"The derivative of sine U is cosine of that U variable times the derivative of U."
"That's finding side lengths using sohcahtoa."
"Anyway that's where the Maclaurin series for sine comes from."
"For my first amazing fact, let's have another look at Euler's way of writing sine as an infinite product up there."
"Complex numbers provide a visceral connection between geometry, trigonometry, and algebra."
"Cosine of 30 degrees: the long side of that, from everything we talked through in the last lecture, is square root of 3 over 2, square root of 3 over 2."
"Wow, what a change! The lines are now completely straight, no warping in sight. Trigonometry saves the day, that's beautiful."
"The cosine rule is E squared equals... Stick these numbers in, it's actually quite simple to use when you know it."
"It's why we combine the idea of a unit circle and our trigonometry."
"What quadrant your angle is in, that's telling you what's positive and what's not."
"It's a clever trick, this trig sub, the way it uses Pythagorean identities and gets rid of the radical."
"Sine theta equals opposite over hypotenuse, cosine of theta equals adjacent over hypotenuse, tan of theta equals opposite over adjacent."
"Wouldn't it be nice if x is equal to tangent theta? If that's the case, then we can use this identity. So that's the idea of the trig substitution."
"For solving triangles, all of the relationships that you can find about the trigonometric identities are usually about scaling the hypotenuse to a length of one and then just applying the trigonometry stuff."
"Radians are incredibly useful in advanced mathematics and science, especially in calculus and trigonometry."
"SOHCAHTOA is really easy to remember. I mean, someone says trigonometry, I think SOHCAHTOA."
"Cosine of zero is one, cosine of 90 is zero, cosine of 180 is negative one, and cosine of 360 is one."
"You need to be able to find the exact value of ratios for angles that are bigger than 90° using special triangles, the unit circle, and the CAST rule."
"The elements of this unit circle can be written as follows: z equals cosine of theta plus i sinus theta."
"...we're going to show you the trigonometric functions like tangent sine and cosine all the goodies that these videos come out so stay tuned if you have any idea as far as what we could do in later videos let us know in the comment section below."
"...so as with this example just a small change in a differential equation can turn a regular exponential solution into a hyperbolic trig one."
"The derivative of x raised to the sine x is x sine x times (cosine x ln x + sine x over x)."
"The final answer is 3x squared cosine x minus x cubed sine x."
"Sine of an angle will equal cosine of its complementary angle."
"Since the sine and cosine functions are defined by points on the circle, we can see that if we keep traveling around the circle, we're going to repeat the values."
"If you take the point on the unit circle that you get with the angle theta, then the point on the circle corresponding to negative theta is the reflection of the original point across the x-axis."
"So I take my two inches, multiply by the sign of the angle, and the answer I get is 0.2481 inches."
"I felt like I was on top of trigonometry."
"The trick is to be able to pick up on the correct pieces of information that you need in order to calculate the question so if you all set off your class for like trigonometry or calculus don't worry you'll be fine."
"Sine and cosine, so we already talked about SoH-CaH-Toa."
"So that's why trigonometric substitution actually works, because you're relating a value that you're given in your integral to these associated components inside a right triangle and that allows you to simplify that function that you're trying to integrate."
"So this becomes 4 times cos^2(theta) underneath our square root sign."
"Trust me, if you understand this video, you will definitely understand trigonometry better."
"The angle between a line and a plane is going to just be the line's direction dotted with the plane's normal, divided by the modulus of the normal multiplied by the modulus of the direction."
"So, if I was to plug these into my sine rule formula like we did before, we'd have 6.7 over sine 95, and that would be equal to x over sine 20."
"The average will just be cosine of twice the angle."
"The sign and cosine give you circular motion, and this defines the vertical and horizontal dimensions."
"The sine of 0 is 0. There is no force."
"For right angle triangles we have Pythagoras and SOHCAHTOA, but for non-right angle triangles we can use the sine rule or the cosine rule."
"One minus sin Theta times one plus sin Theta simplifies down to one minus sin squared Theta."
