Algebra (all content)
Using the Pythagorean trig identity
The Pythagorean identity tells us that no matter what the value of θ is, sin²θ+cos²θ is equal to 1. This follows from the Pythagorean theorem, which is why it's called the Pythagorean identity! We can use this identity to solve various problems. Created by Sal Khan.
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- Why is it written sin^2(θ) + cos^2(θ) = 1 instead of sin(θ)^2 + cos(θ)^2 = 1? The second way seems to better represent how the math is done, so why is it notated the first way?(16 votes)
- Because when you put an exponent on a set of parentheses, it means to raise the value inside the parentheses to the power. Hence, sin(θ)^2 means "take the value of θ, square it, and THEN find the value of the sine function." which is very different from sin^2(θ) which means "find the value of the sine function for θ and then square the result". Note that sin^2(θ) and [sin(θ)]^2 are equivalent expressions. Also, sin(θ^2) and sin(θ)^2 are equivalent expressions.(49 votes)
- Sal said in4:25that that point is clearly a negative value, I don't understand how this works. Can somebody explain to me or tell me where I can learn about this?(5 votes)
- Any of the unit circle lessons really. You just need to get an intuition for when certain angles cross over the axes.
Hopefully you understand that an angle of 0 lies ont he positive x axis. On the unit circle specifically it is point (1,0) so it's positive x and 0 y
the moment you give it an angle at all x stays positive, and now y is positive too. This is because up until 90 degrees (or pi/2 radians) the circle is in quadrant 1
at the right angle when it reaches the y axis y is still positive, but now x is 0
quadrant 2 has x negative now, since it is on the left of the y axis. if it's easier you can remember x = 1 is on the right of the y axis, and x = -1 is on the left.
at 180 degres, or pi radians we are at the x axis again, but on the negative side. again, y is 0 but x is negative rather than an angle of 0.
third quadrant is the quadrant on the negative x axis side and negative y axis side, so both x and y are negative here.
270 degrees/ 3pi/2 radians is the negative y axis, so x is 0 and y is negative. for the axes specifically, if it helps to remember if you are ON an axis then the non axis coordinate is 0. So here the y axis has the y coordinate negative and the x coordinate 0.
Finally quadrant 3 has the x axis positive and y axis negative, then after this it is back to the positive x axis.
Does that help at all? In the video the question starts with saying the angle is -3pi/2 and -pi. Since they are negative they are measured counterclockwise, were -pi/2 is the negative y axis rather than the positive one. So this means -3pi/2 is the positive y anxis and -pi is the negative x axis. the space between these two points is quadrant 2, which as I described above has a negative x value and positive y.(18 votes)
- At2:12, why does the square root of 3/4 become the square root of 3 over 2?(5 votes)
- Because 4 is a perfect square and becomes 2 when you take its square root, while 3 is not a perfect square, so it stays root 3 as the numerator while root 4 is simplified to 2.(16 votes)
- The assumption of x = cos θ and y = sin θ is valid as long as it is a unit circle including the pythagorean trig identity of cos^2 θ + sin^2 θ = 1. In the above problem, it is not mentioned that we are dealing with unit circle.
My question - Can the pythagorean trig identity be used for any circle ?(6 votes)
- (𝑟 cos 𝜃, 𝑟 sin 𝜃) are the coordinates for a point on a circle with radius 𝑟.
cos²𝜃 + sin²𝜃 = 1 is true for any circle.(8 votes)
- How can one input the square root of a number for the exercises?(0 votes)
- You type, "sqrt" and use parentheses () around the number you're taking the root of.(13 votes)
- why would we change 1/2 to 1/4 at about1:09(3 votes)
- 1/2 times 1/2 = 1/4 at0:56he mentions why he puts it there and then you must solve it out. So it equals 1/4(7 votes)
- I'm a bit confused by how sine can be equal to 1/2 on a unit circle. if sine(theta) = opposite/hypotenuse, then doesn't that mean that the hypotenuse = 2 in this case? Isn't that impossible on a unit circle, where the hypotenuse is always 1?(2 votes)
- If the opposite = 1/2 and the hypotenuse = 1, then 1/2 / 1 = 1/2
Hope this helps.(5 votes)
- In the actual problems, I can't figure out how to determine if the answer should be positive or negative with the quadrants because it isn't constant. For example, I have had separate problems where an angle in Quadrant IV was positive in one problem but an angle in Quadrant IV was negative in another problem. The hints don't explain how this is determined and I couldn't find a pattern. (Any pattern that began to form was broken pretty quickly.) Does anybody know how to use quadrants to determine the sign of the answer?(3 votes)
- Whether a trigonometric function is positive is indicated by ASTC
A- All are positive in the first quadrant
S- Sine is the only positive trigonometric function in the second quadrant
T-Tangent is the only positive trigonometric function in the third quadrant
C- Cosine is the only positive trigonometric function in the 4 quadrant.
So the question you mention the trigonometric was positive in 4th quadrant was cosine. While ones that were negative were sin and tan.
