Calculating side lengths or angles in right-angled or general triangles.
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Right-angled
triangle
The ratio of these sides to each other defines the
trigonometric functions.
\[\begin{aligned}
\sin(\alpha) & :=\frac{{\text{opposite
}}}{{\text{hypothenuse}}}\\
\cos(\alpha) & :=\frac{{\text{adjacent }}}{{\text{
hypothenuse}}}\\
\tan(\alpha) & :=\frac{{\text{opposite }}}{{\text{adjacent
}}}\\
\cot(\alpha) & :=\frac{{\text{adjacent }}}{{\text{opposite
}}}
\end{aligned}\]
Pythagorean theorem: \(a^2+b^2=c^2\)
General triangle
Based on our definition, we can calculate, for example, the sine of
an angle in a general triangle by drawing
right-angled auxiliary triangles:
Further useful results for calculating side lengths or angles in
general triangles:
Sine rule: \[\frac{\sin
(\alpha)}{a} = \frac{\sin(\beta)}{b} =
\frac{\sin(\gamma)}{c}\]
Cosine rule: \[a^2 + b^2
- 2ab\cos(\gamma) = c^2\]
Unit
circle: values for sine and cosine
Unit circle in the plane is a circle with radius
\(1\) around the points \((0,0)\).
The drawn triangle has a right angle and the sine
and cosine values can be read off:
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