\dfrac{12}{5} \)
• \( \cot x = -\dfrac{5}{12} \)

Significance for Board and Competitive Exams

Problems involving determination of all trigonometric ratios from a given reciprocal ratio frequently appear in CBSE board examinations and competitive entrance tests such as JEE, NEET and BITSAT.

These problems strengthen conceptual understanding of:

• Reciprocal trigonometric identities
• Pythagorean identities
• Quadrant sign conventions
• Geometric interpretation of trigonometric ratios

A strong command of these relationships helps students solve trigonometric equations, simplify expressions and tackle calculus applications involving trigonometric functions in higher mathematics.

Concept Used

When the value of a trigonometric ratio such as tangent is given along with the quadrant of the angle, the remaining trigonometric functions can be determined using trigonometric identities and quadrant sign rules.

The quotient identity used is

\[ \tan x = \dfrac{\sin x}{\cos x} \]

Another important identity used is

\[ \sec^2 x = 1 + \tan^2 x \]

The signs of trigonometric functions depend on the quadrant in which the angle lies.

• First Quadrant: all trigonometric functions are positive
• Second Quadrant: sine is positive while cosine and tangent are negative
• Third Quadrant: tangent is positive
• Fourth Quadrant: cosine is positive

Since the angle lies in the second quadrant, sine is positive while cosine and tangent are negative.

Solution Roadmap

1. Use the reciprocal relation \( \cot x = \dfrac{1}{\tan x} \).
2. Apply the identity \( \sec^2x = 1 + \tan^2x \) to determine \( \sec x \).
3. Find \( \cos x = \dfrac{1}{\sec x} \).
4. Use the identity \( \sin^2x + \cos^2x = 1 \) to determine \( \sin x \).
5. Determine \( \mathrm{cosec}\ x \) using the reciprocal identity.

Geometric Interpretation

Angles in the second quadrant lie between \(90^\circ\) and \(180^\circ\). In this region of the unit circle, the x-coordinate is negative while the y-coordinate is positive.

Solution

Final Values

• \( \cot x = -\dfrac{12}{5} \)


• \( \sec x = -\dfrac{13}{12} \)


• \( \cos x = -\dfrac{12}{13} \)


• \( \sin x = \dfrac{5}{13} \)


• \( \mathrm{cosec}\ x = \dfrac{13}{5} \)

Significance for Board and Competitive Exams

Problems of this type are frequently asked in CBSE board examinations and competitive entrance tests such as JEE, NEET and BITSAT. They test a student's ability to use trigonometric identities along with quadrant-based sign conventions.

By solving such questions, students strengthen their understanding of:

• Pythagorean trigonometric identities
• Reciprocal and quotient identities
• Quadrant-based sign rules
• Geometric interpretation of trigonometric ratios

These concepts form the foundation for solving trigonometric equations, simplifying identities and studying inverse trigonometric functions in higher mathematics.

Concept Used

Trigonometric functions are periodic in nature. The sine function repeats its values after every full rotation of the circle.

The fundamental periodic identity for sine is

\[ \sin(\theta + 360^\circ) = \sin \theta \]

Therefore, if an angle is greater than \(360^\circ\), it can be reduced to a smaller equivalent angle by subtracting multiples of \(360^\circ\).

Solution Roadmap

1. Express the angle \(765^\circ\) as a sum of a multiple of \(360^\circ\) and a standard angle.
2. Use the periodic identity of the sine function.
3. Evaluate the sine of the resulting acute angle.

Geometric Interpretation

An angle of \(765^\circ\) corresponds to two complete rotations (\(720^\circ\)) followed by an additional \(45^\circ\). On the unit circle, this places the terminal side at the same position as \(45^\circ\).

Solution

Final Value

\[ \sin 765^\circ = \dfrac{1}{\sqrt{2}} \]

Significance for Board and Competitive Exams

Problems involving large angles frequently appear in CBSE board examinations and competitive tests such as JEE, NEET and BITSAT. These questions assess a student's understanding of periodicity and angle reduction techniques.

By solving such problems, students develop the ability to:

• Use periodic properties of trigonometric functions
• Reduce large angles efficiently
• Apply standard trigonometric values
• Visualize angles on the unit circle

These techniques are essential for simplifying trigonometric expressions and solving trigonometric equations in higher mathematics.

Concept Used

Large trigonometric angles can be simplified using periodicity and symmetry properties of trigonometric functions.

The sine function has the periodic property

\[ \sin(\theta + 360^\circ) = \sin \theta \]

Since cosecant is the reciprocal of sine, it also has the same periodicity.

\[ \text{cosec}(\theta + 360^\circ) = \text{cosec}\,\theta \]

Another important identity is the odd function property

\[ \sin(-\theta) = -\sin \theta \]

Therefore,

\[ \text{cosec}(-\theta) = -\text{cosec}\,\theta \]

Solution Roadmap

1. Reduce the large angle using the periodicity of trigonometric functions.
2. Express the resulting angle as a negative standard angle.
3. Use the odd property of sine and cosecant.
4. Substitute the standard value of \( \sin 30^\circ \).

