On a right triangle, find the hypotenuse and opposite side of the adjacent side equals 6, and theta equals 30 degrees

Answer 1

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.