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Geometry and Trigonometry Difficulty: Hard

In the triangle shown, $R S = \sqrt{105}$ $upper R upper S equals StartRoot 105 EndRoot$ . What is the value of $sin R$ $\sin R$ ?

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Explanation

The correct answer is $\frac{52}{53}$ $StartFraction 52 Over 53 EndFraction$ . In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs. The length of the hypotenuse of the right triangle shown is $53$ $53$ . It’s given that $R S = \sqrt{105}$ $upper R upper S equals StartRoot 105 EndRoot$ . Therefore, the length of one of the legs of the triangle shown is $\sqrt{105}$ $StartRoot 105 EndRoot$ . Let $x$ $x$ represent $T S$ , the length of the other leg of the triangle shown. Therefore, $53^{2} = {(\sqrt{105})}^{2} + x^{2}$ , or $2,809 = 105 + x^{2}$ . Subtracting $105$ from both sides of this equation yields $2,704 equals x squared$ . Taking the positive square root of both sides of this equation yields $52 equals x$ . Therefore, $T S$ , the length of the other leg of the triangle shown, is $52$ . The sine of an acute angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The length of the leg opposite angle $R$ is $52$ , and the length of the hypotenuse is $53$ . Therefore, the value of $\sin R$ is $StartFraction 52 Over 53 EndFraction$ . Note that 52/53 or .9811 are examples of ways to enter a correct answer.