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Solve contextual problems involving right triangles

Right-triangle contextual problems are modeled by identifying the right angle and a reference acute angle, then relating known and unknown side lengths with sine, cosine, tangent, or the Pythagorean theorem. The learner interprets quantities such as height, distance, and angle, distinguishes the hypotenuse from the opposite and adjacent sides relative to the reference angle, solves for a missing side or acute angle, and checks the reasonableness and precision of the result; non-right triangles, general trigonometric identities, and advanced angle measures are outside this scope.

Detailed Explanation: Solve contextual problems involving right triangles

A right-triangle context problem usually gives you:

  1. A right angle.
  2. A reference acute angle.
  3. Some known and unknown side lengths.

Use the reference angle to name the sides:

  • Hypotenuse: opposite the right angle.
  • Opposite: across from the reference angle.
  • Adjacent: next to the reference angle, but not the hypotenuse.

Then choose a trigonometric ratio:

sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\qquad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\qquad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Example

A student stands 3535 meters from the base of a tree. The angle of elevation from the ground to the top of the tree is 42∘42^\circ. How tall is the tree?

Assume the ground is level and the student’s viewing point is at ground level.

Step 1: Identify the triangle

The tree forms a vertical side, and the ground forms a horizontal side. These meet at a right angle.

Relative to the 42∘42^\circ angle:

  • The tree’s height is the opposite side.
  • The 3535-meter distance is the adjacent side.
  • The line from the student to the top of the tree is the hypotenuse.

We need the opposite side and know the adjacent side, so use tangent:

tan⁡θ=oppositeadjacent\tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Step 2: Substitute the known values

Let hh represent the height of the tree.

tan⁡42∘=h35\tan 42^\circ=\frac{h}{35}

Step 3: Solve for the height

Multiply both sides by 3535:

h=35tan⁡42∘h=35\tan42^\circ

Using a calculator in degree mode:

h≈35(0.9004)h\approx 35(0.9004) h≈31.5h\approx31.5

Step 4: State and check the answer

The tree is approximately

31.5 meters tall\boxed{31.5\text{ meters tall}}

This is reasonable because the angle is fairly large, 42∘42^\circ, so the tree’s height should be close to—but less than—the horizontal distance of 3535 meters. Always include the correct units and round to a sensible precision.

Learn by doing: Solve contextual problems involving right triangles

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    Trigonometry - Solve Side Lengths from Values

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