Introduction to Trigonometry

8.1 Introduction

Trigonometry is a fundamental branch of mathematics that deals with the relationships between the sides and angles of triangles, especially right triangles. It has a central position in mathematics because it connects geometry with algebra and analysis through the study of these relationships. This section introduces trigonometry by illustrating real-life situations where right triangles can be imagined and used to calculate unknown distances or heights without direct measurement. For example, when a student looks at the top of the Qutub Minar, a right triangle is formed between the student’s eye, the base, and the top of the Minar. Similarly, a girl sitting on a balcony looking down at a flower pot across a river forms another right triangle, which can help estimate the width of the river if the height of the balcony is known. Another example involves a hot air balloon flying in the air, where the altitude of the balloon can be found by imagining right triangles formed with the ground and the line of sight. The term 'trigonometry' comes from the Greek words 'tri' meaning three, 'gon' meaning sides, and 'metron' meaning measure, reflecting its focus on measuring the sides and angles of triangles. Historically, trigonometry originated from ancient civilizations such as Egypt and Babylon and was extensively used by early astronomers to calculate distances of celestial bodies from the Earth. Today, trigonometry underpins many technological and engineering applications. In this chapter, the focus is on defining trigonometric ratios for acute angles in right triangles, extending these definitions to angles of 0° and 90°, calculating ratios for special angles (0°, 30°, 45°, 60°, 90°), and proving fundamental trigonometric identities. The chapter restricts itself to acute angles initially but notes that these concepts can be extended further in advanced studies.

• Trigonometry studies relationships between sides and angles of triangles, especially right triangles.
• Real-life examples include measuring heights and distances without direct measurement.
• The word 'trigonometry' is derived from Greek words meaning 'three sides measure'.
• Historically used in astronomy and geometry since ancient civilizations.
• Focus on trigonometric ratios for acute angles and special angles.
• Foundation for advanced applications in engineering and physical sciences.
• 📌 Trigonometry: The branch of mathematics dealing with the relationships between sides and angles of triangles.
• 📌 Right triangle: A triangle with one angle equal to 90°.
• 📌 Acute angle: An angle less than 90°.

8.2 Trigonometric Ratios

This section formally defines the six fundamental trigonometric ratios for an acute angle in a right triangle. Consider a right triangle ABC with the right angle at B. The angle at A is acute. The side opposite to angle A is BC, the hypotenuse is AC, and the side adjacent to angle A is AB. The trigonometric ratios are defined as follows: - Sine (sin A) = (side opposite to A) / (hypotenuse) = BC / AC - Cosine (cos A) = (side adjacent to A) / (hypotenuse) = AB / AC - Tangent (tan A) = (side opposite to A) / (side adjacent to A) = BC / AB - Cosecant (csc A) = 1 / sin A = (hypotenuse) / (side opposite to A) = AC / BC - Secant (sec A) = 1 / cos A = (hypotenuse) / (side adjacent to A) = AC / AB - Cotangent (cot A) = 1 / tan A = (side adjacent to A) / (side opposite to A) = AB / BC These ratios express the relationships between the sides of the triangle relative to the angle A. It is important to note that the ratios csc A, sec A, and cot A are reciprocals of sin A, cos A, and tan A respectively. Also, tan A = sin A / cos A and cot A = cos A / sin A. The section also explains that these ratios depend only on the angle and not on the size of the triangle. This is demonstrated by considering similar triangles formed by drawing perpendiculars from points on the hypotenuse, showing that the ratios remain constant for the same angle. Historical context is provided, mentioning Aryabhata's early work on sine (called jya) and the evolution of the terms sine, cosine, and tangent through translations and mathematicians like Edmund Gunter and Sir Jonas Moore. The section concludes with a note on notation conventions, emphasizing that sin A denotes the sine of angle A, not a product, and introduces the use of the Greek letter theta (θ) as a common symbol for an angle.

• Six trigonometric ratios defined for an acute angle in a right triangle.
• Sine, cosine, and tangent relate sides opposite, adjacent, and hypotenuse.
• Cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent.
• Trigonometric ratios depend only on the angle, not the triangle size.
• Historical origins from Aryabhata and later European mathematicians.
• Notation conventions clarify that sin A is a function, not a product.
• 📌 Sine (sin): Ratio of opposite side to hypotenuse.
• 📌 Cosine (cos): Ratio of adjacent side to hypotenuse.
• 📌 Tangent (tan): Ratio of opposite side to adjacent side.