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Chapter 4: Problem 21

Find the perimeter of a right triangle that has hypotenuse of length 6 and a \(40^{\circ}\) angle.

Short Answer

Expert verified
The perimeter of the right triangle is approximately 14.454 units.

Step by step solution

01

Calculate the length of the adjacent side using the cosine function

\ In a right triangle, the cosine function can be used to find the length of the adjacent side to the given angle. The formula is: \[cos(\theta) = \frac{adjacent}{hypotenuse}\] where 胃 is the angle measurement and in this case, 胃 = 40掳. We are given the hypotenuse, which is 6, and we want to find the length of the adjacent side. Rearranging the formula to solve for the adjacent side: \[adjacent = hypotenuse \cdot cos(\theta)\] Now, plug in the given values: \[adjacent = 6 \cdot cos(40^{\circ})\] Calculate the value: \[adjacent \approx 6 \cdot 0.766 = 4.596\] So, the length of the adjacent side is approximately 4.596 units.
02

Calculate the length of the opposite side using the sine function

\ To find the length of the opposite side, we can use the sine function: \[sin(\theta) = \frac{opposite}{hypotenuse}\] where 胃 is the angle measurement, which is 40掳 in this case. We are given the hypotenuse, which is 6, and we want to find the length of the opposite side. Rearranging the formula to solve for the opposite side: \[opposite = hypotenuse \cdot sin(\theta)\] Now, plug in the given values: \[opposite = 6 \cdot sin(40^{\circ})\] Calculate the value: \[opposite \approx 6 \cdot 0.643 = 3.858\] So, the length of the opposite side is approximately 3.858 units.
03

Calculate the perimeter of the triangle

\ Now that we have the length of all three sides, we can find the perimeter by adding the lengths together: \[perimeter = adjacent + opposite + hypotenuse\] Plug in the values: \[perimeter \approx 4.596 + 3.858 + 6\] Calculate the value: \[perimeter \approx 14.454\] The perimeter of the right triangle is approximately 14.454 units.

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