A right triangle has one side that measures 4 in. The angle opposite that side measures 80°. What is the length of the hypotenuse of the triangle? Round to the nearest tenth.
Answer (1)
We have a right triangle with a side of 4 inches and an opposite angle of 80 degrees.
Use the sine function to relate the opposite side and the hypotenuse: sin ( 8 0 ∘ ) = c 4 .
Solve for the hypotenuse c : c = s i n ( 8 0 ∘ ) 4 .
Calculate c and round to the nearest tenth: 4.1 in.
Explanation
Problem Analysis We are given a right triangle with one side measuring 4 inches and the angle opposite that side measuring 80 degrees. We need to find the length of the hypotenuse, rounded to the nearest tenth of an inch.
Set up the equation Let a be the length of the side opposite the given angle, and let c be the length of the hypotenuse. We have a = 4 inches and the angle A = 8 0 ∘ . We can use the sine function to relate the side and the hypotenuse: sin ( A ) = c a We want to find c , so we can rearrange the equation to solve for c :
c = sin ( A ) a
Plug in the values Now, we plug in the given values: a = 4 and A = 8 0 ∘ .
c = sin ( 8 0 ∘ ) 4 We know that sin ( 8 0 ∘ ) ≈ 0.9848 . Therefore, c = 0.9848 4 ≈ 4.0617 .
Round to the nearest tenth Rounding to the nearest tenth, we get c ≈ 4.1 inches.
Final Answer The length of the hypotenuse is approximately 4.1 inches.
Examples
Imagine you are building a ramp for skateboarding. You know the height the ramp needs to reach (4 inches) and the angle of the ramp (80 degrees). Using trigonometry, specifically the sine function, you can calculate the length of the ramp (the hypotenuse) needed to achieve that height and angle. This ensures your ramp is safe and functional for skateboarding.
