Trigonometry
Transcript
Now we're gonna have two lessons on trigonometry, just to review these mathematical basics. If you're a pro on trigonometry already, you don't need to watch these lessons. In this one, we're just gonna review the elements of trigonometry. So we're gonna start at the very beginning, but go quickly here. So as you probably understand, all right triangles, with say, a 41 degree angle are similar.
There's a rule in geometry, angle angle similarity. And so if we know there's a right angle, and we know there's a 41 degree angle, then all triangles, regardless of whether they're big or small, are similar to each other. That means the ratios are the same. Thus the fixed ratios will be the same in every right triangle with a 41-degree angle.
So let's just look at this. Here is a representative 41 degree right triangle with all three sides labeled. And so, we have the side hypotenuse, opposite and adjacent. We need to give names to the three important ratios. Now this should be very familiar. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
Notice we're gonna be very careful. If we're just writing the word, if we just mean sine, cosine or tangent in general, we're gonna write out the word. Whenever we write the abbreviation sin, cos, or tan, we always write that in parentheses with an angle. These are what are known as functions in mathematics.
They are always operating on a particular angle, so it's very important to keep track of that. Now as you may remember, you can remember all of those ratios just with this one mnemonic SOHCAHTOA. This is probably very familiar to you, again, brief review, SOHCAHTOA. Sine means, sine is opposite over hypotenuse.
Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. That is what SOHCATOA means. For a general angle, mathematicians typically use the Greek letter theta, so you'll be seeing quite a few thetas in these lessons. We can use this to make general statements true for any angle.
So for example, we can rewrite the ratios in terms of theta, these are the general definitions for any angle. And again, SOHCAHTOA is what summarizes these relationships. So that is the very basic of sine, cosine and tangent. And all of these are true for acute angles. They're all true when the angle is less than 90 degrees.
The next lesson, we'll talk about what happens when the angles are more than 90 degrees, but right now we're just gonna focus on right triangle trigonometry. Now all that is the basics, but we have to focus on two special triangles, the 45-45-90 triangle and the 30-60-90 triangle. And the reason we focus on this is because all the ratios in these two triangles can be deduced from elementary geometry.
If we have an odd angle, say a 41 degree angle, you'd need a calculator to know sine and cosine of 41 degrees. And of course you don't get a calculator on the MCAT. But with 45-45-90 triangles and 30-60-90 triangles, those are sines and cosines that you should be able to figure out without a calculator. And the MCAT absolutely loves giving you problems that have 30 degrees, a force at 30 degrees, a velocity at 45 degrees, this sort of thing, so that you have to rely on these particular ratios.
So you're gonna be seeing these ratios show up quite a bit in these lessons. First of all let's talk about the 45-45-90 triangle. And so this is it, and so it has two equal legs, so in other words, it's an isosceles triangle. And then the hypotenuse, square root of 2, we can get that from the Pythagorean theorem.
So sometimes this is known as the 1-1-root 2 triangle. It's also sometimes known as the isosceles right triangle. The only way that a right triangle could possibly be isosceles is if it's a 45-45-90 triangle. So we can deduce some ratios from this. We know all the proportions.
We can just use SOHCAHTOA to find the values of the functions for 45 degrees. So, the sine of 45, opposite over hypotenuse. This is 1 over hypotenuse, root 2, and typically in mathematics, as you may remember from math, we typically try to avoid fractions that have a radical in the denominator, and so we do something call rationalize. We just multiply both the numerator and the denominator by that radical.
So we're gonna multiply by root 2 over root 2, and what we get is root 2 over 2. Now that should be a familiar looking number, that is the sine of 45 degrees. As it happens, it's also the cosine of 45 degrees cuz the cosine, the opposite and adjacent are equal in value. They're both equal to 1, so this is also gonna be root 2 over 2. And then of course the tangent, opposite over adjacent, that's gonna be 1 over 1, which is 1.
And so those are three values you need to know. I'll just point out, notice that if the angle is less than 45 degrees, you get a tangent less than 1, and if it's greater than 45 degrees, you got a tangent greater than 1, and if it's at 45 degrees, the tangent is exactly 1. So very important, this relationship between the opposite and the adjacent and what that tells you about the angle.
