Browsing category trigonometry

Ask Uncle Colin: A peculiar triangle

Dear Uncle Colin, I have a triangle. All I know is that its angles, $\alpha$, $\beta$ and $\gamma$, satisfy $\cos(\alpha)=\frac{1}{4}$ and $\gamma = 30º$ – and I have to find $\tan(\beta)$. Help! – Can't Obviously See It, Need Explanation Hi, COSINE, and thanks for your message! This is one that

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Ask Uncle Colin: A Cosec Proof

Dear Uncle Colin I'm stuck on a trigonometry proof: I need to show that $\cosec(x) – \sin(x) \ge 0$ for $0 < x < \pi$. How would you go about it? – Coming Out Short of Expected Conclusion Hi, COSEC, and thank you for your message! As is so often

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Ask Uncle Colin: Simultaneous Trigonometry

Dear Uncle Colin, I'm normally pretty good at simultaneous equations, but I can't figure out how to solve this for $a$ and $b$. $\cos(a)-\cos(b) = x$ $\sin(a)-\sin(b) = y$ – Any Random Circle Hi, ARC, and thanks for your message! This is, it turns out, a bit trickier than it

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Are you sure that’s a right angle?

What's that, @pickover? Shiver in ecstasy, you say? Just for a change. Shiver in ecstasy. The sides of a pentagon, hexagon, & decagon, inscribed in congruent circles, form [a] right triangle. — Cliff Pickover (@pickover) May 20, 2017 That's neat. But why? Let's suppose the circles all have radius

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Revisiting some missing solutions

“You know how you’re always putting things like ‘just to keep @RealityMinus3 happy’ in your posts?” “Of course, sensei!” “Well… you remember that post about missing solutions in a trig problem?” “Ut-oh.” What follows is a guest post by Elizabeth A. Williams, who is @RealityMinus3 in real life. This thing

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Ask Uncle Colin: Some missing solutions

Dear Uncle Colin, When I solve $2\tan(2x)-2\cot(x)=0$ (for $0 \le x \le 2\pi$) by keeping everything in terms of $\tan$, I get four solutions; if I use sines and cosines, I get six (which Desmos agrees with). What am I missing? – Trigonometric Answers Not Generated – Expecting 'Nother Two

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The Mathematical Ninja and Cosines

As the student was wont to do, he idly muttered "So, that's $\cos(10º)$…" The calculator, as calculators are wont to do when the Mathematical Ninja is around, suddenly went up in smoke. "0.985," with a heavy implication of 'you don't need a calculator for that'. As the student was wont

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Ask Uncle Colin: an arctangent mystery

Dear Uncle Colin, In an answer sheet, they've made a leap from $\arctan\left(\frac{\cos(x)+\sin(x)}{\cos(x)-\sin(x)}\right)$ to $x + \frac{\pi}{4}$ and I don't understand where it's come from. Can you help? — Awful Ratio Converted To A Number Hello, ARCTAN, and thank you for your message! There's a principle I want to introduce

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Ask Uncle Colin: Trigonometric inverses and picking the correct quadrant

Dear Uncle Colin, When I have an angle in the second quadrant, I can find it just fine using $\cos^{-1}$ – but using $\sin^{-1}$ or $\tan^{-1}$ gives me an angle in the fourth quadrant. I don't understand why this is! — I Need Verbose Explanations; Radians Seem Excellent Hi, INVERSE,

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Ask Uncle Colin: A Triangle That’s Not As Nice As It Looks

Dear Uncle Colin, I need to find an angle! ABC is a triangle with median AD, while angles BAD and CAD are 110º and 20º, respectively. What's angle ACB? — Angle Being Evasive, LOL Hi, ABEL, and thanks for your question! Even if you've used degrees. For heaven's sake, get

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