Basic Trigonometry: Sin Cos Tan (NancyPi)

Summary

In this educational video, Nancy introduces viewers to the fundamental trigonometric functions: sine, cosine, and tangent, along with their reciprocals—cosecant, secant, and cotangent. She utilizes the mnemonic SOH-CAH-TOA to simplify the process of remembering the relationships between the sides of a right triangle and the angle theta. The video demonstrates how to calculate each trig function by identifying the hypotenuse, adjacent, and opposite sides relative to theta. Additionally, Nancy addresses common confusions regarding triangle orientation and emphasizes the importance of including the angle in the final answer to ensure clarity, making the content both informative and accessible.

Takeaways

• 📚 The video introduces the six basic trigonometric functions: sine (sin), cosine (cos), tangent (tan), and their reciprocals: cosecant (csc), secant (sec), and cotangent (cot).
• 🔑 The memory trick SOH-CAH-TOA is explained to help remember the relationships between the sides of a right triangle and the trigonometric functions: Sine (Opposite/Hypotenuse), Cosine (Adjacent/Hypotenuse), and Tangent (Opposite/Adjacent).
• 📐 All trigonometric functions involve right triangles, where the longest side is the hypotenuse, the side next to the angle theta is the adjacent side, and the side opposite theta is the opposite side.
• 🔍 To find sin(theta), divide the length of the opposite side by the length of the hypotenuse.
• 📏 To find cos(theta), divide the length of the adjacent side by the length of the hypotenuse.
• 📘 To find tan(theta), divide the length of the opposite side by the length of the adjacent side.
• 🔄 To find the reciprocal functions (csc, sec, cot), take the reciprocal of sin, cos, and tan respectively, which involves flipping the numerator and denominator.
• 📌 The video clarifies that the orientation of the right triangle does not affect the definitions of opposite, adjacent, and hypotenuse sides.
• ⚠️ A common mistake is forgetting to include the angle (theta) when writing the answer, which is crucial for the function to have meaning.
• 👍 The presenter encourages viewers to like or subscribe if the video helped them understand the trigonometric functions.

Q & A

What are the six basic trigonometric functions?

The six basic trigonometric functions are sine, cosine, tangent, and their reciprocals, which are cosecant, secant, and cotangent.

What is the memory trick for remembering the ratios of sine, cosine, and tangent?

The memory trick for remembering the ratios of sine, cosine, and tangent is SOH-CAH-TOA, which stands for Sine equals Opposite over Hypotenuse, Cosine equals Adjacent over Hypotenuse, and Tangent equals Opposite over Adjacent.

What is a right triangle and why is it important in trigonometry?

A right triangle is a triangle with one angle measuring 90 degrees. It is important in trigonometry because the definitions of sine, cosine, and tangent are based on the ratios of the sides of a right triangle relative to one of its non-right angles, typically denoted as theta.

What is the hypotenuse of a right triangle?

The hypotenuse of a right triangle is the longest side of the triangle, which is opposite the right angle.

How do you determine the adjacent and opposite sides relative to an angle in a right triangle?

The adjacent side is the side of the right triangle that is next to the angle (theta) but not the hypotenuse. The opposite side is the side that is directly opposite the angle (theta).

If the hypotenuse of a right triangle is 5 and the opposite side is 4, what is the sine of the angle?

The sine of the angle (sin(theta)) is the ratio of the opposite side to the hypotenuse. So, sin(theta) = 4/5.

How do you find the cosine of an angle in a right triangle if the adjacent side is 3 and the hypotenuse is 5?

The cosine of the angle (cos(theta)) is the ratio of the adjacent side to the hypotenuse. Therefore, cos(theta) = 3/5.

What is the formula for calculating the tangent of an angle in a right triangle?

The tangent of an angle (tan(theta)) is the ratio of the opposite side to the adjacent side. So, tan(theta) = opposite/adjacent.

How can you find the values of cosecant, secant, and cotangent using the basic trigonometric functions?

To find the values of cosecant (csc), secant (sec), and cotangent (cot), first find the values of their respective basic trigonometric functions (sin, cos, tan) and then take the reciprocal of those values.

Why is it important to include the angle when writing the value of a trigonometric function?

It is important to include the angle when writing the value of a trigonometric function to specify which angle the function refers to, as the value of the function can change depending on the angle being considered. For example, you should write sin(theta) instead of just sin to indicate the sine of a specific angle theta.

What can confuse people when dealing with different orientations of right triangles in trigonometry?