So now we'll talk about the 30-60-90 triangle and the way really to understand this, the way to think about this, is start with an equilateral triangle. Suppose we have an equilateral triangle. All the sides on this equilateral triangle are 2, so there's three sides of length 2, and then we cut it down the middle. Well if we cut it down the middle, obviously we have this, we still have this original side of the equilateral triangle that has a length of 2.
This side got cut in half so it's a length of 1. And then we can use the Pythagorean theorem to find the length of this height, and that turns out to be root 3. And so if you visualize the equilateral triangle that is from which the 30-60-90 triangle came, that will help you remember all of the ratios in it.
And so sometimes this is known as the 1-root 3-2 triangle. That's one way to think about it, or 1-2-root 3, however you want to say it, but remember that 2 is the hypotenuse. Also sometimes it's useful to think about this in terms of a half equilateral triangle. That's really what a 30-60-90 triangle is.
It is an equilateral triangle cut in half, and it's very important to remember that. And so first of all, we can use SOHCAHTOA to find ratios for both 30 degrees and 60 degrees. We'll do 30 degrees first. So the sine of 30 degrees, starting from that 30 degree angle, the opposite is 1 and the hypotenuse is 2.
So this is just 1 over 2, one-half, sine of 30 is one-half. The cosine, well the adjacent is root 3 and the hypotenuse is 2, so cosine of 30 is root 3 over 2. Again these are numbers that should look very familiar from trigonometry. The tangent, opposite over adjacent, that's gonna be 1 over root 3. Again, a radical in the denominator, multipy by root 3 over 3.
And what we get is root 3 over 3. The tangent of 30 degrees, that's a number less than 1. So notice that, because we have an angle less than 45 degrees, the cosine is bigger than the sine and the tangent is less than 1. So, now let's look at 60 degree angles. Here, when we switch over to the other side of the triangle, from the point of view of the 60 degree angle, the opposite is the root 3 side.
And the adjacent, the hypotenuse of course is still the 2, so we get root 3 over 2. Now, the adjacent is 1, so the cosine of 60 degrees is going to be one-half. And if we do the tangent, that's going to be opposite over adjacent, that's root 3 over 1, which of course is just root 3. So notice that now we're dealing with an angle greater than 45 degrees. So now it's true that the sine is greater than the cosine, and it's true that the tangent of 60 is greater than 1.
It's important to practice these ratios so that you're familiar with them. The sine, cosine and tangent of 30, 60 and 90 because, again, these could show up anywhere on the physics problems on the MCAT. Some handy decimal approximations, it doesn't hurt to learn these. Square root of 2 is approximately 1.4, so it means root 2 over 2 is approximately 0.7.
Square root of 3 is approximately 1.7, square root of 3 over 2 is approximately 0.866, and you can really approximate that by 13 over 15 if that helps you. And the square root of 3 over 3 is approximately 0.577. So again very handy to have some approximations on hand because, of course, you won't have a calculator.
Here's a practice problem, of course, this is not very MCAT-like. But it's a practice problem, just to make sure that we're comfortable with these ratios. Pause the video, and then we'll talk about this. Okay, so we have a rectangle that's 10 by 4. We have a 30 degree angle over here and we have a 45 degree angle here.
And we want the length of the base. Well certainly the length from B to E, that's just 10, that's easy. Over on the right side, well we know DE equals EF, because again that's an isosceles right triangle, a 45-45 triangle. So if DE equals 4, then EF equals 4 also. And so the length from B to F is just 14.
It's tricky on the left side because there we have a 30-60-90 triangle. And so we use the ratios there, the ratio of AB to BC is square root of 3. And so that means that AB has to be the square root of 3 bigger than BC, and so if BC is 4, AB is 4 times root 3. And so when we put all this together, we get 14 + 4 root 3.
And that is exactly what we get in answer choice B. In summary, we talked about the basic SOHCAHTOA ratios. Those are the rules of right triangle trigonometry. We also talked about the properties for the two special right triangles, 45-45-90 and 30-60-90. And again, the MCAT will expect you to know those ratios because you won't have a calculator.
