Heights and Distances

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An angle of elevation is the angle formed between the horizontal line and the line of sight when an observer looks at an object situated above the horizontal level. It is measured upward from the hori…

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An angle of depression is the angle formed between the horizontal line and the line of sight when an observer looks at an object situated below the horizontal level. It is measured downward from the h…

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You are measuring an angle of depression because you are looking downward from the horizontal line to observe an object (the car) below your eye level.

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The main trigonometric ratios are: • sin θ = opposite/hypotenuse • cos θ = adjacent/hypotenuse • tan θ = opposite/adjacent Tangent is most commonly used in heights and distances problems.

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Tangent ratio (tan θ = opposite/adjacent) is most useful because in heights and distances problems, we typically know one side (height or distance) and an angle, and need to find the other side. Tange…

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Step 1: Make all the possible right angles in the given problem. This involves identifying where right triangles can be formed in the given situation.

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When a figure involves more than one triangle, you should assume a common side (say x) connecting these right triangles and use this common side to set up equations for both triangles. This helps in c…

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Let height = h, initial distance = x From first position: tan 30° = h/x → h = x/√3 From second position: tan 45° = h/(x-20) → h = x-20 Solving: x/√3 = x-20 x - x√3 = -20√3 x(1-√3) = -20√3 x = 20√3/(√3…