I just rlly don't have a clue how I should proceed with this question.

A picture is hanging 3 meters high and is hanging so the bottom of the picture is 1 meter above you eye level. How far from the wall on which the picture is hanging should you stand so that the angle of vision occupied by the picture is (pi/6)

I know that you draw a triangle which is divided into a 3m part and 1m part. The non-right 3m triangle has angle (P) (pi/6) and the right triangle with 1m has angle (P) as theta. I have to use compound angle identities. But how?

Let

x = the distance from the wall
a = angle of elevation of the bottom of the picture
b = angle subtended by the picture

Then we have

1/x = tan(a)
4/x = tan(a+b)

we want b = pi/6. So,

(tana + tanb)/(1-tana*tanb) = 4/x

Now just plug in 1/x for tan(a) and solve for x

To solve this question, you can use trigonometry and the concept of compound angle identities. Here's a step-by-step explanation on how to proceed:

1. Draw a diagram: Sketch a right-angled triangle with the vertical side representing the height of the picture (3m) and the horizontal side representing the distance you need to find.

2. Define variables: Let the distance from the wall be x.

3. Identify the angles: The angle formed between the horizontal and the hypotenuse of the triangle (eye level to the top of the picture) is π/6. The angle formed between the horizontal and the vertical side of the triangle is θ.

4. Apply trigonometric functions: In this case, you can use tangent to relate the angle θ and the distance x. Tangent is defined as the ratio of the opposite side to the adjacent side in a right triangle.

tangent(θ) = opposite/adjacent = 3/(x+1)

5. Simplify the equation: Multiply both sides of the equation by (x+1) to get rid of the fraction:

(x+1) * tangent(θ) = 3

6. Apply the compound angle identity for tangent: You mentioned using compound angle identities. The tangent of a sum or difference of two angles is given by:

tangent(A ± B) = (tangent(A) ± tangent(B))/(1 ∓ tangent(A) * tangent(B))

In this case, apply the compound angle identity to the left side of the equation:

((x+1) * tangent(θ))/(1 - tangent(θ)) = 3

7. Solve for x: Cross-multiply the equation to eliminate the denominator:

(x+1) * tangent(θ) = 3 - 3 * tangent(θ)

Expand the equation:

x * tangent(θ) + tangent(θ) = 3 - 3 * tangent(θ)

Rearrange the equation:

x * tangent(θ) = 3 - 4 * tangent(θ)

Divide both sides of the equation by tangent(θ) to isolate x:

x = (3 - 4 * tangent(θ))/tangent(θ)

8. Substitute the given angle: Replace tangent(θ) with its value, which you can determine using a calculator or trigonometric table based on the given angle, π/6.

x = (3 - 4 * tan(π/6))/tan(π/6)

Simplify the expression:

x = (3 - 4 * √(3)/3)/(√(3)/3)

x = (9 - 4√(3))/(√(3))

x ≈ 2.732 meters

So, you should stand approximately 2.732 meters away from the wall in order to have the picture occupy an angle of π/6 in your line of sight.