Read full transcriptThere's a rule in geometry, angle angle similarity. And so if we know there's a right angle, and we know there's a 41 degree angle, then all triangles, regardless of whether they're big or small, are similar to each other. That means the ratios are the same. Thus the fixed ratios will be the same in every right triangle with a 41-degree angle.
So let's just look at this. Here is a representative 41 degree right triangle with all three sides labeled. And so, we have the side hypotenuse, opposite and adjacent. We need to give names to the three important ratios. Now this should be very familiar. Sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent.
Notice we're gonna be very careful. If we're just writing the word, if we just mean sine, cosine or tangent in general, we're gonna write out the word. Whenever we write the abbreviation sin, cos, or tan, we always write that in parentheses with an angle. These are what are known as functions in mathematics.
They are always operating on a particular angle, so it's very important to keep track of that. Now as you may remember, you can remember all of those ratios just with this one mnemonic SOHCAHTOA. This is probably very familiar to you, again, brief review, SOHCAHTOA. Sine means, sine is opposite over hypotenuse.
Cosine is adjacent over hypotenuse. Tangent is opposite over adjacent. That is what SOHCATOA means. For a general angle, mathematicians typically use the Greek letter theta, so you'll be seeing quite a few thetas in these lessons. We can use this to make general statements true for any angle.
So for example, we can rewrite the ratios in terms of theta, these are the general definitions for any angle. And again, SOHCAHTOA is what summarizes these relationships. So that is the very basic of sine, cosine and tangent. And all of these are true for acute angles. They're all true when the angle is less than 90 degrees.
The next lesson, we'll talk about what happens when the angles are more than 90 degrees, but right now we're just gonna focus on right triangle trigonometry. Now all that is the basics, but we have to focus on two special triangles, the 45-45-90 triangle and the 30-60-90 triangle. And the reason we focus on this is because all the ratios in these two triangles can be deduced from elementary geometry.
If we have an odd angle, say a 41 degree angle, you'd need a calculator to know sine and cosine of 41 degrees. And of course you don't get a calculator on the MCAT. But with 45-45-90 triangles and 30-60-90 triangles, those are sines and cosines that you should be able to figure out without a calculator. And the MCAT absolutely loves giving you problems that have 30 degrees, a force at 30 degrees, a velocity at 45 degrees, this sort of thing, so that you have to rely on these particular ratios.
So you're gonna be seeing these ratios show up quite a bit in these lessons. First of all let's talk about the 45-45-90 triangle. And so this is it, and so it has two equal legs, so in other words, it's an isosceles triangle. And then the hypotenuse, square root of 2, we can get that from the Pythagorean theorem.
So sometimes this is known as the 1-1-root 2 triangle. It's also sometimes known as the isosceles right triangle. The only way that a right triangle could possibly be isosceles is if it's a 45-45-90 triangle. So we can deduce some ratios from this. We know all the proportions.
We can just use SOHCAHTOA to find the values of the functions for 45 degrees. So, the sine of 45, opposite over hypotenuse. This is 1 over hypotenuse, root 2, and typically in mathematics, as you may remember from math, we typically try to avoid fractions that have a radical in the denominator, and so we do something call rationalize. We just multiply both the numerator and the denominator by that radical.
So we're gonna multiply by root 2 over root 2, and what we get is root 2 over 2. Now that should be a familiar looking number, that is the sine of 45 degrees. As it happens, it's also the cosine of 45 degrees cuz the cosine, the opposite and adjacent are equal in value. They're both equal to 1, so this is also gonna be root 2 over 2. And then of course the tangent, opposite over adjacent, that's gonna be 1 over 1, which is 1.
And so those are three values you need to know. I'll just point out, notice that if the angle is less than 45 degrees, you get a tangent less than 1, and if it's greater than 45 degrees, you got a tangent greater than 1, and if it's at 45 degrees, the tangent is exactly 1. So very important, this relationship between the opposite and the adjacent and what that tells you about the angle.
