Word problem: Wes saw fireworks at 28 degrees; Abby (who was standing 20 m away from him) saw the same fireworks at a 40 degree angle. Where was Abby standing? (The answer is 61.8m, but I don't know how to figure it out or sketch the triangle).
sketch a right-angled triangle.
On the horizontal base mark W for Wes.
let the fireworks be at P and let Q be the point below P, so that angle Q = 90°
let the angle at W be 28°
let A for Abby be a point between W and Q so that angle PAQ = 40° , and AW = 20
Look at triangle PWA , angle WAP = 180-40 = 140°
,
which makes angle WPA = 12°
By the Sine Law, we can find AP
AP/sin28 = 20/sin12
AP = 20sin28/sin12 = ....
Now in the right-angled triangle AQP
cos 43 = AQ/AP
where AQ is the distance that Abby is from the base of the fireworks.
To solve this word problem, we can use trigonometry and the properties of right triangles. Here's how you can find the answer:
1. Sketch a triangle: Begin by drawing a right triangle. Label one of the acute angles as 28 degrees and the other as 90 degrees (since it is a right triangle). Label one of the sides as 20 m (the distance Wes is standing away from the fireworks).
2. Identify the unknown side: Let's call the unknown side, the distance Abby is standing away from the fireworks, as 'x'. Label this side on your triangle.
3. Determine the trigonometric ratio to use: Since we have an angle and we want to find a side length, we should use the tangent function, which is defined as the opposite side (in this case, x) divided by the adjacent side (in this case, 20 m).
4. Setup the equation: The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, tan(40 degrees) = x/20.
5. Solve for x: To find x, cross-multiply and solve the equation. Multiply both sides of the equation by 20 to get: x = 20*tan(40 degrees).
6. Calculate the value of x: Use a calculator to find the tangent of 40 degrees, then multiply it by 20. The result is x = 61.8 m.
Therefore, Abby was standing approximately 61.8 meters away from the fireworks.