## 17Calculus Precalculus - Trigonometry

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All of trigonometry is based on right triangles with the help of the unit circle. We have found some great sites and videos that explain trig intuitively and have lots of examples.
We do not attempt to cover all of trigonometry, just what you need for calculus.

A lot of what they teach in trigonometry is not directly used in calculus. That is to say, they have you work a lot problems in order to become comfortable with trig but the problems do not directly apply to calculus. That said, trig IS REQUIRED for calculus. However, this video playlist will go over most of the trig that you need calculus. This is a great series of videos that will help to refresh your memory or give you an idea of what trig you need to know for calculus.

### James Hamblin - crash course trigonometry youtube playlist

video by James Hamblin

Intuitive Understanding of Trig

It is easy to get lost in the details of trig and never figure out why we are even learning it. BetterExplained.com has a great page on understanding trig intuitively. This page will give you a feel for it and some great examples on where trig is used and why we should study it. Before going on, read and study this page thoroughly.

Right Triangles

Most of the time you will be working with right triangles, which are at the heart of trigonometry. Remember from geometry that right triangles are triangles with one of the interior angles measuring 90o. This is the same as saying that one of the sides is perpendicular to another. Also remember from geometry that when you add up all the interior angles of a triangle, you get 180o. So, if you know that one of the angles is 90o, that leaves 180 - 90 = 90 for the other two angles. This is an important thing to remember when working with right triangles. First determine which angle in the 90o angle, then work with the other two whose sum will also be 90. Pretty cool, eh?

 Special Right Triangles

There are several special right triangles that you will see over and over. So it is important to become very familiar with them. They are 30-60-90 and 45-45-90 triangles. The numbers refer to the interior angles. Notice that both of them have one 90o angle. So we could also call them 30-60 and 45-45 right triangles. And, if you think about it, we really only need one angle when talking about right triangles. So by saying we have a 30 degree right triangle, we have enough information about the triangle to know that we have 30-60-90 triangle.

Here are some videos that will give us some feel for these triangles.

### PatrickJMT - Special Right Triangles in Geometry: 45-45-90 and 30-60-90 [13min-13secs]

video by PatrickJMT

### Krista King Math - 45-45-90 Triangles [10min-53secs]

video by Krista King Math

### Krista King Math - 30-60-90 Triangles [10min-6secs]

video by Krista King Math

The Unit Circle

All of trigonometry is based on the unit circle. So it is important for you to understand the following figure and commit it to memory.

The Unit Circle [ source: wikipedia ] Here is a good video that shows where some of the information in this figure comes from.

### Krista King Math - The Unit Circle [18min-15secs]

video by Krista King Math

There is a lot of information here, so here is a video that help you remember what you need to.

### PatrickJMT - A Way to Remember the Entire Unit Circle for Trigonometry [7min-18secs]

video by PatrickJMT

Trig Identities

Here is a list of the trig identities you will use most in calculus.

Set 1 - Basic Identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - Squared Identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - Double-Angle Formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\cos(2t) = \cos^2(t) - \sin^2(t)$$

Set 4 - Half-Angle Formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Remembering trig identities can be difficult. However, there are techniques to help you learn and memorize them. The main way to remember anything is to use it. It also helps to see how other people remember them. Here are some videos where the instructor explains how he remembers them. You may be able to pick up some techniques and new ideas from these videos.

### PatrickJMT - Trigonometric Identities: How to Derive / Remember Them (1) [13min-53secs]

video by PatrickJMT

### PatrickJMT - Trigonometric Identities: How to Derive / Remember Them (2) [9min-43secs]

video by PatrickJMT

### PatrickJMT - Trigonometric Identities: How to Derive / Remember Them (3) [13min-24secs]

video by PatrickJMT

Okay, so these identities may be a bit overwhelming to learn and remember. Here is a fun video that shows the geometric interpretation of all 6 trig functions. He goes through them pretty fast, so you may want to take notes but this gives you another way to remember the trig identities.

### 3Blue1Brown - Tattoos on Math [8min-14secs]

video by 3Blue1Brown

The Pythagorean Theorem

This is a very important theorem that you will use a lot in precalculus and calculus. Here is a good video that explains it in detail.

### Krista King Math - Pythagorean Theorem [3min-7secs]

video by Krista King Math

Really UNDERSTAND Precalculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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 Intuitive Understanding of Trig Right Triangles The Unit Circle Trig Identities The Pythagorean Theorem

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