"It's really important to have a good understanding of those topics before moving on to these harder versions of trigonometry."
"That's going to be 30 degrees and you can see now why we're looking at 30 and 60 because within this triangle I've got 30 degrees and I've got 60 degrees."
"Sine 60 opposite over hypotenuse is root 3 over 2."
"The graph of y equals sine X is reflected in the x-axis."
"Find the exact value of tan 30 times sine 60 and give your answer in its simplest form."
"That's how to find the exact trigonometric values."
"In trigonometry, tan theta is sine theta over cos theta."
"Sine squared theta plus cos squared theta is one."
"But this is equal to yet another trigonometric function called the tangent function."
"We know the equation of a cycloid; it involves sines and cosines."
"So sine X equals zero, okay, so this is the same as saying X equals K PI where K is an integer."
"The derivative of sine is just cosine."
"One plus cotangent squared theta, that is exactly what we want to see, that's cosecant squared theta."
"This first sum is exactly the Taylor expansion of cosine and this second sum which is attached to an i is exactly the Taylor expansion of sine."
"Trigonometric identities are really just a bunch of formulas that help us manipulate or rewrite different kinds of trig expressions."
"All trig identities come back to the relationships between the different parts of a right triangle."
"We wouldn't even really need them at all but we like to establish these trig identities because there are some kinds of trig expressions that seem to pop up over and over again."
"If you're able to do this, then you're definitely on the right track, and you're showing me that you know a lot of sub-skills in trigonometry."
"Recognizing the Pythagorean identity is going to equal to sine squared."
"F of s is approximately equal to a not divided by 2 plus summation from n equal to 1 to Infinity of a sub n cos NX plus b sub n sine n x."
"One of the major formulas which you have to remember or apply is tan squared X plus 1 equals secant squared X."
"Writing tan squared X as equal to secant squared X minus 1 will solve your question."
"The sine of an angle equals the opposite side over the hypotenuse."
"Sine of X equals cosine of the complement."
"Only sine is positive in quadrant two."
"Tangent is positive in quadrant three."
"Cosine is positive in quadrant four."
"Cosine relates to X and tangent relates to the ratio of Y over X for that angle measure."
"Astronomy had a huge part to play in this, but trigonometry is basically the concept of studying polynomials that have three angles, three sides."
"Just remember, when we're looking for an angle, we do the inverse."
"To get Theta by itself, we do the opposite of tan, which is arctan."
"Press equals, and we get 30.6 degrees."
"Sine theta equals opposite over hypotenuse."
"Hipparchus, the ancient Greek astronomer and mathematician, is called the father of trigonometry."
"The sine of angle A over side length A is equal to the sine of angle B over B, which is equal to the sine of angle C over C."
"That's the answer, the square root of three over two, once you take the sine of pi over 3."
"The derivative of cosine is negative sine, so we put on a negative sign, and then the input stays the same."
"The derivative of sin x is just as good as cos x."
"The derivative of cos x gives us negative sin x."
"The derivative of tan of x is equal to sec squared of x."
"The derivative of secant of x is equal to secant of x times tangent of x."
"The derivative of cosecant of x becomes negative cosecant of x times cotangent of x."
"When is sine equal to zero? Sine is equal to zero at integer multiples of Pi."
"This is one of the most important little trig identities that's relevant for Fourier series."
"Euler's formula says that if I have some complex number Theta times i, I take e to that power, and that's equal to cosine Theta as a new real part, and the new imaginary part is going to be sin Theta."
"With radians we can use cosine and sine to actually determine the x and the y direction."
"Cosine 30 is the square root of 3 over 2, and once we square that, I₂ is three-fourths of I₁."
"The cosine of an angle equals the adjacent over the hypotenuse."
"The sine of an angle equals the opposite over the hypotenuse."
"Area tends to equal a half AB sine C."
"Two possible values of cos theta are going to be plus minus four over five."
"The fundamental identity of trigonometry is that \( \sin^2 x + \cos^2 x \) is equivalent to 1."
"Therefore, we can say that 10 cosine theta minus 3 sine theta equals the square root of 109 cosine theta plus sixteen point seven."
