Finding the Value of Tan P Given sin P 3/5

When faced with the problem of finding the value of (tan P) given that (sin P frac{3}{5}) and (P) is an acute angle, we can use the principles of right triangle geometry and the Pythagorean Theorem to solve it. This guide will walk you through the steps to determine the tangent of angle (P).

Step-by-Step Solution

Understanding the Given Information

We are given (sin P frac{3}{5}). In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse. Hence, we can set up the triangle as follows:

(sin P frac{text{opposite}}{text{hypotenuse}} frac{3}{5})

Applying the Pythagorean Theorem

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Therefore, we can use this theorem to find the length of the adjacent side of the triangle.

Let's denote:

The length of the opposite side as (3) The length of the hypotenuse as (5) The length of the adjacent side as (x)

According to the Pythagorean Theorem:

(text{hypotenuse}^2 text{opposite}^2 text{adjacent}^2)

Substituting the known values:

(5^2 3^2 x^2)

Calculating the squares:

(25 9 x^2)

Solving for (x^2):

(x^2 25 - 9)

(x^2 16)

Taking the positive square root (since (P) is an acute angle and the adjacent side must be positive):

(x sqrt{16} 4)

Calculating tan P

Now that we have the lengths of all sides of the triangle, we can find the value of (tan P). The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. Therefore:

(tan P frac{text{opposite}}{text{adjacent}} frac{3}{4})

Visualizing the Solution

3-4-5 Triangle

A quick way to remember this solution is to use the 3-4-5 right triangle, a well-known Pythagorean triplet where (3^2 4^2 5^2). This visualization can simplify the problem and make the calculation more intuitive:

Hypotenuse: 5 Opposite: 3 Adjacent: 4

Using these values, we can directly apply the definition of tangent:

(tan P frac{3}{4})

Generalized Formulation

To generalize, if a right triangle has an opposite side of length 3 and a hypotenuse of length 5, then it has an adjacent side of length 4, since (3^2 4^2 5^2). Hence, the tangent of angle (P) is:

(tan P frac{3}{4})

Conclusion

In summary, by using the principles of right triangle geometry and the Pythagorean Theorem, we were able to determine that if (sin P frac{3}{5}), then (tan P frac{3}{4}). This problem can be a valuable learning tool for students working on trigonometric problems. Understanding these concepts can greatly enhance your problem-solving skills in trigonometry.

If you have any further questions or need more assistance with similar problems, definitely refer to the resources provided or consult with a mathematics tutor. Keep practicing and stay curious!