Trigonometry | Part -1

Nine Standard >> Trigonometry | Part -1

Introduction to Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the angles and sides of triangles. While originally developed to study angles in geometry, trigonometry plays a major role in physics, engineering, astronomy, and other scientific fields.

Rotation Angle

A rotation angle is formed when a ray rotates about its endpoint. If the initial side of the ray lies along the positive x-axis and it rotates to a terminal side, the angle it makes is measured in radians or degrees.

Positive angle: Formed by counterclockwise rotation.
Negative angle: Formed by clockwise rotation.

The range of rotation angles is:
\( -\infty < \theta < \infty \)

Difference Between Trigonometric Angle and Geometrical Angle

Geometrical Angle	Trigonometric Angle
Measured only between 0° and 180°.	Measured in a complete range \( -\infty < \theta < \infty \).
Formed by intersecting lines or segments in geometry.	Formed by rotation of a ray on a coordinate plane.
Typically acute, right, obtuse, or straight angles.	Can be positive, negative, and extend beyond full rotations.
Commonly used in static shapes (like triangles).	Commonly used in dynamic and periodic systems (like waves).

Trigonometric Ratios of an Acute Angle in a Right-Angled Triangle

Right Angle

Consider a triangle ONP, where:

O is the origin and center of a circle,
P is a point on the circumference of the circle,
OP = r is the radius of the circle,
N lies on the x-axis such that line segment PN is perpendicular to it (i.e., ∠PNO = 90°).

Let ∠PON = \( \theta \). This is the angle formed between the radius and the x-axis. In triangle ONP, the basic trigonometric ratios of angle \( \theta \) are defined as:

\( \sin \theta \): Ratio of the length of the perpendicular (\( PN \)) to the hypotenuse (\( OP \)):
\( \sin \theta = \frac{PN}{OP} = \frac{opposite}{hypotenuse} \)
\( \cos \theta \): Ratio of the base (\( ON \)) to the hypotenuse (\( OP \)):
\( \cos \theta = \frac{ON}{OP} = \frac{adjacent}{hypotenuse} \)
\( \tan \theta \): Ratio of the perpendicular to the base:
\( \tan \theta = \frac{PN}{ON} = \frac{opposite}{adjacent} \)
\( \cot \theta \): Reciprocal of tangent:
\( \cot \theta = \frac{ON}{PN} = \frac{1}{\tan \theta} \)
\( \sec \theta \): Reciprocal of cosine:
\( \sec \theta = \frac{OP}{ON} = \frac{1}{\cos \theta} \)
\( \csc \theta \): Reciprocal of sine:
\( \csc \theta = \frac{OP}{PN} = \frac{1}{\sin \theta} \)

Trigonometry introduces us to angles beyond geometry, where angles can take on any real value. Trigonometric ratios connect angles to side lengths in triangles, and are foundational to many fields including engineering, physics, and navigation.