Alternatively know that cos theta = x, sin theta = y,
tan theta = y/x and then you could figure out where each trigonometric function is positive based on y and x values in the unit circle. So in the 4 quadrant since x is positive cosine is positive since cos theta = x. On the other hand sin theta is negative because the y value is negative in the 4th quadrant. And hence tan theta is negative since tan theta = y/x(4 votes)
- At about6:05Sal mentions that the tangent of an angle is equal to the slope of the terminal side of the angle (if you think of the terminal side as a line). What is the reason for this?(3 votes)
- slope is rise over run, or y/x. tangent is sine over cosine, or y over x, at least when dealing with the unit circle. But either way you get the same ratio.
If you don't get why sine over cosine is y/x remember sine is opposite over hypotenuse, and the opposite side when measuring the angle inside a unit circle, is always the y coordinate, and hypotenuse is always 1. Same reasoning for why cosine is x.
Then you need to remember slope can be scaled. if the slope is 3/2, that's the same as saying 6/4, or anything else that simplifies to 2/3, so you can scale it so it makes a right triangle with a hypotenuse of 1.
Also, it might even be simpler to remember tangent is opposite over adjacent, and opposite is y and adjacent it x.
Hope this helped.(4 votes)
- I still don't quite understand how Sal concluded that the cosine of theta is negative. Can someone explain what Sal did to conclude that theta isn't positive?(3 votes)
- with the limits between -3/2 π < theta < π, that puts the angle in the second quadrant where the adjacent is negative and the opposite is positive. So since cos =adj/hyp, it is negative in the second quadrant.(3 votes)
Let's say that we're told that some angle theta, which is going to be expressed in radians, is between negative 3 pi over 2 and negative pi. It's greater than negative 3 pi over 2 and it's less than negative pi. And we're also told that sine of theta is equal to 1/2. Just from this information can we figure out what the tangent of theta is going to be equal to? And I encourage you to pause the video and try this on your own. In case you're stumped, I will give you a hint. You should use the Pythagorean identity, the fact that sine squared theta, plus cosine squared theta is equal to 1. So let's do it. So we know the Pythagorean identity, sine squared theta, plus cosine squared theta is equal to 1. We know what sine squared theta is. Sine theta is 1/2. So this could be rewritten as 1/2 squared, plus cosine squared theta, is equal to 1. Or we could write this as 1/4 plus cosine squared theta is equal to 1. Or we could subtract 1/4 from both sides, and we get cosine squared theta is equal to-- let's see. You subtract 1/4 from the left hand side, then this 1/4 goes away. That was the whole point. 1 minus 1/4 is 3/4. So what could cosine of theta be? Well, when I square it, I get positive 3/4. So it could be the positive or negative square root of 3/4. So cosine of theta could be equal to the positive or negative square root of 3 over 4, which is the same thing as the positive or negative square root of 3, over the square root of 4, which is 2. So it's the positive or negative square root of 3 over 2. But how do we know which one of these it actually is? Well, that's where this information becomes useful. Let's draw our unit circle. If you're saying, well, why am I even worried about cosine of theta? Well, if you know sine of theta you know cosine of theta. Tangent of theta is just sine of theta over cosine theta. So then you will know the tangent of theta. But let's look at the unit circle to figure out which value of cosine we should use. So let me draw it, the unit circle. That's my y-axis. That is my x-axis. And I will draw the unit circle in pink. So that's my best attempt at drawing a circle. Please forgive me for its lack of perfect roundness. And it says theta is greater than negative 3 pi over 2. So where is negative 3 pi over 2? So let's see. This is negative pi over 2. So this is one side of the angle. Let me do this in a color. So this one side of the angle is going to be along the positive x-axis. And we want to figure out where the other side is. So this right over here that's negative pi over 2. This is negative pi. So it's between negative pi, which is right over here. So let me make that clear. Negative pi is right over here. It's between negative pi and negative 3 pi over 2. Negative 3 pi over 2 is right over here. So our angle theta is going to put us someplace over here. And the whole reason I did this-- so this whole arc right here-- you could think of this as the measure of angle theta right over there. And the whole reason I did that is to think about whether the cosine of theta is going to be positive or negative. We clearly see it's in the second quadrant. The cosine of theta is the x-coordinate of this point where our angle intersects the unit circle. So this point right over here-- actually let me do it in that orange color again-- this right over here, that is the cosine of theta. Now is that a positive or negative value? Well it's clearly a negative value. So for the sake of this example, our cosine theta is not a positive 1. It is a negative 1 So we could write the cosine theta is equal to the negative square root of 3 over 2. So we figured out cosine theta, but we still have to figure out tangent of theta. And we just have to remind ourselves that the tangent of theta is going to be equal to the sine of theta over the cosine of theta. Well they told us the sine of theta is 1/2. So it's going to be 1/2 over cosine of theta, which is negative square root of 3 over 2. And what does that equal? Well that's the same thing as 1/2 times the reciprocal of this. So times negative 2 over the square root of 3. These twos will cancel out and we are left with negative 1 over the square root of 3. Now some people don't like a radical in the denominator, like this. They don't like an irrational denominator. So we could rationalize the denominator here by multiplying by square root of 3 over square root of 3. And so this will be equal to negative square root of 3 over 3 is the tangent of this angle right over here. And that actually makes sense, because the tangent of the angle is the slope of this line. And we see that it is indeed a negative slope.