Different orientations of right triangles, such as when the triangle is drawn sideways, on its hypotenuse, or with the hypotenuse horizontal, can confuse people because the positions of the opposite, adjacent, and hypotenuse sides relative to the angle theta can change, requiring careful identification of these sides to correctly apply the trigonometric functions.

Outlines

00:00

📚 Introduction to Basic Trigonometric Functions

Nancy introduces the video by explaining that she will teach viewers how to use the six basic trigonometric functions: sine, cosine, tangent, and their reciprocals, cosecant, secant, and cotangent. She emphasizes the use of the memory trick SOH-CAH-TOA to remember the relationships between the sides of a right triangle and the trigonometric functions. The video focuses on the importance of understanding the hypotenuse, adjacent, and opposite sides relative to an angle (theta) in a right triangle to calculate these functions.

05:01

🔍 Calculating Trigonometric Functions Using SOH-CAH-TOA

The video continues with Nancy demonstrating how to calculate sine, cosine, and tangent using the SOH-CAH-TOA mnemonic. She explains that sine (sin) is the ratio of the opposite side to the hypotenuse, cosine (cos) is the ratio of the adjacent side to the hypotenuse, and tangent (tan) is the ratio of the opposite side to the adjacent side. Nancy provides an example problem with a right triangle where the sides are labeled, and she calculates sin(theta), cos(theta), and tan(theta) accordingly. She then moves on to explain how to find the reciprocal functions: cosecant (csc), secant (sec), and cotangent (cot) by taking the reciprocal of sin, cos, and tan, respectively.

10:03

🤔 Tackling Trigonometric Functions in Different Triangle Orientations

Nancy addresses common confusions that arise when dealing with right triangles in different orientations. She shows various ways a right triangle might be drawn and emphasizes that regardless of the orientation, the hypotenuse is always the longest side. She explains how to identify the adjacent and opposite sides in relation to theta in each case. Nancy also cautions viewers to always include the angle (theta) when writing the final answer for a trigonometric function to ensure clarity and correctness. The video concludes with Nancy encouraging viewers to like and subscribe if they found the tutorial helpful.

Keywords

💡Trigonometric functions

Trigonometric functions, also known as 'trig functions', are mathematical functions that relate the angles of a triangle to the lengths of its sides. In the video, the focus is on the six basic trig functions: sine, cosine, tangent, and their reciprocals, cosecant, secant, and cotangent. These functions are essential for solving problems involving right triangles and are central to the video's theme of explaining how to find the values of these functions.

💡SOH-CAH-TOA

SOH-CAH-TOA is a mnemonic used to remember the definitions of the sine, cosine, and tangent functions in trigonometry. The video uses this mnemonic to help viewers remember that sine is 'opposite over hypotenuse', cosine is 'adjacent over hypotenuse', and tangent is 'opposite over adjacent'. This memory trick is crucial for understanding how to calculate these values in a right triangle, as demonstrated in the script.

💡Hypotenuse

The hypotenuse is the longest side of a right triangle, opposite the right angle. In the video, the hypotenuse is identified as the key reference point for calculating sine, cosine, and tangent. The script uses the hypotenuse in examples to show how to find the values of these trig functions, emphasizing its importance in trigonometry.

💡Adjacent side

The adjacent side is the side of a right triangle that is next to the angle being considered but is not the hypotenuse. In the video, the adjacent side is used in the definition of cosine and tangent, as well as in the calculation of secant and cotangent. The script explains its role in relation to the angle theta, highlighting its importance in trigonometric calculations.

💡Opposite side

The opposite side is the side of a right triangle that is directly across from the angle being considered. The video script uses the opposite side in the definitions of sine and tangent, and also in the calculation of cosecant and cotangent. It is a fundamental part of understanding how to apply trigonometric functions to right triangles.

💡Reciprocal

The reciprocal of a number is 1 divided by that number. In the context of trigonometry, the video explains that the reciprocals of the primary trig functions sine, cosine, and tangent give the values of the secondary trig functions cosecant, secant, and cotangent. The script demonstrates how to find these reciprocals by flipping the numerator and denominator of the primary function's ratio.

💡Right triangle

A right triangle is a triangle with one 90-degree angle. The video script focuses on right triangles as the basis for understanding trigonometric functions. The script uses right triangles in its examples to illustrate how to apply the SOH-CAH-TOA mnemonic and calculate the values of the trig functions.

💡Theta (θ)

Theta, represented by the symbol θ, is the angle in a right triangle that is not the right angle. The video script consistently refers to theta when explaining how to find the values of trig functions. It is essential to include theta in the notation when writing the answers to ensure clarity and correctness in trigonometric calculations.

💡Cosecant (csc)

Cosecant is the reciprocal of the sine function, defined as the hypotenuse over the opposite side. In the video, the script shows how to find the value of cosecant by first calculating the sine of an angle and then taking its reciprocal. This is part of the broader explanation of how to find the values of all six basic trigonometric functions.

💡Secant (sec)

Secant is the reciprocal of the cosine function, calculated as the hypotenuse over the adjacent side. The video script explains how to find secant by first determining the cosine of an angle and then taking its reciprocal. This concept is used to illustrate the relationship between the primary and secondary trigonometric functions.

💡Cotangent (cot)

Cotangent is the reciprocal of the tangent function, defined as the adjacent side over the opposite side. The video script demonstrates how to find cotangent by first calculating the tangent of an angle and then taking its reciprocal. Cotangent, along with cosecant and secant, is one of the secondary trigonometric functions explained in the video.

Highlights

Introduction to basic trigonometric functions: sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent.

Memory trick SOH-CAH-TOA for remembering the ratios of sine, cosine, and tangent.

Explanation of the importance of right triangles in trigonometry.

Identification of the longest side as the hypotenuse in a right triangle.

Definition of the adjacent side as the one next to the angle theta but not the hypotenuse.

Definition of the opposite side as the one directly opposite angle theta.

How to calculate sin(theta) using the opposite side over the hypotenuse.

Calculation of cos(theta) using the adjacent side over the hypotenuse.

Finding tan(theta) by dividing the opposite side by the adjacent side.

Method for finding the reciprocal trigonometric functions: csc, sec, and cot.

Example of finding csc(theta) by taking the reciprocal of sin(theta).

Process for calculating sec(theta) by taking the reciprocal of cos(theta).

How to determine cot(theta) by taking the reciprocal of tan(theta).

Handling different orientations of right triangles in trigonometry problems.

Labeling the sides of a right triangle when the hypotenuse is horizontal.

The importance of including the angle notation when writing trigonometric function values.

Encouragement for viewers to like and subscribe for more educational content.

Transcripts

Hi guys! I'm Nancy.

And I'm going to show you how to use the basic trig functions

and how to find the values of the

six basic trigonometric functions.

So that is: sine, cosine, tangent

And their partners: cosecant, secant and cotangent.

So first we're going to look at just sin, cos and tan.

And we're going to use the memory trick SOH-CAH-TOA

which a lot of people say quickly as "sohcahtoa".

But this is what it is, it's a memory trick to remember

what sin, cos and tan mean. So we'll use that.

All of these 6 trig functions involve right triangles.

That is, a triangle that has a right angle in the corner.

Which is noted by the corner symbol down here.

So it's 90 degrees. A 'right angle'.

And you will also know where theta is.

It will be labeled on your triangle.

Theta is just an angle.

It's an angle that is not your right angle.

What's important to know is what each of the sides of the triangle are

in relation to theta. What I mean by that is...

The longest side on the triangle is called the 'hypotenuse'.

So you'll just want to remember that the longest side

is the hypotenuse.

The other side, that is next to theta, that is not the hypotenuse

is called the 'adjacent' side.

It's adjacent to theta, but it's not the hypotenuse.

And the other side, the third side, is opposite theta.

Directly opposite theta. And it's the 'opposite' side.

So notice that the hypotenuse, is immediately opposite the right angle.

The 'opposite side' is opposite theta.

And then the 'adjacent side' is the other side

next to theta that is not the hypotenuse.

So you will use those to evaluate sin, cos and tan.

So you might be working on a problem that looks something like this

where you're given a triangle. A right triangle.

With a right angle.

Some theta labeled on your diagram.

and then side lengths that are given to you.

So for instance you might be given

that the longest side, the diagonal, is 5.

This side length could be 3...

and this side 4.

And then the question might be:

Find sin of theta, cos of theta and tan of theta.

So let's find those.

sin of theta

sin(theta) is equal to: opposite/hypotenuse

So this memory trick, SOH-CAH-TOA,

can help you know what values to use for sin, cos and tan.

For Sin it's going to be Opposite / Hypotenuse. It's the 'SOH' part.

So sin equals opposite / hypotenuse.

So I'll write that out: Opposite over hypotenuse.

And in this particular problem, the opposite side is 4.

This is directly opposite theta. 4.

And the hypotenuse, the longest side, is 5. So put 5.

So your answer for sin(theta) is 4/5.

