8 A tent pole measuring 2.1 metres high is secured by ropes in two directions. The ropes are held by pegs $\mathrm{A}$ and $\mathrm{B}$ at angles of $43^{\circ}$ and $39^{\circ}$, respectively, from the horizontal. The line from the base of the pole to peg $A$ is at right angles to the line from the base of the pole to peg B. Round your answers to 2 decimal places in these questions.
a Find the distance from the base of the tent pole to: i $\operatorname{peg} \mathrm{A}$
ii peg B
b Find the angle at peg B formed by peg A, peg B and the base of the pole.
Find the distance between peg A and peg B.
Answer
By solving the above equations, we find that the distance from the base of the tent pole to peg A is approximately \(\boxed{2.25}\) metres, to peg B is approximately \(\boxed{2.59}\) metres, the distance between peg A and peg B is approximately \(\boxed{3.43}\) metres, and the angle at peg B formed by peg A, peg B and the base of the pole is \(\boxed{90}\) degrees.
Step 1 ：Given the height of the tent pole is 2.1 metres and the angles from the horizontal to pegs A and B are 43 degrees and 39 degrees respectively, we can use trigonometry to find the distances from the base of the pole to pegs A and B. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. Therefore, we can set up the following equations: \(\tan(43) = \frac{2.1}{\text{distance to A}}\) and \(\tan(39) = \frac{2.1}{\text{distance to B}}\). Solving these equations will give us the distances to pegs A and B.
Step 2 ：To find the angle at peg B formed by peg A, peg B and the base of the pole, we can use the law of cosines. The law of cosines states that for any triangle with sides of lengths a, b, and c and an angle γ opposite side c, the following equation holds: \(c^2 = a^2 + b^2 - 2ab\cos(γ)\). We know the lengths of sides a and b (the distances to pegs A and B), and we can find the length of side c (the distance between pegs A and B) using the Pythagorean theorem. Solving for γ will give us the angle at peg B.
Step 3 ：Finally, we can use the Pythagorean theorem to find the distance between pegs A and B. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Therefore, we can set up the following equation: \(\text{distance between A and B}^2 = \text{distance to A}^2 + \text{distance to B}^2\). Solving this equation will give us the distance between pegs A and B.
Step 4 ：By solving the above equations, we find that the distance from the base of the tent pole to peg A is approximately \(\boxed{2.25}\) metres, to peg B is approximately \(\boxed{2.59}\) metres, the distance between peg A and peg B is approximately \(\boxed{3.43}\) metres, and the angle at peg B formed by peg A, peg B and the base of the pole is \(\boxed{90}\) degrees.
