In AQRS, the measure of ZS=90°, SR = 12, RQ = 37, and QS = 35. What ratio represents the cotangent of ZQ? R 37 12 S Q 35

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### Understanding Cotangent in Right Triangles

In \( \triangle QRS \), the measure of \( \angle S = 90^\circ \). The side lengths are given as follows: \( SR = 12 \), \( RQ = 37 \), and \( QS = 35 \). We are tasked with determining the ratio that represents the cotangent of \( \angle Q \).

### Triangle Diagram Explanation
The diagram is of a right triangle \( \triangle QRS \) where:
- \( \angle S \) is the right angle (\( 90^\circ \))
- \( SR = 12 \) is the length of the opposite side to \( \angle Q \)
- \( QS = 35 \) is the length of the adjacent side to \( \angle Q \)
- \( RQ = 37 \) is the hypotenuse of the triangle

The triangle is labeled with points \( R \) at the top vertex, \( S \) at the bottom left vertex (right angle), and \( Q \) at the bottom right vertex.

### Calculating Cotangent
The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side. For \( \angle Q \), the adjacent side is \( QS \) and the opposite side is \( SR \).

\[ \text{cot}(\angle Q) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{QS}{SR} = \frac{35}{12} \]

Thus, the ratio that represents the cotangent of \( \angle Q \) is \( \frac{35}{12} \).

This ratio can either be left as a fraction or converted into a decimal for approximation purposes, but in this context, the fraction is the preferred form due to its precise nature.

Students should be familiar with these concepts and apply similar methods to relate trigonometric ratios to different angles in right triangles. Understanding the cotangent function in different contexts is essential for more advanced studies in trigonometry and calculus.
Transcribed Image Text:### Understanding Cotangent in Right Triangles In \( \triangle QRS \), the measure of \( \angle S = 90^\circ \). The side lengths are given as follows: \( SR = 12 \), \( RQ = 37 \), and \( QS = 35 \). We are tasked with determining the ratio that represents the cotangent of \( \angle Q \). ### Triangle Diagram Explanation The diagram is of a right triangle \( \triangle QRS \) where: - \( \angle S \) is the right angle (\( 90^\circ \)) - \( SR = 12 \) is the length of the opposite side to \( \angle Q \) - \( QS = 35 \) is the length of the adjacent side to \( \angle Q \) - \( RQ = 37 \) is the hypotenuse of the triangle The triangle is labeled with points \( R \) at the top vertex, \( S \) at the bottom left vertex (right angle), and \( Q \) at the bottom right vertex. ### Calculating Cotangent The cotangent of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the opposite side. For \( \angle Q \), the adjacent side is \( QS \) and the opposite side is \( SR \). \[ \text{cot}(\angle Q) = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{QS}{SR} = \frac{35}{12} \] Thus, the ratio that represents the cotangent of \( \angle Q \) is \( \frac{35}{12} \). This ratio can either be left as a fraction or converted into a decimal for approximation purposes, but in this context, the fraction is the preferred form due to its precise nature. Students should be familiar with these concepts and apply similar methods to relate trigonometric ratios to different angles in right triangles. Understanding the cotangent function in different contexts is essential for more advanced studies in trigonometry and calculus.
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