Trigonometry Worksheet

Introduction to Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this worksheet, we will explore the basics of trigonometry, including the definitions of sine, cosine, and tangent, and how to apply these concepts to solve problems.

Trigonometric Functions

The three main trigonometric functions are: * Sine (sin): The ratio of the length of the side opposite a given angle to the length of the hypotenuse. * Cosine (cos): The ratio of the length of the side adjacent to a given angle to the length of the hypotenuse. * Tangent (tan): The ratio of the length of the side opposite a given angle to the length of the side adjacent to the angle.

These functions can be used to solve problems involving right triangles, and they are essential in many areas of mathematics and science.

Right Triangle Trigonometry

In a right triangle, the trigonometric functions can be used to find the lengths of the sides and the measures of the angles. The following are some key concepts to remember: * The hypotenuse is the side opposite the right angle. * The opposite side is the side opposite a given angle. * The adjacent side is the side adjacent to a given angle. * The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Some examples of right triangle trigonometry problems include: * Finding the length of the hypotenuse given the lengths of the other two sides. * Finding the measure of an angle given the lengths of the sides. * Finding the length of a side given the measure of an angle and the length of the hypotenuse.

Tables and Formulas

The following table summarizes some of the key trigonometric formulas:

Function	Formula
Sine	sin(a) = opposite side / hypotenuse
Cosine	cos(a) = adjacent side / hypotenuse
Tangent	tan(a) = opposite side / adjacent side

Some other important formulas to remember include: * sin(a) = cos(90 - a) * cos(a) = sin(90 - a) * tan(a) = sin(a) / cos(a)

Graphs of Trigonometric Functions

The graphs of the trigonometric functions are periodic, meaning that they repeat themselves at regular intervals. The following are some key features of the graphs: * The period of a function is the distance between two consecutive points on the graph that have the same value. * The amplitude of a function is the maximum value of the function. * The phase shift of a function is the horizontal distance between the graph of the function and the graph of the same function shifted to the right by one period.

Some examples of graphs of trigonometric functions include: * The graph of y = sin(x), which has a period of 2π and an amplitude of 1. * The graph of y = cos(x), which has a period of 2π and an amplitude of 1. * The graph of y = tan(x), which has a period of π and an amplitude of undefined (since the function approaches infinity at certain points).

Applications of Trigonometry

Trigonometry has numerous applications in various fields, including: * Physics: Trigonometry is used to describe the motion of objects, including the trajectory of projectiles and the vibration of springs. * Engineering: Trigonometry is used to design and analyze structures, including bridges and buildings. * Navigation: Trigonometry is used to determine the position and direction of ships and aircraft.

Some examples of applications of trigonometry include: * Finding the height of a building: By using the angle of elevation and the distance from the building, we can use trigonometry to find the height of the building. * Determining the direction of a ship: By using the angle of the sun or a landmark, we can use trigonometry to determine the direction of the ship. * Analyzing the vibration of a spring: By using trigonometry, we can describe the motion of the spring and determine its frequency and amplitude.

💡 Note: Trigonometry is a fundamental subject that has numerous applications in various fields, and it is essential to understand the basics of trigonometry to solve problems and apply them to real-world situations.

To summarize, trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles, and it has numerous applications in various fields. By understanding the basics of trigonometry, including the definitions of sine, cosine, and tangent, and how to apply these concepts to solve problems, we can solve problems involving right triangles and apply them to real-world situations.