"For small angles, the sine of theta is approximately equal to the opposite over hypotenuse."
"Maximum height occurs when the cosine of \( a \cdot bt + 16.70 \) equals minus one."
"So sec theta is actually equal to one over cos theta because the third letter is C."
"So for sine of A plus B it's sine A cos B plus or minus cos A sine B."
"Show that cos 3A is equal to 4 cos cubed A minus 3 cos A."
"The antiderivative of cosine is going to be sine, but the antiderivative of cosine four theta is sine four theta divided by four."
"The antiderivative of one is going to be theta, and for cosine two theta is going to be sine two theta divided by two."
"Cosine and sine relate to each other by a phase shift."
"Every 2 pi, that's our period; our range is still negative one to one."
"Our x-intercepts happen at every pi over 2 plus k pi."
"Our key points for our cosine are zero one, pi over 2 zero, pi negative 1, 3 pi over 2 zero, and 2 pi positive 1."
"Cos squared plus sine squared is one."
"Sine of theta equals a y-coordinate on the unit circle."
"Cosine theta equals the x-coordinate."
"The limit as theta goes to zero of sine theta over theta is equal to one."
"The period is 360 degrees divided by K."
"The antiderivative of 1 over u squared plus one is the arc tan of u."
"Wherever Cos theta is 0, tan theta doesn't exist because tan is just sine over cosine."
"Anytime you're working with a right triangle... think about right triangle trig again, let's go back to SOHCAHTOA."
"It's really really important when you're doing trigonometry on your calculator... you have to make sure it's in degree mode."
"A good parametric representation would be to let x equal 5 cosine t, y equals 5 sine t."
"Sine of an angle theta equals cosine of 90 degrees minus that angle theta."
"Sine inverse of opposite over hypotenuse equals theta."
"This is a pretty difficult trigonometry problem for two reasons."
"Sine of an angle equals cosine of the complement of that angle."
"The cosine of 80 degrees is the side adjacent which is x divided by the hypotenuse which is 140."
"Right triangle trig is primarily used for two things: finding missing sides when you know one side and an angle or finding missing angles when you know two or more sides."
"For your GCSE, you have to remember the exact values for certain key trigonometric ratios."
"By learning how to generate this table, you understand the three trig ratios, the unit circle, degrees, radians, and what the trig ratios mean."
"This table is the basis of what we need to know before we can actually start using trigonometry."
"The first question on every trigonometry test should be: generate this table."
"Once you know how to generate this, there are two different avenues you can go in trigonometry."
"The instructions stated that the music video had to be based on some sort of song or piece of music and had to explain a trigonometric concept with an example."
"This is an integral that we know the antiderivative is sin u + C."
"It's kind of a handy thing, the one I remember is the sine of one degree is 0.0175."
"E to the i-x is cos x plus i sine x, which is my identity, one of the most remarkable equations in the whole of mathematics."
"Cosine theta sine theta is the same thing as sine theta cosine theta."
"Sine of two theta equals two sine theta cosine theta."
"The Pythagorean identity simply says sine squared theta plus cosine squared theta equals 1."
"Cosine squared theta minus sine squared theta."
"We're going to replace this piece with 1 minus sine squared theta by the Pythagorean identity."
"Tangent squared theta means tangent theta quantity squared."
"We're going to create half angle formulas from our identities that we got stemming from our double angle formulas."
"Can you still use trigonometry if you don't have a right triangle? Yeah, that's why we have the law of sines and the law of cosines."
"If you know two angles and one side, you can find the missing sides with law of sines."
"If you know two angles and one side, you can use the law of sines."
"When you have opposite and adjacent, you're not using sine, you're using tan."
"Tan of the angle is equal to the opposite over the adjacent."
"In the tree of trig, there are quite a few branches, man."
"The sine rule is for all triangles."
"Sine squared x plus cos squared x equals one."
"I am going to cover trigonometry in one video."
"We can express any periodic function as the sum of a bunch of cosines and sines and then a constant shift."
