Finding Trigonometric Ratios in a Right-Angled Triangle

In this article, we will explore how to find the values of sin P, cos P, and tan P in a right-angled triangle PQR, where PQR is right-angled at Q. Given that PR 25 cm and PQ 5 cm, we will use the Pythagorean theorem to determine the lengths of the sides and then calculate the trigonometric ratios.

Understanding the Triangle

Consider a right-angled triangle PQR where PQR is right-angled at Q. This means that angle Q is 90 degrees, making PR the hypotenuse of the triangle. The given lengths are PR 25 cm and PQ 5 cm. We are tasked with finding the values of sin P, cos P, and tan P.

The Pythagorean Theorem and Side Calculations

To find the length of the third side, QR, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.

Step 1: Assume that QR x. Then, PR 25 - x.

The Pythagorean theorem can be written as:

[ PQ^2 QR^2 PR^2 ]

Substituting the known values:

[ 5^2 x^2 (25 - x)^2 ]

This simplifies to:

[ 25 x^2 625 - 5 x^2 ]

Subtracting (x^2) from both sides:

[ 25 625 - 5 ]

Further simplifying:

[ 5 600 ]

[ x 12 text{ cm} ]

Therefore, QR 12 cm and PR 25 - 12 13 cm.

Calculating Trigonometric Ratios

Now that we know the lengths of all sides, we can calculate the trigonometric ratios for angle P:

sin P

[ sin P frac{QR}{PR} frac{12}{13} ]

cos P

[ cos P frac{PQ}{PR} frac{5}{13} ]

tan P

[ tan P frac{QR}{PQ} frac{12}{5} ]

Summary of Results:

(sin P frac{12}{13}) (cos P frac{5}{13}) (tan P frac{12}{5})

Alternate Solution

Using the primitive Pythagorean triplet 5-12-13, we get:

QR 12 cm PR 13 cm

Thus, the trigonometric ratios are:

(sin P frac{QR}{PR} frac{12}{13}) (cos P frac{PQ}{PR} frac{5}{13}) (tan P frac{QR}{PQ} frac{12}{5})

Additional Method

Using the given equation:

[ PQR 25 text{ cm}, , PQ 5 text{ cm} ]

Let QR x. Then PR 25 - x.

The Pythagorean equation is:

[ 5^2 x^2 (25 - x)^2 ]

This simplifies to:

[ 25 x^2 625 - 5 x^2 ]

Subtracting (x^2) from both sides:

[ 25 625 - 5 ]

Solving for x:

[ 5 600 ]

[ x 12 text{ cm} ]

QR 12 cm, and PR 25 - 12 13 cm.

Thus, the trigonometric ratios are:

(sin P frac{QR}{PR} frac{12}{13}) (cos P frac{PQ}{PR} frac{5}{13}) (tan P frac{QR}{PQ} frac{12}{5})

Conclusion

In summary, knowing the sides of a right-angled triangle allows us to calculate its trigonometric ratios efficiently. Understanding the Pythagorean theorem is crucial for these calculations. The results for the trigonometric ratios in this triangle are:

(sin P frac{12}{13}) (cos P frac{5}{13}) (tan P frac{12}{5})

This method can be applied to similar problems involving right-angled triangles and trigonometric ratios.