Trigonometric ratios: labelling sides and choosing sin, cos or tan

Picture a ladder leaning against a wall. The ladder, the wall and the ground make a right-angled triangle. A friend at the foot of the ladder calls the wall the opposite side, meaning the side across from her. A friend at the top calls the same wall the adjacent side, meaning the side next to him. One wall, two names, and both friends are right. How that works is what this video is about, and every trigonometry question after it depends on it.

This is video two of eleven in Pythagoras, Trigonometry and Coordinate Geometry. If the hypotenuse, the longest side, still feels new, Pythagoras' theorem in two dimensions is the one to go back to.

Trigonometry starts by naming the three sides of a right-angled triangle, and one name never changes. The hypotenuse faces the right angle, and it's always the longest side. Then one of the other two angles gets marked. We usually call it theta, a Greek letter that stands for an angle. The opposite side is the side facing that marked angle. Every angle faces its own side - look straight across from theta, and the side you see is the opposite. That leaves one side, running from theta to the right angle. It's called the adjacent side, and adjacent means next to. The hypotenuse touches theta too, but it already has its name. Doing it in this order means you never guess. Hypotenuse first, from the square corner. Opposite second, from the marked angle. Adjacent is whatever is left.

Back to the ladder, drawn twice. In the first copy, the marked angle sits at the foot of the ladder, between the ladder and the ground. The ladder faces the square corner, so the ladder is the hypotenuse. Looking straight across from the foot, you see the wall, so the wall is opposite. The ground is left over, which makes it adjacent. In the second copy, the marked angle moves up to the top of the ladder, between the ladder and the wall. With the angle at the top, which side is opposite: wall, ground or ladder? The ground. From the top, looking straight across, the ground is what you face. Opposite and adjacent have swapped places, and the wall is now adjacent, touching the angle at the top. The hypotenuse never moved. The labels belong to the marked angle, not to the triangle, and that is how both friends were right. One examiner's report, listing what students should be able to do, includes this line. "know how to use the trigonometric functions correctly and how to label triangles correctly" A side labelled from the wrong angle spoils the working, even when the ratio's name is right. The fix is to label all three sides from the marked angle, hypotenuse first, before any ratio is chosen.

With the sides labelled, each trigonometric ratio is one side divided by another. The sine of theta is the opposite divided by the hypotenuse. The cosine of theta, cos for short, is the adjacent divided by the hypotenuse. The tangent of theta, tan for short, is the opposite divided by the adjacent. Take the first letters and you get SOH, CAH, TOA. Sine is opposite over hypotenuse, cos is adjacent over hypotenuse, tan is opposite over adjacent. These divisions are worth having because, for a given angle, they never change. Every right-angled triangle with a thirty-degree angle has the same sine of thirty, however big it is drawn. Choosing the ratio comes from two sides: the one you know, and the one you want. The ratio that uses exactly those two is the one. Say the ladder's length is known, the wall's height is wanted, and the angle at the foot is marked. The ladder is the hypotenuse and the wall is opposite, and opposite with hypotenuse means sine. That gives the equation: sine theta equals the wall's height over the ladder's length. An equation with real sides in it, not a chant. Now the ground is known and the wall's height wanted. Which ratio, and why not sine? Tan. From the foot, the wall is opposite and the ground is adjacent, and tan is opposite over adjacent. Sine would need the ladder, which this question never mentions. An examiner's report on another paper puts the aim in one sentence. "Using trigonometry involves setting up an equation to solve, as with any basic equation." Chanting the letters is not the working. The fix is to write the line out in full, sine theta equals opposite over hypotenuse, with the real sides put in. Solving that equation is the job of the next video, Using trigonometry to find missing sides and angles.

Some angles have sines and cosines you can find exactly, with no calculator at all. Two small triangles hold every one of them. Start with an equilateral triangle whose sides are all two units long. All three of its angles are sixty degrees. Cut it straight down the middle. Each half is a right-angled triangle with a hypotenuse of two and a bottom side of one. The bottom angle is still sixty degrees, and the top angle is halved to thirty. The height comes from Pythagoras. Two squared take away one squared is four take away one, which is three, and that makes the height root three. From the thirty-degree angle, which side is opposite: one, two or root three? The side of one. Looking straight across from the angle at the top, you face the short bottom side. That makes sine thirty one over two, a half. The adjacent side to thirty is root three. That makes cos thirty root three over two, and tan thirty one over root three. From the sixty-degree angle the sides swap roles, as they did on the ladder. Sine sixty is root three over two, cos sixty is a half, and tan sixty is root three. The second triangle is half a square. Take a square with sides of one and cut it along its diagonal. Each half has two sides of one and two angles of forty-five degrees. Its hypotenuse is root two, because one squared plus one squared is two. Using the half square, what is tan forty-five? One. From either forty-five angle, the opposite and the adjacent are both one, and one over one is one. Sine forty-five and cos forty-five are both one over root two, which some tables write as root two over two. Zero and ninety come from squashing the triangle. As the marked angle shrinks towards zero, the opposite shrinks to nothing and the adjacent stretches to the full hypotenuse. So sine zero is zero, cos zero is one, and tan zero is zero. As the angle grows towards ninety, the opposite stretches to the full length of the hypotenuse, and the adjacent shrinks to nothing. That makes sine ninety one and cos ninety zero. Tan ninety would divide by nothing, so it has no value at all. Those two triangles rebuild the whole table in under a minute, which is quicker than memorising fourteen separate numbers.

One exact value can finish a question on its own. A kite string is eight metres long and makes an angle of thirty degrees with the ground, and the kite's height is wanted. Which ratio links the string and the kite's height? Sine. The string is the hypotenuse, and the height faces the marked corner, so it's opposite over hypotenuse. That gives the equation: sine thirty equals the height over eight. Sine thirty is a half, from the half equilateral triangle, so the height is half of the string. Half of eight metres is four metres. The kite is four metres up, with no calculator anywhere. One examiner's report, on a question built like this one with a hypotenuse of ten, says this. "Those students who tried to use trigonometry to solve this question, frequently calculate the correct answer. Many did not remember that sine thirty equals nought point five therefore got no further than sine thirty equals x divided by ten or x equals ten sine thirty." Their method was sound, and the missing piece was one value. The fix is the half equilateral triangle, sides of one and two, which gives you sine thirty in seconds. This chapter's handle carries all of it: every angle faces its own side - look straight across. From the marked angle, the side you see is the opposite, and the other labels follow.

The ladder's wall had two names. Now the labels meet triangles this video never drew. Right angle at C, angle marked at A. Which side is opposite: A B, B C or A C? B C. Standing at A and looking straight across, B C is the side you face. Here's another. You know the adjacent and want the hypotenuse. Which ratio? Cos. It's the one ratio built from the adjacent side and the hypotenuse. Now a harder one: a ten-metre slide at sixty degrees. How far along the ground? Five metres. The slide is the longest side and the ground touches the angle, so cos sixty equals the ground distance over ten. Cos sixty is a half, and half of ten is five.

When the labels feel like they're in the bag, tap the thumbs-up, and this video is marked as finished. If they're still wobbling, save the video for later. Label one triangle out loud, hypotenuse first, and it tends to stick.

Next in the chapter: Using trigonometry to find missing sides and angles.

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