The equation $\sin \left(25^{\circ}\right)=\frac{9}{c}$ can be used to find the length of $\overline{ AB }$. What is the length of $\overline{ AB }$? Round to the nearest tenth.
Answer (1)
We have the equation sin ( 2 5 ∘ ) = c 9 .
Multiply both sides by c : c ⋅ sin ( 2 5 ∘ ) = 9 .
Divide both sides by sin ( 2 5 ∘ ) : c = s i n ( 2 5 ∘ ) 9 .
Calculate c and round to the nearest tenth: 21.3 in.
Explanation
Analyze the problem We are given the equation sin ( 2 5 ∘ ) = c 9 , which relates the sine of a 2 5 ∘ angle to the ratio of 9 and the length c of A B . Our goal is to find the value of c , rounded to the nearest tenth.
Multiply by c To solve for c , we first multiply both sides of the equation by c :
c ⋅ sin ( 2 5 ∘ ) = 9
Isolate c Next, we divide both sides by sin ( 2 5 ∘ ) to isolate c :
c = sin ( 2 5 ∘ ) 9
Calculate c Now, we calculate the value of c :
c = sin ( 2 5 ∘ ) 9 ≈ 0.4226 9 ≈ 21.2958
Round to the nearest tenth Finally, we round the value of c to the nearest tenth: c ≈ 21.3 Therefore, the length of A B is approximately 21.3 inches.
Examples
Imagine you're building a ramp for skateboarding. You know the angle of elevation should be 2 5 ∘ , and the vertical height of the ramp needs to be 9 inches. Using trigonometry, specifically the sine function, you can calculate the length of the ramp (the hypotenuse of the right triangle formed by the ramp, the ground, and the vertical height). This ensures your ramp is safe and functional for skateboarding.
