Understanding Trigonometric Ratios Given tan A 2/3

When we are given that tan A 2/3, it implies that in a right triangle, the ratio of the opposite side to the adjacent side is 2 to 3. This can be visualized where the opposite side is 2 units and the adjacent side is 3 units, leading to an angle A in a right triangle with these side lengths.

Step-by-Step Guide to Finding Trigonometric Ratios

First, let's determine the hypotenuse of the right triangle using the Pythagorean theorem:

Pythagorean Theorem

The Pythagorean theorem states that for a right triangle with sides a, b, and hypotenuse c, the relationship is c^2 a^2 b^2. Substituting the values of the opposite (2) and adjacent (3):

h^2 2^2 3^2 4 9 13

h sqrt{13}

Now that we have the hypotenuse, we can find all the other trigonometric ratios.

Calculating Sine and Cosine

- **Sine (sin A)**: The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.

sin A opposite / hypotenuse 2 / sqrt{13} ≈ 0.5547

- **Cosine (cos A)**: The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.

cos A adjacent / hypotenuse 3 / sqrt{13} ≈ 0.8321

Calculating Cosecant and Secant

- **Cosecant (csc A)**: The cosecant is the reciprocal of sine.

csc A 1 / sin A sqrt{13} / 2 ≈ 1.8028

- **Secant (sec A)**: The secant is the reciprocal of cosine.

sec A 1 / cos A sqrt{13} / 3 ≈ 1.5359

Calculating Cotangent

- **Cotangent (cot A)**: The cotangent is the reciprocal of tangent.

cot A 1 / tan A 3 / 2 1.5

Summary

Summarizing all the trigonometric ratios given that tan A 2/3: - sin A 2 / sqrt{13} - cos A 3 / sqrt{13} - tan A 2 / 3 - csc A sqrt{13} / 2 - sec A sqrt{13} / 3 - cot A 3 / 2

Quadrant Analysis

From the given information, we assumed that the angle A is in the first quadrant, where all trigonometric ratios are positive. However, if A is in the third quadrant, where both sine and cosine are negative, the results would be different but the ratios would remain the same in absolute values, just with a negative sign. - If A is in Quadrant I: - sin A 2 / sqrt{13} - cos A 3 / sqrt{13} - tan A 2 / 3 - csc A sqrt{13} / 2 - sec A sqrt{13} / 3 - cot A 3 / 2 - If A is in Quadrant III: - sin A -2 / sqrt{13} - cos A -3 / sqrt{13} - tan A 2 / 3 - csc A -sqrt{13} / 2 - sec A -sqrt{13} / 3 - cot A 3 / 2

Conclusion

Given that tan A 2/3, we can determine the values of all the trigonometric ratios. The ratios depend on the quadrant in which the angle A is located. Understanding these relationships helps in solving trigonometric equations and simplifying complex trigonometric expressions.