Final answer:
The ratios for sin A, cos A, and tan A in a right triangle are found using the lengths of the triangle's sides. They are given by the formulas sin A = opposite/hypotenuese, cos A = adjacent/hypotenuese, and tan A = opposite/adjacent. Values should be simplified to their lowest terms.
Step-by-step explanation:
To find the ratios for sin A, cos A, and tan A for a given right triangle, we use the definitions of these trigonometric functions based on the sides of the triangle. For a right triangle with an angle A, the sine of angle A (sin A) is the ratio of the length of the side opposite to A (often referred to as 'opposite') to the length of the hypotenuse. The cosine of angle A (cos A) is the ratio of the length of the side adjacent to A (often referred to as 'adjacent') to the hypotenuse. The tangent of angle A (tan A) is the ratio of the length of the opposite side to the length of the adjacent side.
For example, if in a right triangle, the lengths of the adjacent side (Ax), the opposite side (Ay), and the hypotenuse (A) are known, then:
- sin A = Ay / A
- cos A = Ax / A
- tan A = Ay / Ax
The values for sin A, cos A, and tan A should be given in simplified form, either as integers or simplified fractions, depending on the specific lengths of the sides of the provided triangle.
The Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (A) is equal to the sum of the squares of the lengths of the other two sides (Ax and Ay), can be used to verify that the values for sin A, cos A, and tan A are consistent with the triangle's dimensions.
It is important to reduce the ratios to their simplest form in order to answer the student's question accurately.