A Problem to Review SOH CAH TOA and the Pythagorean Theorem for use in Physics

Summary

In this educational video, Mr. P reviews essential physics concepts with his students Bo, Billy, and Bobby. They discuss the significance of SOH CAH TOA and the Pythagorean Theorem in solving right triangle problems. The class delves into the correct application of these formulas, emphasizing the importance of including the angle's Greek letter 'theta' to avoid errors. Through a step-by-step problem-solving approach, they calculate the hypotenuse and side lengths, and determine angles using trigonometric ratios. The lesson underscores the necessity of precision, correct calculator settings, and the proper use of significant figures in scientific calculations.

Takeaways

• 😀 The class begins with a review of SOH CAH TOA and the Pythagorean Theorem, emphasizing their importance in physics.
• 📐 Mr. P introduces a right triangle with sides labeled X, Y, and the hypotenuse H, and angles labeled theta-1 and theta-2.
• 🔢 The students learn that the sine function relates the opposite side to the hypotenuse in a right triangle, and cosine relates the adjacent side to the hypotenuse.
• 📚 Mr. P corrects a common mistake among students by emphasizing that the variable 'theta' should not be omitted from SOH CAH TOA equations.
• 📈 The class demonstrates how to calculate the hypotenuse using the cosine function and the given angle and side.
• 📉 The students are reminded to use the correct mode on their calculators to match the units of the given angles (degrees).
• 🔍 The Pythagorean Theorem is applied to find the missing side of the triangle, with a focus on using unrounded numbers to maintain accuracy.
• 📏 The tangent function is used to find the second angle of the triangle, with a step-by-step approach to ensure accuracy.
• 👨‍🏫 Mr. P uses humor and analogies to stress the importance of following each step in calculations to avoid errors.
• 🔗 The lesson concludes with a reminder that SOH CAH TOA and the Pythagorean Theorem are only applicable to right triangles.

Q & A

What is the main topic of the class discussion in the script?

The main topic of the class discussion is the review of SOH CAH TOA and the Pythagorean Theorem in the context of physics, specifically using them to solve problems involving right triangles.

What does the acronym 'SOH CAH TOA' stand for and why is it important?

SOH CAH TOA stands for 'Sine Opposite Hypotenuse, Cosine Adjacent Hypotenuse, Tangent Opposite Adjacent'. It is important because it is a mnemonic used to remember the definitions of sine, cosine, and tangent in trigonometry, which are fundamental to solving right triangle problems in physics.

What is the significance of the Greek letter 'theta' in the context of the script?

In the script, the Greek letter 'theta' (θ) is used to denote angles in right triangles. It is significant because it helps to specify which angle is being referred to when using trigonometric ratios, ensuring the correct sides of the triangle are used in calculations.

Why does Mr. P emphasize the inclusion of 'theta' in the SOH CAH TOA mnemonic?

Mr. P emphasizes the inclusion of 'theta' to ensure students do not skip steps and to avoid mistakes when solving problems. It is a reminder that the trigonometric functions are functions of an angle, and thus the angle must be specified in the equations.

What is the correct equation for 'SOH' according to the script?

The correct equation for 'SOH' is 'sine theta equals opposite over hypotenuse', which is written as sin(θ) = opposite/hypotenuse.

How does Mr. P correct the students' misunderstanding of the SOH CAH TOA mnemonic?

Mr. P corrects the students by emphasizing that the mnemonic should include 'theta' to represent the angle, and he explains that omitting 'theta' can lead to mistakes in problem-solving.

What is the role of the Pythagorean Theorem in the script?

In the script, the Pythagorean Theorem is used to find the missing sides of a right triangle, specifically to calculate the length of side Y after finding the hypotenuse using trigonometric functions.

Why does Mr. P stress the importance of showing work in solving the triangle problem?

Mr. P stresses the importance of showing work to ensure that students follow a logical step-by-step process, which helps to avoid skipping steps and makes it easier to identify and correct errors.

What mistake does Bobby make when calculating the hypotenuse, and how does Mr. P correct it?

Bobby initially calculates the hypotenuse as 'Negative 354', which is incorrect. Mr. P corrects this by reminding him to ensure his calculator is in degree mode and to use the correct formula, resulting in the correct answer of 5.6 meters.

