History of Trigonometry

Trigonometry is known to have been used during the ancient Egyptian period, which dates back before Christ. However, the founder of trigonometry is considered to be Battani. Battani lived in the Harran district of Şanlıurfa, Turkey, and used the concepts of sine and cosine while performing astronomical calculations. Furthermore, Battani was the first scientist to prove the cosine theorem. The term “sine” is derived from the Arabic word “Cayb,” which means “curvature, embrace, pocket.”

The terms tangent and cotangent were first defined by Abu Wafa al-Burjani. The word tangent is derived from the Arabic word “lamis,” which means “touch,” and was later changed to tangent in French. Abu Wafa al-Burjani also proved the Sine Theorem.

Basic Concepts

To understand trigonometry, we must know the fundamental concepts. Let’s start by explaining these concepts.

Angle

• Line: A straight line made up of points that can be extended in both directions as much as desired. A line only has length but cannot be measured.
• Ray: A ray is a set of points on a line that starts from a point A and extends in one direction.
• Line Segment: A line segment is a set of points between two specific points on a line.

Definition of an Angle

An angle is formed by two rays that do not lie on the same line and share a common starting point. The rays are the sides of the angle, and the common point is the vertex of the angle.

Measurement of an Angle

The measure of an angle is the extent of its opening. The angle and its measurement are not the same. A number between 0 and 180 degrees is assigned to each angle.

0° < A < 180°

Degree

The degree is the unit used to express the measure of an angle. A full rotation of a circle is considered to be 360 degrees. A quarter circle is 90 degrees, a half circle is 180 degrees, and a full circle is 360 degrees.

The degree unit is commonly used in daily life because it is more intuitive. For example, a quarter circle is 90 degrees, while a full circle is 360 degrees.

Example: If an angle is 90°, it covers one-quarter of a circle.

Radian

Radian is another unit for measuring angles and is more suitable for mathematical calculations. On the unit circle, the measure of an angle is proportional to the length of the arc. If an angle rotates by the length of the arc, it is called 1 radian.

The full circle is measured in 2π radians. So, 180° equals π radians, and 90° equals π/2 radians.

Example: 90° = π/2 radians.

Radian is preferred in fields like trigonometry and calculus because calculations with this unit yield more direct and simplified results.

Principal Measurement

The principal measurement of an angle is the standard measure used to reduce the angle’s measure to between 0° and 360°. If an angle has multiples of 360° added or subtracted, its principal measurement is found.

Principal Measurement Calculation:

• Positive Angle: If an angle has a multiple of 360° added, its principal measurement is found by subtracting multiples of 360°. For example, for 450°, its principal measurement is 450° – 360° = 90°.
• Negative Angle: If the angle is negative, 360° is added to find its principal measurement. For example, for -30°, its principal measurement is -30° + 360° = 330°.

Unit Circle Trigonometry

In the Cartesian coordinate system, the unit (trigonometric) circle is a circle with a radius of 1 unit and its center at the intersection of the coordinate axes.

1.In the Cartesian coordinate system, the unit (trigonometric) circle is a circle with a radius of 1 unit and its center at the intersection of the coordinate axes. The equation of the unit circle is:

x² + y² = 1

2.Let’s consider the point A on the unit circle. The length of the x-coordinate of this point is called the cosine of the angle α and is denoted as cos(α). The length of the y-coordinate is called the sine of the angle α and is denoted as sin(α).

3. Let’s draw the triangle OAB outside the unit circle, where |OB| = 1. Let the point A be at (x, y), and let OAB^ = α. The length of the y-coordinate of point A is called the tangent of angle α, and it is denoted as tan(α) or tg(α).

Now, let’s draw the triangle OCD outside the unit circle, where |OC| = 1 cm. Let point D be at (x, y), and let COD^ = α. The length of the x-coordinate of point D is called the cotangent of angle α, and it is denoted as cot(α) or ctg(α).

4. Consider the tangent line CB at point A on the unit circle. If |OA| = 1, and OAB^ = α, the length of the x-coordinate of point B is called the secant of angle α and is denoted as sec(α). The length of the y-coordinate of point C is called the cosecant of angle α and is denoted as csc(α) or cosec(α).

Trigonometric Ratios of Acute Angles in Right Triangles

In a right triangle, the side opposite the right angle is called the “Hypotenuse,” the side adjacent to the angle is called the “Adjacent Side,” and the side opposite the angle is called the “Opposite Side.”

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Part 2