Question
Find 2 sides of a right triangle if the adjacent side is 6, and theta= 30 degrees
Answers
GPT 3.5
To find the other two sides of a right triangle with an adjacent side of 6 and an angle of 30 degrees, we can use the trigonometric functions.
The adjacent side (A), which is given, is 6.
The angle (θ), which is given, is 30 degrees.
Let's label the opposite side as O and the hypotenuse as H.
The trigonometric function that relates the adjacent side (A) and the hypotenuse (H) is cosine (cos).
So, we can use the formula: cos(θ) = A/H
Substituting the given values, we have: cos(30°) = 6/H
cos(30°) is equal to √3/2, so the equation becomes: √3/2 = 6/H
To solve for H, we can cross multiply: H * √3/2 = 6
Multiplying both sides by 2/√3: H = 6 * 2/√3
Simplifying: H = 12/√3 = (12/√3) * (√3/√3) = 12√3/3 = 4√3
Therefore, the hypotenuse (H) is 4√3.
To find the opposite side (O), we can use the Pythagorean theorem, which states that the sum of the squares of the two legs (O and A) is equal to the square of the hypotenuse (H).
Using the formula: H^2 = O^2 + A^2
Substituting the known values: (4√3)^2 = O^2 + 6^2
Simplifying: 48 = O^2 + 36
Rearranging and subtracting 36 from both sides: O^2 = 48 - 36 = 12
Taking the square root of both sides: O = √12 = √(4 * 3) = √4 * √3 = 2√3
Therefore, the opposite side (O) is 2√3.
To summarize:
The hypotenuse (H) is 4√3.
The opposite side (O) is 2√3.
The adjacent side (A), which is given, is 6.
The angle (θ), which is given, is 30 degrees.
Let's label the opposite side as O and the hypotenuse as H.
The trigonometric function that relates the adjacent side (A) and the hypotenuse (H) is cosine (cos).
So, we can use the formula: cos(θ) = A/H
Substituting the given values, we have: cos(30°) = 6/H
cos(30°) is equal to √3/2, so the equation becomes: √3/2 = 6/H
To solve for H, we can cross multiply: H * √3/2 = 6
Multiplying both sides by 2/√3: H = 6 * 2/√3
Simplifying: H = 12/√3 = (12/√3) * (√3/√3) = 12√3/3 = 4√3
Therefore, the hypotenuse (H) is 4√3.
To find the opposite side (O), we can use the Pythagorean theorem, which states that the sum of the squares of the two legs (O and A) is equal to the square of the hypotenuse (H).
Using the formula: H^2 = O^2 + A^2
Substituting the known values: (4√3)^2 = O^2 + 6^2
Simplifying: 48 = O^2 + 36
Rearranging and subtracting 36 from both sides: O^2 = 48 - 36 = 12
Taking the square root of both sides: O = √12 = √(4 * 3) = √4 * √3 = 2√3
Therefore, the opposite side (O) is 2√3.
To summarize:
The hypotenuse (H) is 4√3.
The opposite side (O) is 2√3.