Answer :
To determine which reciprocal ratio represents the adjacent side over the opposite side in a right-angled triangle, we need to understand the definitions of the basic trigonometric functions and their reciprocals.
1. Sine (sin): The ratio of the opposite side to the hypotenuse.
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
2. Cosine (cos): The ratio of the adjacent side to the hypotenuse.
[tex]\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
3. Tangent (tan): The ratio of the opposite side to the adjacent side.
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Next, we look at the reciprocal functions:
1. Cosecant (csc): The reciprocal of sine.
[tex]\[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}} \][/tex]
2. Secant (sec): The reciprocal of cosine.
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} \][/tex]
3. Cotangent (cot): The reciprocal of tangent.
[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \][/tex]
Given these definitions, the reciprocal ratio that represents the adjacent side over the opposite side is cotangent.
Thus, the correct answer is:
O cotangent
1. Sine (sin): The ratio of the opposite side to the hypotenuse.
[tex]\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \][/tex]
2. Cosine (cos): The ratio of the adjacent side to the hypotenuse.
[tex]\[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \][/tex]
3. Tangent (tan): The ratio of the opposite side to the adjacent side.
[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
Next, we look at the reciprocal functions:
1. Cosecant (csc): The reciprocal of sine.
[tex]\[ \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\text{hypotenuse}}{\text{opposite}} \][/tex]
2. Secant (sec): The reciprocal of cosine.
[tex]\[ \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\text{hypotenuse}}{\text{adjacent}} \][/tex]
3. Cotangent (cot): The reciprocal of tangent.
[tex]\[ \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\text{adjacent}}{\text{opposite}} \][/tex]
Given these definitions, the reciprocal ratio that represents the adjacent side over the opposite side is cotangent.
Thus, the correct answer is:
O cotangent
