Right triangle trigonometry
RIGHT TRIANGLE TRIGONOMETRY
The word Trigonometry can be broken into the parts Tri, gon, and metry, which means
“Three angle measurement,” or equivalently “Triangle measurement.” Throughout this unit, we will learn new ways of finding missing sides and angles of triangles which we would be unable to find using the Pythagorean Theorem alone.
The basic trigonometric theorems and definitions will be found in this portion of the text, along with a few examples, but the reader will frequently be directed to refer to detailed
“tutorials” that have numerous examples, explorations, and exercises to complete for a more thorough understanding of each topic.
One comment should be made about our notation for angle measurement. In our study of
Geometry, it was standard to discuss the measure of angle A with the notation m∠A . It is a generally accepted practice in higher level mathematics to omit the measure symbol
(although there is variation from text to text), so if we are discussing the measure of a 20o angle, for example, we will use the notation ∠A = 20 rather than m∠A = 20 .
Special Right Triangles
In Trigonometry, we frequently deal with angle measures that are multiples of 30o, 45o, and 60o. Because of this fact, there are two special right triangles which are useful to us as we begin our study of Trigonometry. These triangles are named by the measures of their angles, and are known as 45o-45o-90o triangles and 30o-60o-90o triangles. A diagram of each triangle is shown below:
45o
hypotenuse
leg
45o
leg
longer leg 30o
hypotenuse
60o shorter leg
Tutorial:
For a more detailed exploration of this section along with additional examples and exercises, see the tutorial entitled “Special Right Triangles.”
The theorems relating to special right triangles can be found below, along with examples of each.
Right Triangle Trigonometry
Special Right Triangles
Theorem: In a 45o-45o-90o triangle, the legs are congruent,
The word Trigonometry can be broken into the parts Tri, gon, and metry, which means
“Three angle measurement,” or equivalently “Triangle measurement.” Throughout this unit, we will learn new ways of finding missing sides and angles of triangles which we would be unable to find using the Pythagorean Theorem alone.
The basic trigonometric theorems and definitions will be found in this portion of the text, along with a few examples, but the reader will frequently be directed to refer to detailed
“tutorials” that have numerous examples, explorations, and exercises to complete for a more thorough understanding of each topic.
One comment should be made about our notation for angle measurement. In our study of
Geometry, it was standard to discuss the measure of angle A with the notation m∠A . It is a generally accepted practice in higher level mathematics to omit the measure symbol
(although there is variation from text to text), so if we are discussing the measure of a 20o angle, for example, we will use the notation ∠A = 20 rather than m∠A = 20 .
Special Right Triangles
In Trigonometry, we frequently deal with angle measures that are multiples of 30o, 45o, and 60o. Because of this fact, there are two special right triangles which are useful to us as we begin our study of Trigonometry. These triangles are named by the measures of their angles, and are known as 45o-45o-90o triangles and 30o-60o-90o triangles. A diagram of each triangle is shown below:
45o
hypotenuse
leg
45o
leg
longer leg 30o
hypotenuse
60o shorter leg
Tutorial:
For a more detailed exploration of this section along with additional examples and exercises, see the tutorial entitled “Special Right Triangles.”
The theorems relating to special right triangles can be found below, along with examples of each.
Right Triangle Trigonometry
Special Right Triangles
Theorem: In a 45o-45o-90o triangle, the legs are congruent,