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How do you know which method to use to solve triangles?

Last updated date: 20th Jun 2024
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Hint: To solve a triangle means, finding the length of its sides and measure of its angle. We can solve the triangle by using either Law of Sines which states that \[\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\]. Or Law of Cosine which states that \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}-2(bc)\cos A\], here \[A,B\And C\]are the angles of the triangle and \[a,b\And c\] are the length of sides opposite to them respectively.

Complete step by step answer:
To solve a triangle, we can use the Sine rule or Cosine rule. It depends on the given components of the triangle that is the number of sides or number of angles we know, which rule should be used.
We will now see the condition for which the Sine rule should be used to solve a triangle, see the following figure,
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As we can see from the above \[\Delta ABC\], \[\angle C={{33}^{\circ }},\angle B={{110}^{\circ }}\And b=20\]. We know the measure of 2 angles and the length of 1 side of the triangle. We know that the sum of all angles of a triangle equals to \[{{180}^{\circ }}\]. Using this we can find the measure of the third angle as,
  & \Rightarrow \angle A={{180}^{\circ }}-\angle C-\angle B \\
 & \Rightarrow \angle A={{180}^{\circ }}-{{33}^{\circ }}-{{110}^{\circ }} \\
 & \Rightarrow \angle A={{37}^{\circ }} \\
Now, from the figure we have the measure of \[\angle B\] and length of side \[b\]. We can find the value of \[\dfrac{\sin B}{b}\] and use it in the Sine rule \[\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\]. The remaining components can be found using this relation.
We will see the condition for which the Cosine rule should be used to solve a triangle, see the figure given below,
seo images

From the above figure, we can say that \[\angle O={{43}^{\circ }},m=30cm\And n=20cm\]. We know the measure of an angle and the length of sides opposite to the other two angles. Thus, we can easily use the Cosine rule for this triangle to find the other components.

There can be other conditions also where a particular rule should be used other than these two. If one of the angles is a right angle then the Cosine rule becomes \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\] which is called the Pythagoras theorem.