How does Mr. P handle the situation when Billy tries to skip steps in solving for theta-2?

When Billy tries to skip steps, Mr. P interrupts and emphasizes the importance of taking every step, using a metaphor about falling down stairs to illustrate the point. He then guides Billy to correctly substitute variables and numbers into the equation before solving.

Outlines

00:00

📚 Introduction to SOH CAH TOA and the Pythagorean Theorem

The script opens with a classroom scene where Mr. P is reviewing essential trigonometric and geometric concepts with his students. The focus is on the SOH CAH TOA mnemonic for solving right triangles and the Pythagorean Theorem. Mr. P emphasizes the importance of including the angle theta in trigonometric ratios to avoid mistakes. The class discusses the meaning of 'opposite' and 'adjacent' in relation to the angle theta and the hypotenuse of a right triangle. The dialogue is interspersed with humor and light-hearted moments to keep the students engaged.

05:01

🔍 Applying Trigonometry to Solve a Right Triangle

In this segment, the students apply the concepts of cosine and the Pythagorean Theorem to solve a specific right triangle problem. Bo starts by setting up the cosine equation for theta-1, and after a reminder from Mr. P to show work, the class proceeds to solve for the hypotenuse using the cosine ratio. The students then use the Pythagorean Theorem to find the length of side Y. Mr. P stresses the importance of using unrounded numbers to maintain accuracy and significant figures. The process highlights the correct mathematical steps and the common pitfalls to avoid, such as skipping steps or using incorrect modes on the calculator.

10:03

📐 Final Calculations and Wrapping Up the Lesson

The final part of the script involves finding the second angle, theta-2, using the tangent ratio. Billy correctly sets up the equation, and the class discusses the importance of subscripts in trigonometric problems. The solution process includes taking the inverse tangent to find the angle measure. Mr. P uses this opportunity to reiterate the importance of following each step carefully and not skipping any part of the process. The lesson concludes with a reminder to use the right triangle properties and the Pythagorean Theorem appropriately. The script ends with a note about the availability of lecture notes and a light-hearted reminder to enjoy them responsibly.

Keywords

💡SOH CAH TOA

SOH CAH TOA is a mnemonic used to remember the primary trigonometric ratios for right triangles: Sine, Cosine, and Tangent. In the video, it is used to teach students how to relate the angles of a right triangle to the lengths of its sides. The mnemonic stands for Sine Opposite over Hypotenuse, Cosine Adjacent over Hypotenuse, and Tangent Opposite over Adjacent. The video emphasizes the importance of including the angle (theta) in these ratios to avoid mistakes, as students often mistakenly omit it, leading to errors in problem-solving.

💡Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that states in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. The theorem is crucial in the video for solving problems involving right triangles and is used to find the length of the sides of the triangle once one side and an angle are known.

💡Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. In the video, the concept is central as it discusses various properties and methods for solving right triangles using trigonometric ratios and the Pythagorean Theorem. The video uses a right triangle to demonstrate how to apply SOH CAH TOA and the Pythagorean Theorem.

💡Hypotenuse

The hypotenuse is the longest side of a right triangle, which is opposite the right angle. In the video, the hypotenuse is symbolized as 'H' and is used in the Pythagorean Theorem to solve for unknown sides. The script emphasizes that the hypotenuse is always opposite the right angle, which is a critical distinction when applying trigonometric functions.

💡Trigonometric Ratios

Trigonometric ratios, specifically sine, cosine, and tangent, are the relationships between the angles and sides of a right triangle. These ratios are essential in the video for solving triangles and are encapsulated by the mnemonic SOH CAH TOA. Each ratio is defined in relation to an angle (theta) and the sides of a triangle, and the video demonstrates how to apply these ratios to find unknowns in a triangle.

💡Significant Figures

Significant figures refer to the number of meaningful digits in a number, which is important for maintaining accuracy in scientific calculations. The video script mentions the need to round the hypotenuse to two significant figures, illustrating the importance of precision in calculations and the impact of rounding on the final result.

