There are two basic approaches to trigonometry:
Both approaches were introduced in Introduction to Trigonometry.Be sure to read this prior section, since it covers important notation and conventions.
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![]() using the familiar $\,30^\circ$-$60^\circ$-$90^\circ\,$ triangle: ![]()
$$
\begin{alignat}{2}
\sin 30^\circ&\ \ =\ \ && \frac{s}{2s} &\ \ =\ \ && \frac 12\cr\cr
\cos 30^\circ &\ \ =\ \ && \frac{\sqrt 3s}{2s} &\ \ =\ \ && \frac{\sqrt 3}2\cr\cr
\sin 60^\circ &\ \ =\ \ && \frac{\sqrt 3s}{2s} &\ \ =\ \ && \frac{\sqrt 3}2\cr\cr
\cos 60^\circ &\ \ =\ \ && \frac{s}{2s} &\ \ =\ \ && \frac 12
\end{alignat}
$$
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Consider a right triangle with legs of lengths $\,5\,$ and $\,7\,.$
Let $\,\theta\,$ be the largest acute angle.
Find $\,\sin\theta\,,$ $\,\cos\theta\,,$ and $\,\tan\theta\,.$
(It is not necessary to simplify radicals or to rationalize denominators.)
SOLUTION: By the Pythagorean Theorem: ${\text{HYP}}^2 = 5^2 + 7^2$ Therefore: $\text{HYP} = \sqrt{5^2 + 7^2} = \sqrt{74}$ The largest acute angle is opposite the longest leg. Thus: $$ \begin{gather} \cssId{s63}{\sin\theta = \frac{7}{\sqrt{74}}}\cr\cr \cssId{s64}{\cos\theta = \frac{5}{\sqrt{74}}}\cr\cr \cssId{s65}{\tan\theta = \frac{7}{5}} \end{gather} $$ NOTE: Suppose that the right triangle had legs of lengths $\,500\,$ and $\,700\,$ (instead of $\,5\,$ and $\,7\,$). Then, you would first ‘shrink’ the triangle to a more manageable size, by dividing the lengths by $\,100\,.$ Similar triangles have the same angles! Don't work with big numbers when you don't have to! Or, suppose the right triangle had legs of lengths $\,0.00005\,$ and $\,0.00007\,$ (instead of $\,5\,$ and $\,7\,$). You would first ‘stretch’ the triangle by multiplying the lengths by $\,100{,}000\,.$ Similar triangles have the same angles! Don't work with very small numbers when you don't have to! |
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On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
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