the leg opposite the 50 degree angle in a right triangle measures 8 meters. find the length of the hypotenuse
hypotenuse c = a/sinA = 8/sin50 = 10.44 (meters)
To find the length of the hypotenuse in a right triangle, given one leg and an angle, you can use the trigonometric function cosine. In this case, the leg opposite the 50-degree angle measures 8 meters. Let's label the hypotenuse as "h" and the other leg as "x".
Using the cosine function, we can write the equation as:
cos(50°) = x / h
Rearranging the equation, we get:
h = x / cos(50°)
Plugging in the value of x (8 meters) and evaluating the cosine function, we can find the length of the hypotenuse:
h = 8 / cos(50°)
Calculating the value of cos(50°) using a calculator, we find:
h ≈ 8 / 0.6428 ≈ 12.445 meters
Therefore, the length of the hypotenuse is approximately 12.445 meters.
To find the length of the hypotenuse in a right triangle, you can use the Pythagorean theorem. The theorem 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.
Let's label the sides of the right triangle as follows:
- The leg opposite the 50-degree angle as side A
- The leg adjacent to the 50-degree angle as side B
- The hypotenuse as side C
Given that side B (the leg opposite the 50-degree angle) measures 8 meters, and we need to find the length of side C (the hypotenuse).
Using the Pythagorean theorem, we have:
A^2 + B^2 = C^2
Substituting the known values:
A^2 + 8^2 = C^2
To solve for C, we need to find the value of A. Since the triangle is a right triangle, the sum of the interior angles is 180 degrees. Therefore, the remaining angle (other than 90 and 50 degrees) is 180 - 90 - 50 = 40 degrees.
Since the sides of a triangle opposite to its angles are proportional, we can use the following relationship:
A / sin(40 degrees) = 8 / sin(50 degrees)
Now, let's solve for A by rearranging the equation:
A = (8 * sin(40 degrees)) / sin(50 degrees)
Using a scientific calculator or trigonometric tables, we can find the approximate value of sin(40 degrees) and sin(50 degrees) and substitute them into the equation to find the value of A.
Once we have the value of A, we can substitute it back into the Pythagorean theorem equation to find the value of C by taking the square root of both sides:
C = sqrt(A^2 + 8^2)