Geometric Interpretation

Adding \(1440^\circ\) (four complete revolutions) to the given angle does not change the position of the terminal side. The angle therefore corresponds to \(-30^\circ\), which lies in the fourth quadrant.

Solution

Final Value

\[ \text{cosec}(-1410^\circ) = -2 \]

Significance for Board and Competitive Exams

Angle reduction problems involving very large or negative angles are common in CBSE board examinations and competitive entrance tests such as JEE, NEET and BITSAT.

Through such questions students learn to:

• Apply periodic properties of trigonometric functions
• Use odd and even function identities
• Reduce very large angles to standard angles
• Evaluate trigonometric functions efficiently

These techniques are essential for simplifying trigonometric expressions and solving trigonometric equations in advanced mathematics.

Concept Used

The tangent function is periodic. Its values repeat after every \( \pi \) radians.

The periodic identity is

\[ \tan(\theta + \pi) = \tan \theta \]

Therefore, when evaluating tangent of large angles, the angle can be reduced by subtracting multiples of \( \pi \) until a standard angle is obtained.

Solution Roadmap

1. Express the angle \( \dfrac{19\pi}{3} \) as a sum of a multiple of \( \pi \) and a standard angle.
2. Use the periodic property of tangent.
3. Evaluate the tangent of the resulting standard angle.

Geometric Interpretation

The angle \( \dfrac{19\pi}{3} \) represents multiple complete rotations around the unit circle followed by an additional angle of \( \dfrac{\pi}{3} \). Since tangent repeats every \( \pi \), the terminal side coincides with that of \( \dfrac{\pi}{3} \).

Solution

Final Value

\[ \tan \dfrac{19\pi}{3} = \sqrt{3} \]

Significance for Board and Competitive Exams

Angle reduction using periodicity is an important skill in trigonometry. Problems involving large radian measures frequently appear in CBSE board examinations and competitive tests such as JEE, NEET and BITSAT.

Through such questions students learn to:

• Use periodic properties of trigonometric functions
• Reduce large radian angles efficiently
• Recognize standard trigonometric angles
• Apply unit circle reasoning

These techniques are fundamental for simplifying trigonometric expressions and solving advanced trigonometric equations in higher mathematics.

Concept Used

Trigonometric functions are periodic. The sine function repeats its values after every \(2\pi\) radians.

The periodic identity of sine is

\[ \sin(\theta + 2\pi) = \sin \theta \]

Thus, when evaluating sine of very large or negative angles, multiples of \(2\pi\) can be added or subtracted to obtain an equivalent angle within a standard interval.

Another important identity is the odd property of sine

\[ \sin(-\theta) = -\sin \theta \]

Solution Roadmap

1. Reduce the large negative angle using periodicity of sine.
2. Express the resulting angle as a standard angle.
3. Substitute the known value of the sine function.

Geometric Interpretation

The angle \( -\dfrac{11\pi}{3} \) corresponds to several clockwise rotations on the unit circle. By adding multiples of \(2\pi\), the angle can be brought to the equivalent position \( \dfrac{\pi}{3} \).

Solution

Final Value

\[ \sin \left(-\dfrac{11\pi}{3}\right) = \dfrac{\sqrt{3}}{2} \]

Significance for Board and Competitive Exams

Angle reduction in radian measure is an important concept in trigonometry and frequently appears in CBSE board examinations and competitive entrance tests such as JEE, NEET and BITSAT.

Through such questions students develop the ability to:

• Use periodic properties of trigonometric functions
• Reduce large radian angles efficiently
• Identify equivalent angles on the unit circle
• Apply standard trigonometric values

Mastery of these techniques is essential for simplifying trigonometric expressions and solving trigonometric equations in higher mathematics.

Concept Used

The cotangent function is periodic. Its values repeat after every \( \pi \) radians.

The periodic identity for cotangent is

\[ \cot(\theta + \pi) = \cot \theta \]

Therefore, when evaluating cotangent of large or negative angles, multiples of \( \pi \) or \( 2\pi \) can be added or subtracted to obtain an equivalent angle whose value is known.

Another useful identity is

\[ \cot \theta = \dfrac{\cos \theta}{\sin \theta} \]

Solution Roadmap

1. Reduce the given angle using periodicity of the cotangent function.
2. Express the resulting angle as a standard angle.
3. Substitute the known trigonometric value.

Geometric Interpretation

The angle \( -\dfrac{15\pi}{4} \) represents multiple clockwise rotations around the unit circle. By adding \(4\pi\), the terminal side moves to the equivalent position \( \dfrac{\pi}{4} \).

Solution

Final Value

\[ \cot \left(-\dfrac{15\pi}{4}\right) = 1 \]

Significance for Board and Competitive Exams

Problems involving large radian angles are frequently asked in CBSE board examinations and competitive entrance tests such as JEE, NEET and BITSAT. These questions evaluate a student's understanding of periodicity and angle reduction techniques.

By solving such problems, students develop the ability to:

• Apply periodic properties of trigonometric functions
• Reduce large radian angles efficiently
• Identify equivalent angles on the unit circle
• Use standard trigonometric values correctly

These skills are essential for simplifying trigonometric expressions and solving advanced trigonometric equations in higher mathematics.