TNDOT Friday 14: Trigonometry

Let’s get back to work, and to fun. The TNDOT got derailed by the VCE exams, and the exams screw-up, so let’s try to get it back on track. TNDOT 12 is still there: not of that much interest but people can contribute, and I’ll try to polish that one off with solutions in the next few days.

The one that is open and interesting, from a few weeks back, is TNDOT 13, comprising the first three questions from the 1913 Matric trigonometry Honours exam: a few people have commented and appear to have come up with solutions, although not much has been posted; people are definitely welcome to and encouraged to post solutions. This one is a continuation of that 1913 exam: the next three questions. As background, the other trig TNDOT posts are TNDOT 1, TNDOT 2, TNDOT 4 and TNDOT 8.

Have fun.

4. Find the general solution of

$\color{blue}\boldsymbol{\sin^{-1}x = \cos^{-1}a.}$

Shew that

$\color{blue}\boldsymbol{2\tan^{-1}\left(\sqrt{\dfrac{a-b}{a+b}}\tan\dfrac{x}{2}\right)\, = \, \cos^{-1}\left(\dfrac{b + a \cos x}{a + b\cos x}\right).}$

Find a solution of

$\color{blue}\boldsymbol{\tan^{-1}ax + \dfrac12\sec^{-1}bx \,=\,\dfrac{\pi}{4}.}$

5. State and prove De Moivre’s Theorem.

If $\color{blue}\boldsymbol{\sin\theta =\dfrac{1}{\sqrt2}+ t}$ , where $\color{blue}\boldsymbol{t}$ is small, shew how to find $\color{blue}\boldsymbol{\theta}$ in terms of $\color{blue}\boldsymbol{t}$ .

By this means, find $\color{blue}\boldsymbol{\sin 46^{\circ}}$ , and state the accuracy of the result.

6. Distinguish between an equation and an identity.

Shew that

$\color{blue}\boldsymbol{\cos^2\theta + \cos^2(\phi + \theta)-2\cos \phi \cos \theta\cos(\phi + \theta)\,=\,\frac12 }$

is not an equation in $\color{blue}\boldsymbol{\theta}$ and will not give a value of $\color{blue}\boldsymbol{\theta}$ in terms of $\color{blue}\boldsymbol{\phi}$ .

Find the value of $\color{blue}\boldsymbol{\phi}$ , and shew that it is independent of the value of $\color{blue}\boldsymbol{\theta}$ .

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3 Replies to “TNDOT Friday 14: Trigonometry”

Very nice questions, all of which offer different things. Early thoughts:

Q4 – nice. Need to think a bit about why one part talks about “general” solution and the other says to find “a” solution.

Q5 – not sure about this one. “Accuracy” seems to want a numerical answer – but what form? Hopefully not a percentage…

Q6 – interesting. I’m guessing the distinction between identities and equations was covered properly in textbooks of the time…

So “an equation in $\theta$ ” is not $\theta$ in an equation.

marty says:

November 24, 2024 at 4:26 pm

Not in 1913, apparently.

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