Trigonometry

Unit: 7

Book Icon Class 9: Optional Math

Trigonometric Ratios, Trigonometric Ratios of Special Angles, Identities of Trigonometric Ratios, Trigonometric Ratios of Any Angles, Trigonometric Ratios of Compound Angles

Trigonometric Ratios

These ratios are the fundamental relationships in trigonometry and are used to relate the angles of a triangle to the lengths of its sides. There are six trigonometric ratios.

Basic Trigonometric Ratios: sin, cos, tan

Reciprocal Trigonometric Ratios: cosec, sec, cot

For a given angle \(\theta\):

- Perpendicular: The side opposite the angle \(\theta\). (Opposite)

- Base: The side next to the angle \(\theta\). (Adjacent Side)

- Hypotenuse: The longest side, opposite the right angle.

1. Sine (sin): The ratio of the length of the Perpendicular to the Hypotenuse.

\[ \sin(\theta) = \frac{\text{Perpendicular}}{\text{Hypotenuse}} = \frac{p}{h} \]

2. Cosine (cos): The ratio of the length of the Base to the Hypotenuse.

\[ \cos(\theta) = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{b}{h} \]

3. Tangent (tan): The ratio of the length of the Perpendicular to the Base.

\[ \tan(\theta) = \frac{\text{Perpendicular}}{\text{Base}} = \frac{p}{b} \]

4. Cosecant (cosec/csc): The reciprocal of sine. It is the ratio of the Hypotenuse to the Perpendicular.

\[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{Hypotenuse}}{\text{Perpendicular}} = \frac{h}{p} \]

5. Secant (sec): The reciprocal of cosine. It is the ratio of the Hypotenuse to the Base.

\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{Hypotenuse}}{\text{Base}} = \frac{h}{b} \]

6. Cotangent (cot): The reciprocal of tangent. It is the ratio of the Base to the Perpendicular.

\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{Base}}{\text{Perpendicular}} = \frac{b}{p} \]

Trigonometric Ratio of Special Angles

The values of trigonometric ratios for \(0^0\), \(30^0\), \(45^0\), \(60^0\), and \(90^0\), are as follows.

Angle	\[0^0\]	\[30^0\]	\[45^0\]	\[60^0\]	\[90^0\]
Ratio	\[0^0\]	\[30^0\]	\[45^0\]	\[60^0\]	\[90^0\]
\[\text{sin}\]	\[0\]	\[\frac{1}{2}\]	\[\frac{1}{\sqrt{2}}\]	\[\frac{\sqrt{3}}{2}\]	\[1\]
\[\text{cos}\]	\[1\]	\[\frac{\sqrt{3}}{2}\]	\[\frac{1}{\sqrt{2}}\]	\[\frac{1}{2}\]	\[0\]
\[\text{tan}\]	\[0\]	\[\frac{1}{\sqrt{3}}\]	\[1\]	\[\sqrt{3}\]	\[\infty\]

Identities of Trigonometric Ratios

What are identities?

- In mathematics, Identities are equations that are true for all values of the variable(s) within their domain. Trigonometric identities are equalities involving trigonometric functions that hold true for all valid angles. These are essential for solving trigonometric equations, simplifying expressions, and proving relationships in geometry and calculus.

Here are the Trigonometric Identities categorized by type:

\(\begin{aligned} & \text{1. Pythagorean Identities } \\ & a. sin^{2}\theta + cos^{2}\theta = 1 \\ & b. 1 + tan^{2}\theta = sec^{2}\theta \\ & c. 1 + cot^{2}\theta = cosec^{2}\theta \\ \\ & \text{2. Reciprocal Identities} \\ & a. cosec \theta = \frac{1}{sin\theta} \\ & b. sec \theta = \frac{1}{cos\theta} \\ & c. cot \theta = \frac{1}{tan\theta} \\ \\ & \text{3. Quotient Identities} \\ & a. tan\theta = \frac{sin\theta}{cos\theta} \\ & b. cot\theta = \frac{cos\theta}{sin\theta} \\ \end{aligned}\)

Trigonometric ratios of any angle

In the previous chapter, we learned the Trigonometric Ratio of Special Angles like \(0^\circ , 30^\circ , 45^\circ , 60^\circ \text{and, } 90^\circ \) . Now in this chapter, we will explore techniques and methods for Trigonometric Ratios of Any Angles. Below is the list of trigonometric functions \( \sin, \cos, \tan, \csc, \sec, \cot \) for the given angles in terms of \( \theta \):

\(\begin{aligned} & A. (90^\circ - \theta) \\ & 1. \sin(90^\circ - \theta) = \cos\theta \\ & 2. \cos(90^\circ - \theta) = \sin\theta \\ & 3. \tan(90^\circ - \theta) = \cot\theta \\ & 4. \csc(90^\circ - \theta) = \sec\theta \\ & 5. \sec(90^\circ - \theta) = \csc\theta \\ & 6. \cot(90^\circ - \theta) = \tan\theta \\ \\ \end{aligned}\)