💡Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin, arccos, and arctan, are used to find an angle when the ratio of the sides of a right triangle is known. In the video, the inverse tangent function is used to solve for an angle (theta-2) when the ratio of the opposite side to the adjacent side is given. This function is crucial for finding angles in right triangles when sides are known.

💡Radian and Degree Mode

Radian and degree are two different units for measuring angles. The video script highlights the importance of ensuring that the calculator is set to the correct mode (degree) when working with angles measured in degrees, as using the wrong mode can lead to incorrect calculations.

💡Adjacent and Opposite Sides

Adjacent and opposite sides are relative terms used in trigonometry to describe the sides of a right triangle in relation to a given angle. The video explains that the adjacent side is next to the angle, and the opposite side is directly across from it. Correctly identifying these sides is crucial for applying trigonometric ratios accurately.

💡Error Checking

Error checking is the process of verifying the accuracy of calculations and ensuring that steps are not skipped in mathematical procedures. The video script emphasizes the importance of error checking by showing the consequences of using rounded numbers too early in calculations and the necessity of following each step carefully to avoid mistakes.

Highlights

Introduction to the importance of reviewing SOH CAH TOA and the Pythagorean Theorem in physics.

Definition of a right triangle and its significance in physics.

Explanation of the Greek letter theta and its use in physics for angles.

Clarification of the correct usage of SOH CAH TOA, emphasizing the inclusion of theta in the equations.

The significance of using the correct sides as opposite and adjacent to the angle in question.

Instruction on how to use cosine theta to find the hypotenuse of a right triangle.

Emphasis on showing work and not skipping steps in problem-solving.

Use of the Pythagorean Theorem to find the length of the side Y in a right triangle.

The importance of using unrounded numbers in calculations to maintain accuracy.

How to find the angle theta-2 using the tangent function.

Advice on using the answer button on calculators for more precise results.

The necessity of having a right triangle to apply SOH CAH TOA and the Pythagorean Theorem.

Mr. P's humorous analogy of skipping steps being like falling down stairs to emphasize the importance of following steps in calculations.

The concept that the hypotenuse is always opposite the right angle in a right triangle.

The correct method to solve for the hypotenuse using cosine and the importance of setting up the equation properly.

The correct approach to solving for an angle using the tangent function and the inverse tangent.

Final summary and reminder of the key learnings from the lecture.

Transcripts

Bo: Hey, guys.

Billy: Hey, Bo!

Bobby: Hi, Bo.

♫ (lyrics) Flipping Physics ♫

Mr. P: Ladies and gentlepeople.

The bell has rung, therefore class has begun.

Therefore, you should be seated in your seat,

ready and excited to review SOH CAH TOA,

and the Pythagorean Theorem.

Billy: Oh, boy!

Bo: Shouldn't we already know those.

Mr. P: Bo, that is certainly true.

Before taking this algebra-based physics class,

you should have already learned this,

however, it is such an important piece of physics

that I feel it necessary to review.

Bo: Okay. That makes sense.

Mr. P: Let's start with a triangle.

Billy, what kind of triangle is this?

Billy: A right triangle.

Mr. P: Bobby, what is a right triangle?

Bobby: A triangle with a right angle?

Mr. P: Does that mean that there are triangles

with wrong angles in them?

(Bo laughs)

Bobby: I suppose not.

A right angle is a 90-degree angle,

so a right triangle is a triangle that has

a 90-degree angle in it.

Mr. P: Good.

This symbol is the symbol for a right angle,

or a 90-degree angle, and that makes this

a right triangle, or a triangle that has a 90-degree angle in it.

Let's identify some parts of this triangle.

Let's call this side X, which has a distance of 4.7 meters,

this side of the triangle, Y,

this angle of the triangle is going to be theta-1,

and that has a measurement of 33 degrees.

This is symbol theta, it's a Greek letter.

It's a Greek letter theta,

and it's commonly used for angles in physics.

Let's call this side H for hypotenuse,

or the side opposite the right angle,

and let's identify this angle as theta-2.

Our goal is to determine the values for Y,

the hypotenuse, and theta-2.

Billy, how would you like to begin?

Billy: The interior angles of any triangle

add up to 180 degrees,

so we just need to ...

Mr. P: Hold up.

