An introduction to high-school and college level trigonometry. This web-book is intended to bridge the gap between basic understanding of trigonometry and some of the more notorious concepts that trigonometry plays a pivital role in.
Trigonometry is the study of triangles. The subject focuses on the properties of triangles, especially of the right triangle. From the name, you might guess that trigonometry is purely about triangles and to an extent that is true. A triangle is a simple, versatile shape that appears everywhere in mathematics and physics.
However, the further one goes into the study of trigonometry, it becomes apparent that the triangle is really a medium through which one approaches other ideas, such as circles, ellipsis and hyperbolas; triangles connect concepts together and decompose complicated ideas into tangible lengths and angles.
"You think it's about triangles, but really it's about circles."Unknown
For example, modern mathematics heavily utilizes the concept of a coordinate system, where points in space are defined relative to an origin. The two ubiquitous coordinate systems, the Cartesian Coordinate System and the Polar Coordinate System, utilize the trigonometry of the right-triangle to define a point in space. Since these coordinate systems appear everywhere in applications, trigonometry plays a pivotal foundational and translational role.
Before diving into the fundementals of trigonometry, here are some high-level motivating questions and applications of trigonometry.
- Modern mathematics heavily utilizes the concept of a coordinate system. A point in the cartesian coordinate system forms the shape of a right-triangle, defined by the length of its adjacent and opposite sides. A point in the polar coordinate system forms the shape of a right-triangle, defined by the angle relative to the positive direction and the length of its hypotenuse.
- A right triangle forms the shape and components of a 2D vector which is a conceptual model used throughout physics.
- Triangulation forms the basis of Global Positioning Systems (GPS) as well as is a neat application of finding where you are on a maping using angles and landmark locations.
- The sine function describes a sinusoidal waveform used throughout physics and math. Waves model a large amount of human’s understanding about light, color, sound and much more.
- Planetary orbits can be defined in terms of triangles (see the construction of an ellipse)
- 3D modeling often uses triangular meshes to compose complex shapes in three dimensions. (Shading, rendering)
- Estimate the curvature of the earth using two obelisks (how tall is an object given the length of its shadow)
A triangle is a three sided polygon. The shape can be defined by three points as shown in the 2D interactive below. Try clicking and dragging some of the blue controls points points around.
There is more to discuss about triangles, but in the interest of laying ground-work for the study of trigonometry instead let’s first look at a list of motivating problems that hopefully explain why this subject is interesting and useful.
- What is the area of any triangle?
- What is the center point of a triangle?
- How to triangulate the position of an object?
- How do I approximate the shape of something geometrically (triangle mesh)?
- What is the surface area of a shape? (related to the one above)
- Construct geodesic dome
- What is the height of an object given the length of its shadow?
An angle is the amount of rotation between two rays. The point where the two rays meet is called the vertex. Angles are a fundemental building block of trigonometry as every triangle’s shape is formed by three angles.
An angle is often thought of in terms of the circle. The angle is measured relative to the fixed ray formed the point at the center of the circle and the point on the rightmost point of the circle. By convention, the positive direction is the counter-clockwise direction.
There are two conventions to measuring the rotation of an angle: degrees and radians. Both are reasonable ways to think about rotation and calculators allow for functions to evaluate angles in both units of measure. It is useful to be comfortable using both radians and degrees.
Radians are a unit that measure angle using the radius of a circle. One radian is equal to the amount of rotation required to travel the length of one radius along the circumference of the circle.
Degrees is a unit of measure for angles. A full rotation is equal to 360 degrees. In the cartesian coordinate system, degrees are measured starting from the rightmost edge of the circle.
An acute angle is an angle that is smaller than 90 degrees or PI fourths.
A Perpendicular angle, sometimes also referred to as a square angle, is exactly 90 degrees or PI fourths.
An obtuse angle is an angle that is larger than 90 degrees or PI fourths.
A right triangle is a special case of a triangle where one of the angles is a perpendicular angle.
The diagonal side is called the hypotenuse, the horizontal side is called the adjacent side, and the vertical side is called the opposite side.
A unique property of the right triangle is defined in Pythagorean’s theorem which relates the squares of the sides together. This can be visualized by drawing the square area of each side next to the right triangle.
The triangle defined by the three angles: 45 degrees, 45 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.
The triangle defined by the three angles: 30 degrees, 60 degrees, and 90 degrees is a special triangle that has meaningful properties in mathematics.
The trigonometric functions relate the angle of the right triangle to different ratios of its sides. These are very useful and found throughout mathematics and physics. We will start with the definitions of cosine, sine, and tangent.
- How tall is the object (solve for a side using an angle and side)?
- Use the arc functions to calculate the angle between two lines.
A circle of radius one, placed at the center of the cartesian coordinate system is a useful way to visualize the properties of the right triangle and the trigonometric functions. Placing the triangle in the coordinate plane relates the adjacent, opposite, and tangent sides of the triangle to the x and y dimensions of a two dimensional plane.
By placing the unit circle at the origin of the cartesian coordinate system, observe that the dimensions of the triangle correspond with any point along the circumference of the circle.
This brings us back to the trigonometric functions and gives new meaning to their input and output.
The law of sines is an equation that relates the three sides of a triangle with the three angles of a triangle using the sine function.
The law of cosines is a more general form of pythagoreans theorem that relates the squares of the sides together using the cosine function.
The trigonometric identites are a set of equations derived from the properties of the right triangle, the trigonometric functions, and Pythagorean’s theorem.
The pythagorean identity relates the sides of the right triangle together using only the angle of the right triangle. The identity is derived using pythagorean’s theorem and the properties of the unit circle. Geometrically, the components of the equation can be visualized by the right triangle below.
The reason why the trigonometric functions cosine and sine comes from the hypotenuse of the right triangle being one. This is probably best demonstrated with the unit circle, which visualizes all right triangles of hypotenuse one.
Substituting the values of the adjacent, opposite, and hypotenuse side of the triangle into the definitions of the trigonometric functions we see that, for right triangles of hypotenuse one, the adjacent side is equal to the cosine of the angle and the opposite side is equal to the sine of the angle.
Finally, to finish deriving the origin of the pythagorean identity we can combine these facts with Pythagorean’s Theorem, which relates the squares of the sides of a right triangle together.
Substituting in the adjacent and opposite side we are left with Pythagorean’s identity.
See the rest the trigonometric identities.
In the real world, a triangles come in all shapes in sizes. Often a programmer is converting a triangle from data into a recognizable form. A common way to represent a triangle is with two vectors, where a vector is formed by the magnitude and direction between two points in space.
When a triangle is represented as two vectors, you can calculate the area of any triangle using the formula for the area of the parellogram formed between two vectors.
Then we can divide the area betwen two vectors, which forms a parallelogram by two.
Vectors, linear algebra.
Do we model space using vectors and operations as matrices? Or do we introduce the complex plane and introduce a new type of number that can model rotation and other such transformations? What about three-dimensional space and calculus?
Linear Algebra or The Complex Plane.