Trigonometric ratios of obtuse angles (Higher)

Press cos, one hundred and twenty, equals on a calculator, and the display reads minus one over two, a negative fraction. Every cosine so far has been one side divided by another, and a length is never negative. Past ninety degrees, something new is going on. One picture, a crane arm swinging up past upright, explains that minus sign and tells you which ratios pick one up.

This is video six of eleven in Pythagoras, Trigonometry and Coordinate Geometry. The exact values of sine thirty and cos sixty do the work here, and Trigonometric ratios, labelling sides and choosing sin, cos or tan is where they come from.

This is Higher-tier material, and only some specifications list it by name, so check your own paper.

Every ratio so far came from an angle inside a right-angled triangle, so every angle was under ninety degrees. An angle between ninety and a hundred and eighty degrees is called obtuse, and ordinary triangles can have one. Picture a crane arm one metre long, hinged at the cab, lying flat and pointing forwards. Now lift it through an angle, theta. While theta is acute, meaning under ninety degrees, the arm is the hypotenuse of a right-angled triangle, its longest side. With a hypotenuse of one, opposite over hypotenuse is the opposite itself, the tip's height. Adjacent over hypotenuse is the tip's forward reach. So, that gives a definition that works for any angle. Sine theta is the height of the arm's tip, and cos theta is how far forward the tip reaches. Try ninety degrees, straight up. What are the tip's height and forward reach? Height one, forward reach zero. That makes sine ninety equal one, and cos ninety equal zero. Now keep lifting, to a hundred and twenty. Is the forward reach positive or negative? Negative. The tip has swung back behind the cab, so it reaches backwards, and cos one hundred and twenty is negative. The tip is still above the hinge, though. Its height stays positive, so sine one hundred and twenty is positive. So, for every obtuse angle, sine keeps its plus sign and cosine flips to minus. Tan is sine divided by cos, so tan flips to minus as well.

Now, the crane arm gives exact values too. Lift it to thirty degrees, then to a hundred and fifty, and the two positions are mirror images either side of upright. A hundred and fifty is thirty short of a hundred and eighty, so the arm leans back at the same thirty-degree slope. The tip sits at the same height both times. Same height means same sine. Sine of a hundred and eighty take away x equals sine x. Given that sine thirty is nought point five, sine one hundred and fifty is sine of a hundred and eighty take away thirty. That is sine thirty, so sine one hundred and fifty is nought point five as well. Right, cosine next. You know cos sixty is nought point five, and a hundred and twenty is the mirror image of sixty. What is cos one hundred and twenty: a half, minus a half, or one and a half? Minus a half. The arm at a hundred and twenty matches the arm at sixty in size, but it reaches backwards. Cos of a hundred and eighty take away x equals minus cos x. Side by side, the two rules differ by one thing. Sine of a hundred and eighty take away x is sine x. Cos of a hundred and eighty take away x is minus cos x. Tan, meanwhile, follows cos. Tan of a hundred and eighty take away x is minus tan x, which makes tan one hundred and thirty-five equal minus tan forty-five, which is minus one.

A calculator, by the way, handles obtuse angles directly. Press sine, one hundred and fifty, equals, and the display reads one over two. Press cos, one hundred and twenty, equals, and the display reads minus one over two. That minus sign - that's the calculator agreeing with the crane arm. Here is a student's line: cos one hundred and fifty equals cos thirty, about nought point eight six six. What's wrong with that line? It uses the sine's mirror rule on cosine. Past ninety, cosine flips sign, so the true value is minus that number. Now, the opposite slip puts a minus sign on sine. Sine is the tip's height above the hinge, so between zero and a hundred and eighty degrees it is never negative. Every angle faces its own side - look straight across. In a triangle, an obtuse angle faces the longest side, and it's the one angle whose cosine is negative. The cosine rule, in the next video, uses exactly that.

The crane arm, then, settled the minus sign. These last questions use angles this video never touched. Without a calculator, is cos one hundred and ten positive or negative? It comes out negative. A hundred and ten is obtuse, and cosine flips to minus past ninety. Now sine: sine forty-five is one over root two. What is sine one hundred and thirty-five? One over root two as well. A hundred and thirty-five is a hundred and eighty take away forty-five, and sine keeps its sign. One more, then we're done: tan sixty is root three. What is tan one hundred and twenty? Minus root three. It's the mirror of sixty, with tan flipping to minus, as cos does.

Once you're comfortable with it, a quick thumbs-up takes this video off the list you still need to watch. If the signs still feel slippery, save it for later. The cosine rule uses them straight away, so they come round again.

Next in the chapter: The cosine rule.

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