Trivium
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Know the exact algebraic representation
of the output value of any trigonometric function
evaluated at any integer multiple
of \(\frac{\pi}{6}\) (30°) or \(\frac{\pi}{4}\) (45°).
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Sketch the graph of any composite
of a trigonometric function with linear functions,
and conversely recognize the graph of such a function
and write down its formula.
Problems
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What’s the smallest positive value \(t\)
such that \(\cos(t) = \frac{1}{2}?\)
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What’s the smallest positive value \(t\)
such that \(\cos(t) = -\frac{1}{2}?\)
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What’s the smallest positive value \(t\)
such that \(\tan(t) = \sqrt{3}?\)
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Suppose that \(\sin(t) = \frac{3}{4}\) for some value of \(t.\)
What are all the possible positive values of \(\cos(t)?\)
What are all the possible positive values of \(\tan(t)?\)
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Suppose that \(\tan(t) = 5\) for some value of \(t.\)
What are all the possible positive values of \(\sin(t)?\)
What are all the possible positive values of \(\cos(t)?\)
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Does there exist a value of \(t\)
such that \(\cos(t) = 2?\)
If so, approximately what is it?
If not, why not?
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Typically only the sine and cosines functions
are taught to be lengths in the context of the “unit circle,”
while other trig functions are defined algebraically in terms of sine and cosine.
However, many these functions can also be thought of
as lengths on the “unit circle.”
The diagram here shows some of these functions,
as well as a few antiquated functions
like versine and exsecant, as lengths.
Recalling some basic geometric knowledge,
confirm that the definitions of
tangent and secant and cotangent and cosecant
in terms of sine and cosine
agree with the lengths of the line segments
labelled with those trig functions in this diagram.