"The inner product of everything with cosine X is the integral of f of X times cosine X."
"The cosine of A equals minus zero point two seven four approximately."
"For small values of theta, sine theta is approximately the same as theta."
"Cosine theta is approximately the same as 1 minus theta squared all over 2."
"E to the I Theta is equal to cos theta plus i sin theta."
"The limit as H goes to zero of sine of H over H is equal to one."
"The derivative of the sine function is simply the cosine."
"Sine of 90 - θ is cos θ because when you change θ and 90 - θ, what's opposite becomes adjacent and what's adjacent becomes opposite."
"The sine rule is really useful whenever you've got an angle and a side opposite each other."
"The cosine rule can be really useful to find missing angles as well as missing sides."
"If you want to find the area of a triangle and if you know two sides and the angle in between them, you can just do a half a b sine c."
"For right angle triangles, you need to be thinking Pythagoras and you need to be thinking trigonometry."
"For the sine rule, you need an angle and two sets of angles and corresponding sides."
"Cosine of theta x squared plus cosine theta y squared plus cosine theta Z squared is just going to be equal to one."
"You just have to remember that Alpha is inverse tan of the second number over the first number all the time."
"We need to rewrite sine squared using the double angle."
"Cosine of any negative angle is the same as cosine of the positive version of that angle."
"It's beautiful, the first infinite sum converges to a familiar trig function."
"We know that one plus tan squared x is sec squared x."
"The first infinite sum converges to a familiar trig function."
"The tan of the angle is equal to the opposite divided by the adjacent."
"The cos of the angle is equal to the adjacent divided by the hypotenuse."
"The sine of 41 times x is equal to 3."
"The cos of 52 multiplied by 14 is equal to 8.62 centimeters to two decimal places."
"Tangent of theta, the central angle, is given by the y-coordinate over the x-coordinate."
"Trigonometry is very powerful and there's just no substitute for solving particular problems involving angles and sides of triangles."
"This is an important trig identity that you'll want to know."
"Just remember, cosine has the plus, sine has the minus."
"Trigonometry relates an angle to actually a ratio of y over one or the opposite over the hypotenuse."
"Sine of some angle is going to equal the y-coordinate of a point on the unit circle."
"Sine and cosine are phase shifts of one another."
"If sine of A is equal to cosine of B, that means that A plus B is equal to 90."
"Tan 30 degrees is equal to 1 over root 3."
"Remember that this one sine a minus b is sine a cos b minus cos a sine b."
"Sine 2A is equal to 2 cos A sine A."
"Cos squared theta will be equal to 1 minus sine squared theta."
"Sine 3 theta is equal to 3 sine theta minus 4 sine cubed theta."
"Tan squared 22.5 degrees is equal to 3 minus 2 root 2."
"Sine A plus sine B is equal to 2 sine(A+B/2) multiplied by cos(A-B/2)."
"Cos A plus cos B is 2 cos(A+B/2) multiplied by cos(A-B/2)."
"Negative 2 sine(A+B/2) multiplied by sine(A-B/2) is for cos A minus cos B."
"In trigonometry, identities can either be derived or proved or shown."
"By comparison, tan a is equal to 1, meaning that a is 45 degrees."
"Cosine theta plus I sine theta, anyone know what this is called? This is Euler's formula."
"The integral of minus sine of x is cosine of x."
"The sine of pi over 4 is just simply going to be the square root of 2 over 2."
"Tangent is sine over cosine; cotangent is cosine over sine."
"If two angles add up to 90, really the two acute angles in a right triangle, then the sine of one is the cosine of the other. It's really that simple."
"A lot of trigonometry you'll be surprised, I would say like 50/50, 50% of what you're going to be doing is going to be without a calculator."
"It's just easier for our brains at this point to think about cosine."
"You've got to make sure you've got the right trig ratio, never forget that mnemonic SOHCAHTOA."
"The period of a sine or cosine graph is equal to 2 pi divided by K."
"Tan 30 degrees is equal to my opposite over my adjacent."
"If sine is one-to-one on that domain, then we can find an inverse for it."