In /_\WXY,y=1.3 inches, x=1.1 inches and /_X=51^(@). Find all possible values of /_Y, to the nearest 10th of a degree. Answer:

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In  /_\WXY,y=1.3 inches,  x=1.1 inches and  /_X=51^(@). Find all possible values of  /_Y, to the nearest 10th of a degree. Answer:

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Apply Law of Sines: To find the possible values of /_Y, we can use the Law of Sines, which relates the ratios of the lengths of sides of a triangle to the sines of the opposite angles. The Law of Sines states that for any triangle ABC with sides a, b, and c opposite angles A, B, and C respectively, the following is true: (sin(A)/a) = (sin(B)/b) = (sin(C)/c) In our case, we have triangle WXY with sides w, x, and y opposite angles W, X, and Y respectively. We are given that side y = 1.3 inches, side x = 1.1 inches, and angle X = 51 degrees. We want to find angle Y. First, we will find the ratio for angle X and side x using the Law of Sines.Calculate Ratio for Angle X: We calculate the sine of angle X: sin(X) = sin(51 degrees) Using a calculator, we find that: sin(51 degrees) ≈ 0.7771 (rounded to four decimal places) Now we can set up the ratio for angle X and side x: (sin(X)/x) = (0.7771/1.1)Set Up Equation for Angle Y: Next, we calculate the ratio: (0.7771/1.1) ≈ 0.7065 (rounded to four decimal places) This ratio should be equal to the ratio of (sin(Y)/y) according to the Law of Sines.Calculate Sin(Y): Now we set up the equation for angle Y and side y using the Law of Sines: (sin(Y)/y) = 0.7065 (sin(Y)/1.3) = 0.7065 To find sin(Y), we multiply both sides by y (1.3 inches): sin(Y) = 0.7065 * 1.3Find Acute Angle Y: We calculate the value of sin(Y): sin(Y) = 0.7065 * 1.3 ≈ 0.91845 (rounded to five decimal places) However, we must be cautious here because the sine function can have two different angles that have the same sine value, one acute (less than 90 degrees) and one obtuse (between 90 and 180 degrees), within the range of 0 to 180 degrees which are the possible values for an angle in a triangle.Find Obtuse Angle Y: First, we find the acute angle Y that corresponds to the sine value we calculated: Y = arcsin(0.91845) Using a calculator, we find that: Y ≈ 66.4 degrees (rounded to the nearest 10th of a degree) This is the acute angle solution for angle Y.Check Validity of Acute Angle: To find the obtuse angle solution, we use the fact that sin(180 degrees - Y) = sin(Y). Therefore, the obtuse angle solution for Y is: Y_obtuse = 180 degrees - Y_acute Y_obtuse = 180 degrees - 66.4 degrees Y_obtuse ≈ 113.6 degrees (rounded to the nearest 10th of a degree)Check Validity of Obtuse Angle: We must now check if the obtuse angle solution is valid for a triangle. The sum of the angles in any triangle is 180 degrees. We already have one angle, X, which is 51 degrees. Adding the acute angle solution for Y, we have: 51 degrees + 66.4 degrees = 117.4 degrees This leaves us with: 180 degrees - 117.4 degrees = 62.6 degrees for angle W Since this is a valid angle measurement, the acute angle solution for Y is valid.Now we check the obtuse angle solution. Adding the obtuse angle solution for Y to angle X, we have: 51 degrees + 113.6 degrees = 164.6 degrees This leaves us with: 180 degrees - 164.6 degrees = 15.4 degrees for angle W Since this is also a valid angle measurement, the obtuse angle solution for Y is also valid.

In $\triangle \mathrm{WXY}, y=1.3$ inches, $x=1.1$ inches and $\angle \mathrm{X}=51^{\circ}$ . Find all possible values of $\angle \mathrm{Y}$ , to the nearest $10$ th of a degree. $\newline$ Answer:

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In $\triangle \mathrm{WXY}, y=1.3$ inches, $x=1.1$ inches and $\angle \mathrm{X}=51^{\circ}$ . Find all possible values of $\angle \mathrm{Y}$ , to the nearest $10$ th of a degree. $\newline$ Answer: