P and Q are two observation posts on the same horizontal ground asthe foot R of a vertical pole RT. P is 2.5km south of R. Q is east of R and is on a bearing of 065° from P. The angle of elevation of T from P is 40°.calculate, correct to three significant figures The angle of elevation of T from Q
To solve this problem, we can use trigonometry. Let's start by drawing a diagram.
We are given that P is 2.5 km south of R and Q is east of R. Let's assume that the distance between R and Q is d km. Now, let's draw the lines connecting P, R, and Q.
We are also given that the angle of elevation of T from P is 40°. This means that the angle between the ground and the line segment PT is 40°.
Let's call the angle of elevation of T from Q as θ. Now, we have a right angle triangle with the vertical side TR and the horizontal side RQ. We want to find the angle θ, which is the angle between the ground and the line segment TQ.
Using trigonometry, we can write:
tan(θ) = TR / RQ
We can find TR by using the angle of elevation of T from P and the distance between P and R.
TR = PR * tan(40°)
= 2.5 km * tan(40°)
Since we are given the bearing of Q from P, we can find the angle between PR and PQ by subtracting 65° from 180°.
Angle between PR and PQ = 180° - 65°
= 115°
Now, we can use this angle and the distance between P and Q to find RQ using the cosine rule.
RQ² = PR² + PQ² - 2 * PR * PQ * cos(115°)
Substituting the known values, we can calculate RQ.
Now, we can find tan(θ) by dividing TR by RQ.
tan(θ) = TR / RQ
Finally, we can calculate θ by taking the inverse tangent of tan(θ).
θ = arctan(tan(θ))
Now, we can proceed to calculate the values.
Let's find TR:
TR = 2.5 km * tan(40°)
≈ 1.90053 km
Now, let's calculate RQ:
RQ² = (2.5 km)² + d² - 2 * 2.5 km * d * cos(115°)
Now, we can substitute the value of RQ into the equation for tan(θ) and find θ:
tan(θ) = 1.90053 km / RQ
θ ≈ arctan(tan(θ))
Finally, we can calculate the value of θ using a calculator. Round the answer to three significant figures.
Therefore, the angle of elevation of T from Q is approximately ________ degrees.