So now we'll talk about the 30-60-90 triangle and the way really to understand this, the way to think about this, is start with an equilateral triangle. Suppose we have an equilateral triangle. All the sides on this equilateral triangle are 2, so there's three sides of length 2, and then we cut it down the middle. Well if we cut it down the middle, obviously we have this, we still have this original side of the equilateral triangle that has a length of 2.
This side got cut in half so it's a length of 1. And then we can use the Pythagorean theorem to find the length of this height, and that turns out to be root 3. And so if you visualize the equilateral triangle that is from which the 30-60-90 triangle came, that will help you remember all of the ratios in it.
And so sometimes this is known as the 1-root 3-2 triangle. That's one way to think about it, or 1-2-root 3, however you want to say it, but remember that 2 is the hypotenuse. Also sometimes it's useful to think about this in terms of a half equilateral triangle. That's really what a 30-60-90 triangle is.
It is an equilateral triangle cut in half, and it's very important to remember that. And so first of all, we can use SOHCAHTOA to find ratios for both 30 degrees and 60 degrees. We'll do 30 degrees first. So the sine of 30 degrees, starting from that 30 degree angle, the opposite is 1 and the hypotenuse is 2.
So this is just 1 over 2, one-half, sine of 30 is one-half. The cosine, well the adjacent is root 3 and the hypotenuse is 2, so cosine of 30 is root 3 over 2. Again these are numbers that should look very familiar from trigonometry. The tangent, opposite over adjacent, that's gonna be 1 over root 3. Again, a radical in the denominator, multipy by root 3 over 3.
And what we get is root 3 over 3. The tangent of 30 degrees, that's a number less than 1. So notice that, because we have an angle less than 45 degrees, the cosine is bigger than the sine and the tangent is less than 1. So, now let's look at 60 degree angles. Here, when we switch over to the other side of the triangle, from the point of view of the 60 degree angle, the opposite is the root 3 side.
And the adjacent, the hypotenuse of course is still the 2, so we get root 3 over 2. Now, the adjacent is 1, so the cosine of 60 degrees is going to be one-half. And if we do the tangent, that's going to be opposite over adjacent, that's root 3 over 1, which of course is just root 3. So notice that now we're dealing with an angle greater than 45 degrees. So now it's true that the sine is greater than the cosine, and it's true that the tangent of 60 is greater than 1.
It's important to practice these ratios so that you're familiar with them. The sine, cosine and tangent of 30, 60 and 90 because, again, these could show up anywhere on the physics problems on the MCAT. Some handy decimal approximations, it doesn't hurt to learn these. Square root of 2 is approximately 1.4, so it means root 2 over 2 is approximately 0.7.
Square root of 3 is approximately 1.7, square root of 3 over 2 is approximately 0.866, and you can really approximate that by 13 over 15 if that helps you. And the square root of 3 over 3 is approximately 0.577. So again very handy to have some approximations on hand because, of course, you won't have a calculator.
Here's a practice problem, of course, this is not very MCAT-like. But it's a practice problem, just to make sure that we're comfortable with these ratios. Pause the video, and then we'll talk about this. Okay, so we have a rectangle that's 10 by 4. We have a 30 degree angle over here and we have a 45 degree angle here.
And we want the length of the base. Well certainly the length from B to E, that's just 10, that's easy. Over on the right side, well we know DE equals EF, because again that's an isosceles right triangle, a 45-45 triangle. So if DE equals 4, then EF equals 4 also. And so the length from B to F is just 14.
It's tricky on the left side because there we have a 30-60-90 triangle. And so we use the ratios there, the ratio of AB to BC is square root of 3. And so that means that AB has to be the square root of 3 bigger than BC, and so if BC is 4, AB is 4 times root 3. And so when we put all this together, we get 14 + 4 root 3.
And that is exactly what we get in answer choice B. In summary, we talked about the basic SOHCAHTOA ratios. Those are the rules of right triangle trigonometry. We also talked about the properties for the two special right triangles, 45-45-90 and 30-60-90. And again, the MCAT will expect you to know those ratios because you won't have a calculator.
