A right triangle with an angle of 31°31°31° has a hypotenuse of 101010. Calculate the side of the triangle opposite to the 31°31°31° angle (y), and the side adjacent to the 31°31°31° angle (x). Round your answer to 3 decimal places.

x = 8.572 , y = 5.150 x=8.572,y=5.150 x = 8.572 , y = 5.150

A right triangle with an angle of 31 ° 31° 31° has a hypotenuse of 10 10 10 . Calculate the side of the triangle opposite to the 31 ° 31° 31° angle ( y ), and the side adjacent to the 31 ° 31° 31° angle ( x ). Round your answer to 3 decimal places.

Let's give this problem a try. So we're told that a right triangle with an angle of 31 degrees has a hypotenuse of 10. Or as to calculate the side of the triangle opposite to the 31 degree angle, which is y, and the side adjacent to the 31 degree angle, which is x, and to round our answer to three decimal places. Now the best way to approach this problem is going to be to draw a picture first. So what we're gonna do is draw a right triangle. And on this right triangle, we're going to have an angle of 31 degrees, which we'll draw right here. Now it says that the adjacent side of this angle is going to be x, which we see, and we can see that the opposite side is going to be y. And And then this whole triangle has a hypotenuse of 10. Now what we're asked to do in this problem is solve for this x and this y. And we can just use these strategies of trigonometry and the Pythagorean theorem to solve this problem. Now recall SOHCAHTOA. SOHCAHTOA tells us how the trigonometric functions relate to each of the sides of the triangle. And what I'm first going to do is use the sine, because I know that the sine of our angle is going to be opposite over hypotenuse. Well, our angle we know is 31 degrees, and then the opposite side of 31 degrees, that's gonna be y. And then this is all divided by the hypotenuse, which is given to us as 10. So if I want to solve for y, I just need to multiply this 10 on both sides of the equation. That's going to get the tens tens to cancel on the right side, leaving us with just y is equal to 10 times the sine of 31 degrees. Now what you can do is take 10 times the sine of 31, and you can type that into a calculator. If you put this on a calculator, you should get a decimal approximation of 5.15 rounded to three decimal places. So this right here is our solution for y, this missing side. And keep in mind, you wanna make sure that your calculator is in degree mode when solving this problem because we are given degrees. Now next, what we need to do is find x. And the way that I can solve for x is by using the Pythagorean theorem. So we're going to have that x squared plus y squared is equal to 10 squared, because this is like a squared plus b squared equals c squared, where 10 is the hypotenuse. Now the x that we have here, this is the side we're trying to solve for. The y we just figured out is equal to 5.15, and then we have 10 squared on the right side of the equation as our hypotenuse. Now x squared, this remains the same because this is what we're solving for. But the 5.15 squared, this can actually be put into a calculator to get 26.526 approximately is equal to 10 squared, which is a hundred. And what you can do is take this 26.526, and you can subtract it on both sides of the equation. That is going to get the numbers to cancel on the left side, leaving us with just x squared. And then subtracting these two numbers on the right side should give you a value of 73.7 or excuse me, four seven four, and this is still rounded to three decimal places. Then all you need to do from here is take the square root on both sides of the equation, giving us that x is equal to the square root of this number, which should come out to about 8.572. So this is our solution for x. This is our solution for y, and that is how you can find all the missing sides of this right triangle. So hope you found this video helpful. Thanks for watching.

A right triangle with an angle of 31 ° 31° 31° has a hypotenuse of 10 10 10 . Calculate the side of the triangle opposite to the 31 ° 31° 31° angle ( y ), and the side adjacent to the 31 ° 31° 31° angle ( x ). Round your answer to 3 decimal places.

x = 8.572 , y = 5.150 x=8.572,y=5.150 x = 8.572 , y = 5.150

x = 5.150 , y = 8.572 x=5.150,y=8.572 x = 5.150 , y = 8.572

y = 5.000 , x = 8.660 y=5.000,x=8.660 y = 5.000 , x = 8.660

y = 8.660 , x = 5.000 y=8.660,x=5.000 y = 8.660 , x = 5.000

12. Trigonometric Functions

Solving Right Triangles

Business Calculus

12. Trigonometric Functions

Solving Right Triangles

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Multiple Choice

A right triangle with an angle of $31°$ has a hypotenuse of $10$ . Calculate the side of the triangle opposite to the $31°$ angle (y), and the side adjacent to the $31°$ angle (x). Round your answer to 3 decimal places.

$x=5.150,y=8.572$

$x=8.572,y=5.150$

$y=5.000,x=8.660$

$y=8.660,x=5.000$

Verified step by step guidance

Step 1: Recall the trigonometric relationships in a right triangle. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the hypotenuse, and the cosine of an angle is defined as the ratio of the length of the side adjacent to the angle to the hypotenuse. Mathematically, sin(θ) = opposite/hypotenuse and cos(θ) = adjacent/hypotenuse.

Step 2: Identify the given values. The angle θ is 31°, and the hypotenuse is 10. We need to calculate the side opposite the angle (y) and the side adjacent to the angle (x).

Step 3: Use the sine function to calculate the opposite side (y). Substitute the known values into the formula sin(31°) = y/10. Rearrange the equation to solve for y: y = 10 * sin(31°).

Step 4: Use the cosine function to calculate the adjacent side (x). Substitute the known values into the formula cos(31°) = x/10. Rearrange the equation to solve for x: x = 10 * cos(31°).

Step 5: Use a calculator to evaluate sin(31°) and cos(31°), then multiply by 10 to find the values of y and x. Round the results to three decimal places as specified in the problem.

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