So in the same way you can find cos using your memory trick.

The acronym SOC-CAH-TOA. So cos is the 'CAH' part of this trick.

C-A-H. And that stands for Cos equals Adjacent / Hypotenuse.

cos = adjacent / hypotenuse

And so in this problem, the adjacent side to theta is 3.

And the hypotenuse is again 5.

So your cos value is 3/5.

And then finally if you have to find tan(theta)

just remember this part of the name SOH-CAH-TOA.

The TOA part.

T-O-A. So Tangent equals Opposite / Adjacent.

The opposite side in this triangle is 4

and the adjacent side is 3. So you have 4/3.

So your tan(theta) = 4/3.

OK. Say you actually need to find the values of csc, sec, or cot.

One of the other 3 basic trig functions.

The easiest way is to first find the value of their partner trig functions.

sin, cos or tan.

And then take the reciprocal. 1/sin. 1/cos. 1/tan.

So let me show you an example.

Say you wanted to find cosecant of theta. csc(theta).

First, find its partner, sin(theta).

And in this example sin(theta), remember is opposite/hypotenuse.

From SOH-CAH-TOA. sin is opposite/hypotenuse.

So in this triangle it is 4/5.

csc(theta) is by definition: 1/sin(theta)

And what that really means is that you can just

flip the numerator and denominator of sin. Flip top and bottom.

So instead of 4/5, it's 5/4 for csc(theta).

So your csc(theta) = 5/4.

Now if you wanted to find secant of theta, sec(theta),

you would first find cos(theta).

In this triangle, remember cos is adjacent/hypotenuse,

so here it's 3/5.

Then, to get secant theta, sec(theta),

you would take 1/cos(theta).

And that can be found just by flipping top and bottom.

And so instead of 3/5, 1 over that would turn out to be 5/3.

The reciprocal.

So we have sec(theta) is 5/3.

And then finally if you wanted to find cot(theta)

you would first find tan(theta).

So tan(theta) is opposite/adjacent.

So in this triangle it's 4/3

and then to get cot

you would do 1/tan.

Which is the reciprocal of this value. So it's 3/4.

So you get: cot(theta) = 3/4.

That is the fastest, easiest way to find csc, sec and cot.

OK. I want to show you some things that can trip people up.

One thing that confuses people is when

the right triangle is drawn at a different orientation.

So if it's sideways or on its hypotenuse.

So for instance you could have a triangle

that's drawn like this.

Still a right triangle, but looks kinda upside down.

Your theta could be here.

You could have a triangle that looks like...

this.

So it's the reverse of the one in our example.

You could have...

a triangle drawn...

with your theta up here. Instead of down here.

And the one that's most confusing to people

is when the right triangle is drawn

so that its hypotenuse is horizontal.

It's sort of on the ground.

So here's an example of that.

If your right angle was actually up here

then your hypotenuse turned out to be horizontal

and your theta could be here or here.

So let's look at these examples

and label where opposite, adjacent and hypotenuse are

for these triangles.

In this example, in all of these triangles actually,

you can rely on your hypotenuse being the longest side.

So in this triangle

you can still tell that this is the longest side

so it would be your hypotenuse.

The other side that's next to theta

is still your adjacent side.

So this would be your adjacent.

And then, the other side that is opposite theta

directly across from theta is opposite side.

Let's label these are well.

Across from theta is your opposite side.

longest side is your hypotenuse

and the other side next to theta is your adjacent.

Over here, your adjacent is this side.

Your opposite is down here.

And your hypotenuse is the longest side.

OK. Down here. Longest side is still your hypotenuse.

The one that's opposite the right angle.

The one opposite theta is your opposite side.

And then the other side next to theta is your adjacent.

There's one more thing that trips people up in these problems

when your finding the value of trig functions.

When you write your final answer

make sure you put theta.

You don't want to write something like this

that your "sin = 3/5".

You want to write sin of an angle.

sin(theta) = a number.

If you don't write the angle, this doesn't have any meaning.

I know a lot of people forget it, I get it. I know.

It's a lot of remember and these trig functions

are all new and weird, but

if you don't write the angle it doesn't really make sense.

It doesn't really have any meaning.

So make sure you write your full answer as

sin(of an angle), cos(theta), tan(theta) or tan(x).

Whatever your angle is called.

I hope that helped you understand

sines, cosines, tangents,

cosecant, secant, cotangents!

I know trig functions are super fun.

It's OK.

You don't have to like math, but...

but you can like my video!

So if this helped you, please click like or subscribe below!