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Thread: trigonometry

  1. #1
    Member helloying's Avatar
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    trigonometry

    pls see the attactment
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  2. #2
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    Since $\displaystyle TA$ and $\displaystyle TB$ are tangent to the circle, then extending the radius to the point of tangency will form right angles with them, i.e. $\displaystyle \angle OAT = \angle OBT = 90^{\circ}$

    Now, looking at $\displaystyle \triangle TAB$, we know it is isosceles since $\displaystyle TA = TB$. This implies $\displaystyle \angle TAB = \angle TBA$. Using the fact that the angles of a triangle add up to $\displaystyle 180^{\circ}$, you can deduce that $\displaystyle \angle TAB = \angle TBA = 69^{\circ}$

    But: $\displaystyle \begin{aligned}\angle OAB & = \angle OAT - \angle TAB = 90^{\circ} - 69^{\circ} = 21^{\circ}\\ \angle OBA & = \angle OBT - \angle TBA = 90^{\circ} - 69^{\circ} = 21^{\circ} \end{aligned}$

    So we have all the required angles to find $\displaystyle AB$ and $\displaystyle TA$. To find $\displaystyle AB$, use the sine law for $\displaystyle \triangle OAB$. Then to find $\displaystyle TA$, use the sine law again for $\displaystyle \triangle TAB$ (you'll need $\displaystyle AB$ first).
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