In a right triangle the leg opposite to the acute angle of 30° is 7 in. Find the hypotenuse and other leg.
a = opposite leg
b = adjacent leg
c = hypotenuse
tan θ = opposite leg / adjacent leg
tan θ = a / b
sin θ = opposite leg / hypotenuse
sin θ = a / c
θ = 30° , sin θ = 1 / 2 , tan θ = √3 / 3 , a = 7 in
tan θ = a / b
√3 / 3 = 7 / b
Multiply both sides by 3
√3 = 21 / b
Multiply both sides by b
√3 b = 21
Divide both sides by √3
b = 21 / √3 = 7 ∙ 3 / √3 = 7 ∙ √3 ∙ √3 / √3 = 7√3
sin θ = a / c
1 / 2 = 7 / c
Multiply both sides by 2
1 = 14 / c
Multiply both sides by c
c = 14
adjacent leg = 7√3 in
hypotenuse = 14 in
Tan30 = Y/X = 7/X.
X = 7/Tan30 = 12.
sin30 = Y/r = 7/r.
r = 7/sin30 = 14. = hyp.
hypotomuse =7?
Well, in a right triangle, everything is always right. But let's get into the ACU-ally interesting part.
Using the sine of the angle of 30°, we can find the length of the hypotenuse. So, sin(30°) = opposite/hypotenuse. Plugging in the value of the opposite side, we get:
sin(30°) = 7/hypotenuse
Now, sin(30°) is 1/2, so we can rewrite the equation as:
1/2 = 7/hypotenuse
Cross-multiplying, we get:
hypotenuse = 7/(1/2)
hypotenuse = 7 * 2
hypotenuse = 14 inches
Now, to find the length of the other leg, we can use the Pythagorean theorem: a^2 + b^2 = c^2. Since we know the length of the hypotenuse (14 inches) and the length of one leg (7 inches), we can plug those values into the equation:
7^2 + b^2 = 14^2
49 + b^2 = 196
b^2 = 196 - 49
b^2 = 147
Taking the square root of both sides, we get:
b = √147
So, the length of the other leg is approximately 12.12 inches (rounded to two decimal places).
So, the hypotenuse is 14 inches and the other leg is approximately 12.12 inches. Now that's some triangular humor for you!
To find the hypotenuse and the other leg of the right triangle, we can use the trigonometric functions: sine, cosine, and tangent.
Given that the acute angle of 30° is opposite a leg of 7 in, we can use the sine function to find the hypotenuse.
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. So, in this case, we have:
sin(30°) = Opposite / Hypotenuse
Substituting the known values, we have:
sin(30°) = 7 / Hypotenuse
To find the hypotenuse, we can rearrange the equation:
Hypotenuse = 7 / sin(30°)
Calculating the value:
Hypotenuse = 7 / 0.5 = 14 in
So, the length of the hypotenuse is 14 in.
To find the length of the other leg, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Using this theorem, we have:
Hypotenuse^2 = Leg1^2 + Leg2^2
Substituting the known values, we have:
(14 in)^2 = (7 in)^2 + Leg2^2
Simplifying the equation:
196 in² = 49 in² + Leg2^2
Subtracting 49 in² from both sides:
Leg2^2 = 196 in² - 49 in²
Leg2^2 = 147 in²
Taking the square root of both sides:
Leg2 = √147 in
Simplifying the square root:
Leg2 ≈ 12.124 in
So, the length of the other leg is approximately 12.124 in.
In summary, the hypotenuse is 14 in and the other leg is approximately 12.124 in.