Trigonometry

Unit: 20

Book Icon Class 9: Mathematics

Trigonometry, Trigonometric Ratios, Pythagoras Theorem, Trigonometric Ratio of Special 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	\[0^0\]	\[30^0\]	\[45^0\]	\[60^0\]	\[90^0\]
\[\text{sin}\]	\[0\]	\[\frac{1}{2}\]	\[\frac{1}{\sqrt{2}}\]	\[\frac{\sqrt{3}}{2}\]	\[1\]
\[\text{cos}\]	\[1\]	\[\frac{\sqrt{3}}{2}\]	\[\frac{1}{\sqrt{2}}\]	\[\frac{1}{2}\]	\[0\]
\[\text{tan}\]	\[0\]	\[\frac{1}{\sqrt{3}}\]	\[1\]	\[\sqrt{3}\]	\[\infty\]

To find the values for reciprocal trigonometric ratios, such as cosec, sec, and cot, you simply take the reciprocal of the values of the basic trigonometric ratios (sine, cos, and tan) for the given angle.

Example: Cosec of 30°

The cosecant of an angle is the reciprocal of the sine of that angle.

\(\text{So, } \csc(30^\circ) = \frac{1}{\sin(30^\circ)}\)

\(\text{or, } \csc(30^\circ) = \frac{1}{\frac{1}{2}} \)

Therefore: \(\csc(30^\circ) = 2\)

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