Browsing category trigonometry

A trigonometric trick: Secrets of the Mathematical Ninja

"Have you seen this trick?" asked the student. "If you know all three sides of a right-angled triangle, you can estimate the other angles - $A \simeq \frac{86a}{\frac b2 + c}$!" The Mathematical Ninja thought for a moment, and casually threw a set-square into the wall, millimetres from the student's

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A proof of the sine rule

Here's a nice use of circle theorems: ever wondered why the sine rule works?

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Is there a tangent rule?

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?" The answer to that is yes - yes, there is a tan rule. The natural follow-up is "Why don't we learn it?" Let me explain why not! Here's the tangent rule in

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Is there a tangent rule?

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?" The answer to that is yes - yes, there is a tan rule. The natural follow-up is "Why don't we learn it?" Let me explain why not! Here's the tangent rule in

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Is there a tangent rule?

There's a natural question, when you learn about the sine and cosine rules: "Is there a tan rule?" The answer to that is yes - yes, there is a tan rule. The natural follow-up is "Why don't we learn it?" Let me explain why not! Here's the tangent rule in

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Inverse sines near a half: Secrets of the Mathematical Ninja

"So, $\sin(x) = 0.53$," said the student. "32 degrees," said the Mathematical Ninja. The student frowned - the Mathematical Ninja's showing off was starting to wear her down - and typed it into the calculator to check. "$32.005^º$, actually." "I'll take that," said the Mathematical Ninja. "How did you guess

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Exact trig values – Secrets of the Mathematical Ninja

The Mathematical Ninja surreptitiously pressed a button under the table. There was a flash, a sizzle and a slight smell of burning. The student prodded the on-button of his calculator increasingly frantically. "Oh dear," said the Mathematical Ninja. "It must have been a passing electromagnetic storm that's permanently fried the

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Heroic triangles

In a recent Maths Challenge, students were told the area of a triangle ($7$cm$^2$) and the length of two of its sides ($6$cm and $8$cm), and asked how many possible lengths there were for the third side. It's easy enough to show there are two: let the base of the

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The geometry of $sec$

Ask virtually any maths teacher what $\sec(\alpha)$ means, the chances are they'll say "it's $\frac{1}{\cos(\alpha)}$," without missing a beat. Ask them what it means geometrically... well, I don't want to speak for the teaching profession as a whole, but I'd have been stumped until the other day. As with the

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