A park ranger at point A sights a redwood tree at point B at an angle 23 degrees from a fire tower at point C, From the fire tower, the Angle between the ranger and the tree is 123 degrees. If the ranger at point A is 2.3 miles from the fire tower at point C, how far is it from the ranger to the redwood tree at point B?
Draw a triangle and use the law of sines. The angle at B (between ranger and fire tower) is B = 180 - 23 - 123 = 34 degrees. You want the distance from A to B and that is related to angle C as follows:
AB distance/sin C = AC distance/sin B
AB distance = (sin C/sin B)*2.3 miles
= (sin 123/sin34)2.3 = 3.45 miles
a flagpole casts a shadow of 12m.the sun has an angle of elevation of 36. how tall is the flagpole?
since you know the height of the shadow and an angle, you can use a trig function to find the height of the flagpole, assuming that the triangle formed is a right triangle, unless you know the Law of Sines and the Law of Cosines.
To find the distance from the ranger at point A to the redwood tree at point B, we can use trigonometry and the given information about the angles and distances. Let's break down the problem step by step:
Step 1: Draw a diagram of the situation to better visualize it. Place the ranger at point A, the fire tower at point C, and the redwood tree at point B.
Step 2: Label the distance between the ranger and the fire tower as "AC" and the distance between the fire tower and the redwood tree as "CB".
Step 3: Use the given information about the angles to identify two angles in the situation. Let's name the angle between the ranger and the tree as angle "α" and the angle between the ranger and the fire tower as angle "β".
Step 4: We have the value of angle β, which is 123 degrees. With this angle, we can determine the angle between the fire tower and the redwood tree, angle "θ". Since the sum of angles in a triangle is 180 degrees, we can calculate angle θ by subtracting the known angles β (123 degrees) and α (23 degrees) from 180 degrees.
θ = 180 degrees - β - α
θ = 180 degrees - 123 degrees - 23 degrees
θ = 34 degrees
Step 5: Now, we can use the law of sines to find the length of side CB, which is the distance from the ranger (point A) to the redwood tree (point B). The law of sines states that for any triangle:
a / sin(A) = b / sin(B) = c / sin(C)
In our case, we know side AC (2.3 miles), angle θ (34 degrees), and angle β (123 degrees). We want to find side CB, so we can write the equation as:
2.3 miles / sin(123 degrees) = CB / sin(34 degrees)
Step 6: Rearrange the equation to solve for CB:
CB = 2.3 miles * sin(34 degrees) / sin(123 degrees)
By plugging in these values into a calculator or using trigonometric tables, we can find the value of CB.
Step 7: Calculate the value of CB using the equation above to find the distance from the ranger at point A to the redwood tree at point B.