A hi-res stock $ABC$ is such that $/AC/ = 15 cm$, and the $\Varangle BAC = 48^{\circ}$. Calculate $/BC/$ in meters, correct to 3 decimal places. [Hint: $100 cm = 1 m, \tan 48^{\circ} = 1.1106$, $\left.\operatorname{Sin} 48^{\circ} = 0.7431, \operatorname{Cos} 48^{\circ} = 0.6691\right]$
Answer (1)
Assume the triangle is a right triangle with ∠ C = 9 0 ∘ .
Use the tangent function to find BC : BC = A C × tan ( 4 8 ∘ ) = 15 × 1.1106 = 16.659 cm.
Convert BC to meters: BC = 100 16.659 = 0.16659 m.
Round to 3 decimal places: 0.167 m.
Explanation
Problem Analysis We are given a triangle A BC with ∣ A C ∣ = 15 cm and ∠ B A C = 4 8 ∘ . We need to find the length of side ∣ BC ∣ in meters, correct to 3 decimal places. We are also given the values of tan 4 8 ∘ = 1.1106 , sin 4 8 ∘ = 0.7431 , and cos 4 8 ∘ = 0.6691 . Since we don't have enough information to solve the triangle uniquely, we will consider two cases: when the triangle is a right triangle with ∠ C = 9 0 ∘ and when the triangle is a right triangle with ∠ B = 9 0 ∘ .
Case 1: Right triangle with angle C = 90 degrees Case 1: Assume ∠ C = 9 0 ∘ . In this case, we can use the tangent function to find the length of BC :
tan ( ∠ B A C ) = A C BC BC = A C × tan ( ∠ B A C ) BC = 15 × tan ( 4 8 ∘ ) BC = 15 × 1.1106 BC = 16.659 cm Now, we convert the length of BC from centimeters to meters: B C m e t ers = 100 B C c m B C m e t ers = 100 16.659 B C m e t ers = 0.16659 m Rounding to 3 decimal places, we get BC ≈ 0.167 m.
Case 2: Right triangle with angle B = 90 degrees Case 2: Assume ∠ B = 9 0 ∘ . In this case, we can use the sine function to find the length of BC :
sin ( ∠ B A C ) = A C BC BC = A C × sin ( ∠ B A C ) BC = 15 × sin ( 4 8 ∘ ) BC = 15 × 0.7431 BC = 11.1465 cm Now, we convert the length of BC from centimeters to meters: B C m e t ers = 100 B C c m B C m e t ers = 100 11.1465 B C m e t ers = 0.111465 m Rounding to 3 decimal places, we get BC ≈ 0.111 m.
Final Answer Since the problem does not specify whether the triangle is right-angled, and it does not provide enough information to uniquely determine the length of BC , we will assume that ∠ C = 9 0 ∘ . Therefore, the length of BC is approximately 0.167 meters.
Examples
Imagine you are designing a garden and need to calculate the length of a fence ( BC ) given the length of one side of the garden ( A C = 15 cm) and the angle between these two sides ( 4 8 ∘ ). By assuming a right triangle, you can quickly estimate the fence length using trigonometric functions. This helps in planning the layout and purchasing the necessary materials.
