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If we know three sides of an isosceles triangle, can we find the measure of the angles without using a calculator (that means no law of Cosines/Sines).

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    $\begingroup$ Depends on the meaning of find. One can use the definition of sine. $\endgroup$ – André Nicolas Jun 11 '14 at 0:47
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Call the sides of the triangle lengths $a$, $a$, and $b$. Drawing a line of symmetry through the triangle lets us divide the triangle into two right triangles. Hence we can calculate $$\cos(\theta)=\frac{\frac{b}{2}}{a}=\frac{b}{2a}$$ where $\theta$ is the repeated angle in the triangle. This tells us the angles of the triangle are $\theta$, $\theta$, and $90-2\theta$, where $$\theta=\cos^{-1}(\frac{b}{2a}).$$

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Yes. We can use the law of sines.

That is, choose two sides of unequal lengths $a$ and $b$. Let $\alpha$ and $\beta$ denote the measures of their respective opposite angles$. Then we have:

$$\frac{\sin(\alpha)}{a} = \frac{\sin(\beta)}{b}$$

If $a < b$, then $2\alpha + \beta = 180^\circ$ since the triangle is isosceles. From here, we have $\beta = 180^\circ - 2\alpha$. Plug this into the above, and we have an equation with a single unknown.

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  • $\begingroup$ I meant more in the spirit of not using a calculator when I excluded using the law of Cosines, and this method would still require a calculator. $\endgroup$ – Kush Sharma Jun 11 '14 at 1:00
  • $\begingroup$ Oh I understand, sorry about that. I'll leave my answer up anyhow since others might get something out of it. $\endgroup$ – Kaj Hansen Jun 11 '14 at 1:02

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