Let me interrupt you for a minute, Billy.

You are welcome to do it that way.

In fact, it would be easier to do it that way.

However, right now, I'm trying to review SOH CAH TOA,

and the Pythagorean Theorem,

and what you are doing actually obviates that.

On a quiz or test, you should certainly solve it that way.

However, not right now.

Let's actually start by just defining SOH.

Bo, what does 'SOH' mean?

Bo: SOH means sine opposite over hypotenuse.

Mr. P: Like that?

Bo: No, sine equals opposite over hypotenuse.

Mr. P: Right.

How many of you think SOH means

sine equals opposite over hypotenuse.

Great.

This is one of the reasons we have to review this concept

because you are all wrong.

Like most students, you are skipping steps,

just plain old dropping out useful information.

It's not there in the equation.

You're just leaving it out.

Bobby: But that's how I've always done it.

Bo: Yeah.

Mr. P: You have left out a very important

part of the equation.

SOH means sine theta equals opposite over hypotenuse.

Billy: Pish tosh, it's the same thing!

Bo: Pish tosh.

Mr. P: It is absolutely not the same thing.

I've seen it time and time again

that students who leave out the theta

make mistakes when solving problems like this one.

I don't know why you all eschew theta,

however, you just can't,

so please remember to write out the whole equation.

Bobby, what does 'CAH' mean?

Bobby: Adjacent over hypotenuse.

Cosine theta equals adjacent over hypotenuse.

(Billy and Bo laugh)

Mr. P: And Billy, 'TOA'?

Billy: Tangent theta equals opposite over adjacent.

Mr. P: Great.

Bo, what does opposite mean?

Bo: Huh?

Mr. P: What does opposite mean?

Bo: It means opposite.

Mr. P: Opposite what?

Bo: Oppos- ...

Oh, opposite to the theta, to the angle.

Mr. P: Right, opposite to the angle, or your theta.

Notice that the opposite and adjacent

refer to the sides that are opposite and adjacent

to the angle you are referring to.

Be very careful to always check to make sure

you're using the correct sides as opposites, etcetera,

because it does depend on which angle you are using.

The hypotenuse is always opposite the right angle,

so, again, this side here is opposite of this angle,

and this side is adjacent to this angle.

However, if we're talking about this angle or this theta,

this side is adjacent,

and this side would be opposite that theta,

so please be very careful.

Actually, I think we were doing a problem.

Let's get back to that original problem.

Bo, you were working on this.

How would you like to begin?

Bo: (sighs) Well, you know cosine of theta

equals 4.7 over the hypotenuse.

Mr. P: Another basic tenet of how I teach

is that you have to show your work.

This means that you must start with an equation

and variables.

You can't skip steps.

You can't start with numbers.

You must start with variables.

Please try again.

Bo: Okay, we know cosine theta

equals adjacent over hypotenuse.

Mr. P: Yes.

Bo: And we can use theta-1,

that means that our equation would be cosine of theta-1

equals X over the hypotenuse,

now we know theta-1,

we know the values for theta-1 and X,

so can we plug in our values now?

Mr. P: Yes, now we have written out the equation

and substituted in our variables,

we can substitute in our numbers.

Bo: That means that cosine of 33 degrees

equals 4.7 over the hypotenuse.

Mr. P: Billy, how do we solve this equation for the hypotenuse?

Billy: Multiply both sides by the hypotenuse

and divide both sides by the cosine of 33 degrees.

Mr. P: Absolutely.

Notice how the hypotenuse cancels out,

and then cosine of 33 cancels out,

and we end up with the hypotenuse

equals 4.7 divided by the cosine of 33.

Bobby, what do we get for an answer?

Bobby: Negative 354?

Mr. P: Be careful,

you need to make sure your calculator is in degree mode.

Your calculator is in radian mode,

so please be careful of that

because our angles are in degrees,

we need our calculators in degree mode.

Bobby: Oh, dangit!

It's 5.6041,

and because the least number of significant figures

from our givens was 2,

we need to round to 2 sig figs, or 6.5, no 5.6 meters.

Mr. P: Very nice.

Now we need to find either Y or theta-2.

Bobby?

