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In triangle \( SPQ \), what is the included angle of \( QP \) and \( SP \)?

Answer :

The included angle Q between sides QP and SP in triangle SPQ is given by [tex]\(\arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right)\).[/tex]

In a triangle, the included angle between two sides refers to the angle formed by those two sides. In the case of triangle SPQ, the included angle of sides QP and SP is the angle formed by these two sides, which is angle Q.

If we denote the sides of the triangle as follows:

- Side SP as 'a'

- Side QP as 'b'

- Side SQ as 'c'

Then, according to the Law of Cosines, the relationship between these sides and the included angle is given by:

[tex]\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(Q) \][/tex]

In this formula, 'c' is the side opposite the included angle Q. To find the included angle Q, we rearrange the equation:

[tex]\[ \cos(Q) = \frac{a^2 + b^2 - c^2}{2ab} \][/tex]

By taking the inverse cosine (arccos) of this expression, we can find the measure of angle Q:

[tex]\[ Q = \arccos\left(\frac{a^2 + b^2 - c^2}{2ab}\right) \][/tex]

In summary, the included angle of sides QP and SP in triangle SPQ is given by the arccosine of the expression [tex]\(\frac{a^2 + b^2 - c^2}{2ab}\).[/tex]

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Rewritten by : Barada