The approximate length of the side adjacent to angle \(C\) is \(\boxed{{\mathbf{7}}.7{\mathbf{ inches}}}\) .
Further explanation:
The trigonometry ratio used in the right angle triangles.
The cosine ratio can be written as,
\(\cos \theta = \dfrac{length\ of\ the\ side\ adjacent\ to\ \theta }{hypotenuse}\)
Here, base is the length of the side adjacent to angle \(\theta\) and hypotenuse is the longest side of the right angle triangle where the length of side opposite to angle \(\theta\) is perpendicular that is used in the sine ratio.
Step by step explanation:
Step 1:
The attached right angle triangle can be observed from the given information.
First define the hypotenuse and the base of the triangle.
The side \(BC\) is adjacent to angle \(C\) and the side \(AC\) is the hypotenuse of \(\Delta ABC\) .
Therefore, the \({\text{length of the side adjacent to C}}=BC\) and \({\text{hypotenuse}}=10\) .
Step 2:
Since, the cosine ratio is \(\cos \theta = \dfrac{length\ of\ the\ side\ adjacent\ to\ \theta }{hypotenuse}\)
Now put the value \({\text{length of the side adjacent to C}}=BC\) and \({\text{hypotenuse}}=10\) in the cosine ratio.
\(\begin{aligned}\cos C&=\frac{{BC}}{{10}}\\{\text{co}}s40&=\frac{{BC}}{{10}}\\0.766&=\frac{{BC}}{{10}}\\BC&=7.66\\\end{aligned}\)
Therefore, the approximate length of the side adjacent to angle \(C\) is \(7.66{\text{ inches}}\) .
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Answer details:
Grade: High school
Subject: Mathematics
Chapter: Trigonometry
Keywords: Distance, Pythagoras theorem, base, perpendicular, hypotenuse, right angle triangle, units, squares, sum, cosine ratio, adjacent side to angle, opposite side to angle.
