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What is Trigonometry? - Functions, Formulas & Applications

What is Trigonometry? - Functions, Formulas & Applications
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  • 0:00 What Is Trigonometry?
  • 0:41 Trigonometry and Triangles
  • 2:20 Trigonometry and Circles
  • 3:34 How Will It Help Me…
  • 4:54 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

There is more to trigonometry than just sines and cosines. It can help when you need to build certain things or when you need to calculate certain distances. Learn about the usefulness of trigonometry and how triangles and circles are tied together.

What Is Trigonometry?

Trigonometry can be defined as the calculation part of geometry. Trigonometry is where you apply your knowledge of triangles from geometry and use the resulting formulas to help you solve problems.

Many math symbols come from the Greek language and, not surprisingly, the word trigonometry also has its roots in that language. The first part of the word trigon is Greek for 'triangle.' The second part comes from the Greek word metron for 'measure.' Trigonometry has a lot to do with triangles. Sines, cosines, and tangents all come from the measuring of the lowly triangle.

Trigonometry and Triangles

Not just any triangle will do for trigonometry, though. It has to be a right triangle where one of the angles is a 90 degree angle.

What's so special about the right triangle, you say? Well, just looking at it, we see that a right triangle has names for all three sides. But, whoa, hold your horses. The names don't always remain the same! The hypotenuse is always the side across from the right angle; however, the other two sides switch depending on which angle you are referring to. You see, the adjacent side, as the name suggests, is always next to the angle.

trigonometry

So, if our angle was at the blank angle, then the adjacent and opposite sides above would switch.

These three sides make trigonometry. Yes, trigonometry is all about the ratios between the sides. Sine, cosine, and tangent are all different ratios of these three sides.

Trigonometric function Ratio
Sine: sin(x)= Opposite/Hypotenuse
Cosine: cos(x)= Adjacent/Hypotenuse
Tangent: tan(x)= Opposite/Adjacent

This is the foundation of all of trigonometry! An easy way to remember these ratios is by pronouncing the word SOHCAHTOA. It helps me remember the ratios. SOH is for sine with the ratio of opposite over hypotenuse. CAH is for cosine with the ratio adjacent over hypotenuse. And TOA is for tangent with the ratio opposite over adjacent. Do you see how you have shortened the ratios to the first letter of each word? Remember that one word, and it will do you well in trigonometry. While there are other trigonometric functions, they all are related to these basic three.

Trigonometry and Circles

So, how does all this stuff about triangles relate to circles? Because, as you travel all around the circle, you can create all the triangles you need to solve trigonometry problems.

trigonometry

This particular circle with a radius of one is called the unit circle. Placing the circle on the Cartesian coordinate with the center of the circle as the origin, we will get sides that are negative. Taking a ratio with a negative number will give us a negative number in return. So it is with all three trigonometric functions. There are certain angles that will yield a negative answer. You can check this with the unit circle by drawing a triangle with the hypotenuse drawn at the angle. The angle of the triangle follows the Cartesian coordinate rules, getting bigger as you go counterclockwise from the positive side of the x-axis.

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