Trigonometrical Functions and Triangles

Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles. It plays a significant role in geometry, where the properties of triangles are used to make calculations. This unit will delve into the laws of sines and cosines, and how to apply trigonometry in solving real-world problems.

Laws of Sines and Cosines

The law of sines and the law of cosines are fundamental to solving triangles, that is, finding the lengths of their sides and the measures of their angles.

Law of Sines

The law of sines states that the ratio of the length of a side of a triangle to the sine of its opposite angle is the same for all three sides of the triangle. This can be written as:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the opposite angles.

Law of Cosines

The law of cosines generalizes the Pythagorean theorem to all types of triangles, not just right-angled. It states that:

c² = a² + b² - 2ab cos(C)

where a, b, and c are the lengths of the sides of the triangle, and C is the angle opposite side c.

Word Problems Using Trigonometry

Trigonometry is not just about solving triangles. It has a wide range of applications in various fields such as physics, engineering, computer science, and more. Here, we will look at some basic word problems involving height and distance, and navigation.

Height and Distance Problems

Trigonometry can be used to find the height of a building or a mountain, the width of a river, and the distance of a ship from the shore. These problems typically involve using the tangent of an angle to find a missing side length in a right triangle.

Navigation and Surveying Problems

In navigation and surveying, trigonometry can help determine a ship's location and direction, or the distance between two points on a map. These problems often involve the law of sines and the law of cosines.

Advanced Applications of Trigonometry

Trigonometry also has advanced applications in the study of sound and light waves, and in the analysis of harmonic motion.

Harmonic Motion

Harmonic motion describes the motion of a body that swings back and forth about a stable equilibrium position. The sine and cosine functions are used to model this type of motion.

Sound and Light Waves

The sine and cosine functions also model the behavior of sound and light waves. They can describe the amplitude, frequency, and phase of a wave.

By the end of this unit, you should have a solid understanding of how to apply trigonometry to solve real-world problems. The skills you learn here will be invaluable in many fields, including physics, engineering, computer science, and more.