On a right triangle, find the hypotenuse and opposite side of the adjacent side equals 6, and theta equals 30 degrees
To find the hypotenuse and opposite side of a right triangle when the adjacent side is 6 and theta is 30 degrees, we can use the trigonometric ratios.
Let's call the adjacent side "a", the opposite side "b", and the hypotenuse "c".
The given information is:
Adjacent side (a) = 6
Theta (angle between a and c) = 30 degrees
Using the trigonometric ratio for cosine, which relates the adjacent side and the hypotenuse, we have:
cos(theta) = adjacent/hypotenuse
cos(30 degrees) = 6/c
cos(30 degrees) = √3/2 (refer to the unit circle or use a calculator)
Now we can solve for the hypotenuse (c) using the equation:
√3/2 = 6/c
To isolate "c", we'll multiply both sides of the equation by c:
c * (√3/2) = 6
Multiply both sides of the equation by 2/√3 to get the value of c:
c = 6 * (2/√3)
c = 12/√3
c = (12√3)/3 (rationalizing the denominator)
So, the hypotenuse (c) is equal to 4√3.
Next, we can use the trigonometric ratio for sine, which relates the opposite side and the hypotenuse, to find the value of the opposite side (b).
sin(theta) = opposite/hypotenuse
sin(30 degrees) = b/(4√3)
1/2 = b/(4√3)
To isolate "b", multiply both sides of the equation by (4√3):
4√3 * (1/2) = b
2√3 = b
So, the opposite side (b) is equal to 2√3.