Using the info given, solve for the missing value of the right triangle. Round each angle to the nearest degree, each length to the nearest tenth, & each trigonometric value to four decimal places, if necessary.
tan θ = 0.7536
Hypotenuse = 29 miles
Opposite side = ?
To find the length of the opposite side, we can use the formula for tangent:
tan(θ) = opposite/adjacent
In this case, the angle θ is not given, but the value of tangent is provided as 0.7536. We can use the inverse tangent function (arctan) to find the angle:
θ = arctan(0.7536)
Using a calculator, we find that θ is approximately 36.8699 degrees.
Now, we can use the given angle and the length of the hypotenuse to find the length of the opposite side using the sine function:
sin(θ) = opposite/hypotenuse
Opposite side = sin(θ) * hypotenuse
Opposite side = sin(36.8699) * 29
Using a calculator, we find that the length of the opposite side is approximately 17.6 miles (rounded to the nearest tenth).
To solve for the missing value, we can use the tangent function, which is defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
Given:
tan θ = 0.7536
Hypotenuse = 29 miles
To find the length of the opposite side, we can use the formula:
opposite side = tangent (θ) * adjacent side
Since we only know the hypotenuse and opposite side, we need to find the adjacent side in order to calculate the length of the opposite side.
Using the Pythagorean theorem, we can find the length of the adjacent side:
adjacent side = √(hypotenuse^2 - opposite side^2)
Let's substitute the values into the formula:
adjacent side = √(29^2 - (opposite side)^2)
Now, we can solve for the length of the adjacent side and the length of the opposite side.
Let's start with finding the length of the adjacent side:
adjacent side = √(29^2 - (opposite side)^2)
Next, rearrange the equation to solve for the length of the opposite side:
(opposite side)^2 = (adjacent side)^2 - 29^2
(opposite side)^2 = (29^2 - (adjacent side)^2)
opposite side = √[(29^2 - (adjacent side)^2)]
Now, we can substitute the given tangent value into the equation:
opposite side = √[(29^2 - (adjacent side)^2)]
Using the given tangent value, we have:
0.7536 = opposite side / adjacent side
Squaring both sides of the equation, we get:
0.7536^2 = (opposite side)^2 / (adjacent side)^2
0.567 = (opposite side)^2 / (adjacent side)^2
Next, we can substitute (29^2 - (adjacent side)^2) for (opposite side)^2:
0.567 = [(29^2 - (adjacent side)^2)] / (adjacent side)^2
Next, we can multiply both sides by (adjacent side)^2 to get rid of the denominator:
0.567 * (adjacent side)^2 = 29^2 - (adjacent side)^2
0.567 * (adjacent side)^2 + (adjacent side)^2 = 29^2
Combine like terms:
(0.567 + 1) * (adjacent side)^2 = 29^2
(1.567) * (adjacent side)^2 = 29^2
Now, we can isolate (adjacent side)^2:
(adjacent side)^2 = 29^2 / 1.567
(adjacent side)^2 = 832.591518
Taking the square root of both sides, we get:
adjacent side = √832.591518
adjacent side ≈ 28.842
Now, we can substitute the obtained values into the formula to find the length of the opposite side:
opposite side = √(29^2 - (adjacent side)^2)
opposite side = √(29^2 - (28.842)^2)
opposite side ≈ √(29^2 - 840.042364)
opposite side ≈ √(841 - 840.042364)
opposite side ≈ √0.957636
opposite side ≈ 0.978
Therefore, the length of the opposite side is approximately 0.978 miles.