Bobby: Can't we just use the Pythagorean Theorem now?

Mr. P: Yes, how do we know we can use

the Pythagorean Theorem?

Bobby: Because it's a right triangle.

Mr. P: Yes.

A² plus B² equals C².

The Pythagorean Theorem,

and please notice that the C is always opposite

the right angle,

because it represents the hypotenuse.

Bobby: Okay, A² plus B² equals C².

So X² plus Y² equals the hypotenuse²,

and to solve for Y, we subtract X² from both sides

and then take the square root of the whole equation.

That gives us Y equals the square root

of the quantity H² minus X².

Plugging in our numbers,

we get Y equals the square root of the quantity

5.6² minus 4.7².

That works out to be 3.0447,

which rounds to 3.0 meters with two sig figs.

Mr. P: We made a mistake.

Bobby: Dangit, again!

I used the rounded answer to solve the problem.

We need to use the unrounded number.

We need to use 5.6041 instead of 5.6.

that works out to be 3.0522, which rounds to 3.1 meters

with two sig figs and not 3.0.

Mr. P: Yes. Please be careful not to use rounded numbers

when solving a problem.

This is why I always write out my unrounded answer first

and then I round to get my answer.

That way, I have the unrounded answer if I need it.

Now, we have found Y and the hypotenuse.

Billy, could you please help us find theta-2?

Billy: Tangent theta equals oppostie over adjacent,

so tangent of theta-2 equals 4.7 over-

Mr. P: Ah!

Mr. P: Ah!

Billy: 3.0522 ...

Mr. P: Ah!

Ahh!

Steps.

You need to make sure

that you take every single step.

If you do not take every step,

if you start skipping steps,

you are going to fall down the stairs and get hurt.

There's going to be blood on the stairs

and I will have to call the janitor

so that he can get his PPEs to take care of your BBPs.

PPEs to take care of your BBPs.

Personal protective equipment to take care of your

blood borne pathogens.

Don't skip steps.

You are going to get angry with me.

I know this.

You're going to be like,

"But there's a handrail, Mr. P!

"If I hold onto the handrail, can't I just skip one teensy step?"

No.

You started with an equation, which is great.

However, you need to now plug in the variables,

not the numbers.

Class, if you just plug in the numbers,

what would you be doing?

Billy: Skipping a step.

Bobby: Falling down the stairs?

Bo: Using the handrail?

Mr. P: Write out the equation.

Substitute in the appropriate variables,

then substitute in the numbers.

Billy, we're back on you.

Billy: All right, we already have the equation written down,

so now I guess we substitue in variables.

Tangent theta-2 equals X over Y.

Mr. P: Great!

Notice how important that subscript of 2 on the theta is.

If the subscript were a 1 instead,

the equation would have Y over X

instead of X over Y.

Billy: Now we can substitute in numbers

and get 4.7 over 3.0522.

Mr. P: Bobby, how do we solve for theta-2 in this equation?

Bobby: I remember.

Take the inverse tangent of the whole equation,

both sides of the equation.

Mr. P: Yes, Bobby.

Let's walk through what that looks like on the board.

On the left-hand side of the equation,

you get the inverse tangent of tangent of theta-2.

Bo, what does the inverse tangent

of the tangent of theta-2 actually work out to be?

Bo: Oh yeah, the inverse tangent of

the tangent of theta-2 just works out to be theta-2.

Mr. P: Right, theta-2, and on the right-hand side,

we get the inverse tangent of 4.7 over 3.0522.

Bo, what then does the answer work out to be?

Bo: Wow.

It worked out to be exactly 57 degrees.

So with two significant digits,

theta-2 equals 57 degrees.

Billy: Mr. P, why didn't I get exactly 57 degrees?

Mr. P: Bo must have used the answer button

on his calculator rather than retyping the number in

with fewer significant digits.

I always recommend using the answer button

because you get a more precise answer.

Okay, please remember that you must have a right triangle

in order to use SOH CAH TOA

and the Pythagorean Theorem.

Thank you very much for learning with me today,

I enjoyed learning with you.

Voiceover: Lecture notes

are available at FlippingPhysics.com.

Please enjoy lecture notes responsibly.