Solving Trigonometric Expressions in a Right Triangle

  Title: Solving Trigonometric Expressions in a Right Triangle Given the lengths of sides in a right triangle ( \triangle PQR ) where ( PQ = 12 , \text{cm} ), ( PR = 13 , \text{cm} ), and we need to find ( QR ). Applying Pythagoras Theorem According to the Pythagorean theorem in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Given: - ( PQ = 12 , \text{cm} ) - ( PR = 13 , \text{cm} ) Using the Pythagorean theorem: [ PR^2 = PQ^2 + QR^2 ] [ 13^2 = 12^2 + QR^2 ] [ 169 = 144 + QR^2 ] [ QR^2 = 25 ] Taking the square root of both sides: [ QR = 5 , \text{cm} ] Finding ( \tan{P} - \cot{R} ) To find ( \tan{P} - \cot{R} ), we need to determine the tangent of angle ( P ) and the cotangent of angle ( R ). Given: - ( PQ = 12 , \text{cm} ) - ( QR = 5 , \text{cm} ) Using trigonometric ratios: [ \tan{P} = \frac{PQ}{QR} = \frac{12}{5} ] [ \cot{R} = \frac{QR}{PQ} = \frac{5}{12} ] Substitute the values: [ \tan{P} - \cot{R} = \frac{12}{5} - \frac{5}{12} = \frac{144}{60} - \frac{25}{60} = \frac{119}{60} ] Therefore, ( \tan{P} - \cot{R} = \frac{119}{60} ) or approximately 1.9833. Conclusion In this problem, we used the Pythagorean theorem to find the missing side length ( QR ) in a right triangle. Furthermore, we calculated the difference between the tangent of angle ( P ) and the cotangent of angle ( R ) to obtain the value of ( \tan{P} - \cot{R} = \frac{119}{60} ). This process demonstrates the application of trigonometric concepts in solving geometric problems involving right triangles.    

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