\(\begin{aligned} & B. (90^\circ + \theta) \\ & 1. \sin(90^\circ + \theta) = \cos\theta \\ & 2. \cos(90^\circ + \theta) = -\sin\theta \\ & 3. \tan(90^\circ + \theta) = -\cot\theta \\ & 4. \csc(90^\circ + \theta) = \sec\theta \\ & 5. \sec(90^\circ + \theta) = -\csc\theta \\ & 6. \cot(90^\circ + \theta) = -\tan\theta \\ \end{aligned}\)

\(\begin{aligned} & C. ( 180^\circ - \theta ) \\ & 1. \sin(180^\circ - \theta) = \sin\theta \\ & 2. \cos(180^\circ - \theta) = -\cos\theta \\ & 3. \tan(180^\circ - \theta) = -\tan\theta \\ & 4. \csc(180^\circ - \theta) = \csc\theta \\ & 5. \sec(180^\circ - \theta) = -\sec\theta \\ & 6. \cot(180^\circ - \theta) = -\cot\theta \\ \\ \end{aligned} \)

\(\begin{aligned} & D. (180^\circ + \theta ) \\ & 1. \sin(180^\circ + \theta) = -\sin\theta \\ & 2. \cos(180^\circ + \theta) = -\cos\theta \\ & 3. \tan(180^\circ + \theta) = \tan\theta \\ & 4. \csc(180^\circ + \theta) = -\csc\theta \\ & 5. \sec(180^\circ + \theta) = -\sec\theta \\ & 6. \cot(180^\circ + \theta) = \cot\theta \\ \\ \end{aligned} \)

\(\begin{aligned} & E. ( 270^\circ - \theta ) \\ & 1. \sin(270^\circ - \theta) = -\cos\theta \\ & 2. \cos(270^\circ - \theta) = -\sin\theta \\ & 3. \tan(270^\circ - \theta) = \cot\theta \\ & 4. \csc(270^\circ - \theta) = -\sec\theta \\ & 5. \sec(270^\circ - \theta) = -\csc\theta \\ & 6. \cot(270^\circ - \theta) = \tan\theta \\ \\ \end{aligned} \)

\(\begin{aligned} & F. ( 270^\circ + \theta ) \\ & 1. \sin(270^\circ + \theta) = -\cos\theta \\ & 2. \cos(270^\circ + \theta) = \sin\theta \\ & 3. \tan(270^\circ + \theta) = -\cot\theta \\ & 4. \csc(270^\circ + \theta) = -\sec\theta \\ & 5. \sec(270^\circ + \theta) = \csc\theta \\ & 6. \cot(270^\circ + \theta) = -\tan\theta \\ \\ \end{aligned} \)

\(\begin{aligned} & G. ( 360^\circ - \theta) \\ & 1. \sin(360^\circ - \theta) = -\sin\theta \\ & 2. \cos(360^\circ - \theta) = \cos\theta \\ & 3. \tan(360^\circ - \theta) = -\tan\theta \\ & 4. \csc(360^\circ - \theta) = -\csc\theta \\ & 5. \sec(360^\circ - \theta) = \sec\theta \\ & 6. \cot(360^\circ - \theta) = -\cot\theta \\ \\ \end{aligned} \)

\(\begin{aligned} & H. ( 360^\circ + \theta ) \\ & 1. \sin(360^\circ + \theta) = \sin\theta \\ & 2. \cos(360^\circ + \theta) = \cos\theta \\ & 3. \tan(360^\circ + \theta) = \tan\theta \\ & 4. \csc(360^\circ + \theta) = \csc\theta \\ & 5. \sec(360^\circ + \theta) = \sec\theta \\ & 6. \cot(360^\circ + \theta) = \cot\theta \\ \\ \end{aligned}\)

Trigonometric Ratios of Compound Angles

What is a Compound Angle?

A compound angle is the sum or difference of two angles. If \( A \) and \( B \) are two angles, then their sum \( (A + B) \) and difference \( (A - B) \) are called compound angles.

Formulae for Trigonometric Ratios of Compound Angles

\(\begin{aligned} & \text{Formulae for Compound Angles:} \\ & 1. \sin(A + B) = \sin A \cos B + \cos A \sin B \\ & 2. \sin(A - B) = \sin A \cos B - \cos A \sin B \\ & 3. \cos(A + B) = \cos A \cos B - \sin A \sin B \\ & 4. \cos(A - B) = \cos A \cos B + \sin A \sin B \\ & 5. \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}, \quad \text{(if \( 1 - \tan A \tan B \neq 0 \))} \\ & 6. \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}, \quad \text{(if \( 1 + \tan A \tan B \neq 0 \))} \end{aligned}\)

LESSON CONTENTS

Share Now

Feedback Report an Issue

Lesson Completed successfully.