From a Real Situation to a Right Triangle
Many real problems become straightforward once you translate them into a right-triangle model. The key is to identify: (1) a right angle (often the ground meets a wall, or horizontal meets vertical), (2) an angle you know or can measure, and (3) which side represents what physical quantity (height, distance, ramp length, etc.).
Common cues: “angle of elevation” usually means an angle measured up from horizontal; “angle of depression” is measured down from horizontal; “bearing” describes direction relative to north/south and can be converted into perpendicular (east/west) and parallel (north/south) components.
A Repeatable Problem-Solving Layout
1) Draw the diagram
Sketch the situation as a right triangle. Keep it simple: straight lines, a clear right angle, and label the angle given.
2) Mark known and unknown values
Write units on every labeled length (m, ft, km). Circle what you are solving for.
3) Choose the trig ratio that matches your sides
Decide which sides are involved relative to your chosen angle: opposite, adjacent, hypotenuse. Pick the ratio that uses the known side(s) and the unknown side.
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4) Set up an equation
Write the trig equation with the correct ratio and substitute the known values.
5) Solve
Use algebra to isolate the unknown. Be consistent with units. If you compute a length, it will have the same unit as the given length.
6) Interpret and check
- Reasonableness: angles near 0° produce tiny “opposite” compared to “adjacent”; angles near 90° produce a huge “opposite” compared to “adjacent.”
- Size check: the hypotenuse must be the longest side.
- Units: if you input meters, the output should be meters.
- Rounding: round at the end; keep extra digits during intermediate steps to reduce rounding error.
Context 1: Ramps (Slope Angle and Required Length)
Ramp problems usually involve a rise (vertical height), a run (horizontal distance), and the ramp surface (the slanted length). The slope angle is typically measured from the horizontal ground up to the ramp.
Example A: Find ramp length from rise and slope angle
A loading dock is 1.2 m above the ground. A ramp is built at a 6° angle above horizontal. How long must the ramp be (along its surface)?
- Diagram: right triangle with vertical rise = 1.2 m, ramp = hypotenuse, ground = adjacent, angle at ground = 6°.
- Known: opposite = 1.2 m, angle = 6°.
- Unknown: hypotenuse (ramp length).
- Choose ratio: sine relates opposite and hypotenuse.
sin(6°) = opposite / hypotenuse = 1.2 / L => L = 1.2 / sin(6°)Compute: L ≈ 1.2 / 0.104528 ≈ 11.48 m. Rounded: 11.5 m (to the nearest tenth).
Reasonableness check: 6° is small, so the ramp should be much longer than the rise. 11.5 m for a 1.2 m rise matches that expectation.
Example B: Find required horizontal run from rise and slope angle
A wheelchair ramp must rise 0.75 m at an angle of 4°. How much horizontal distance (run) is needed?
- Known: opposite = 0.75 m, angle = 4°.
- Unknown: adjacent (run).
- Choose ratio: tangent relates opposite and adjacent.
tan(4°) = opposite / adjacent = 0.75 / R => R = 0.75 / tan(4°)Compute: R ≈ 0.75 / 0.069927 ≈ 10.73 m. Rounded: 10.7 m.
Reasonableness check: 4° is even smaller than 6°, so the run should be very large compared to the rise. A run around 11 m for a 0.75 m rise is plausible.
Rounding guidance for ramps
- Construction contexts often round to the nearest centimeter (0.01 m) or nearest inch, but follow the problem’s instruction.
- Keep at least 3–4 decimal places in trig values during computation, then round at the end.
Context 2: Shadows (Height from Angle of Elevation)
Shadow problems typically use a vertical object (height), a horizontal shadow length, and a sun ray line (hypotenuse). The angle of elevation is measured from the ground up to the line of sight (or sun ray).
Example C: Find height from shadow length and angle of elevation
A flagpole casts a 18.0 ft shadow. The angle of elevation of the sun is 37°. Estimate the height of the flagpole.
- Diagram: right triangle with adjacent = 18.0 ft (shadow), opposite = height, angle at the tip of the shadow = 37°.
- Known: adjacent and angle.
- Unknown: opposite (height).
- Choose ratio: tangent relates opposite and adjacent.
tan(37°) = height / 18.0 => height = 18.0 * tan(37°)Compute: height ≈ 18.0 × 0.753554 ≈ 13.56 ft. Rounded: 13.6 ft.
Reasonableness check: 37° is moderate; height should be smaller than shadow length but not tiny. 13.6 ft vs 18 ft is reasonable.
Example D: Find distance from angle of elevation and height
You stand 45 m from the base of a building and measure the angle of elevation to the top as 52°. How tall is the building (ignoring your eye height)?
- Known: adjacent = 45 m, angle = 52°.
- Unknown: opposite (height).
- Choose ratio: tangent.
tan(52°) = H / 45 => H = 45 * tan(52°)Compute: H ≈ 45 × 1.279942 ≈ 57.60 m. Rounded: 57.6 m.
Reasonableness check: 52° is fairly steep; height should be larger than the horizontal distance times 1 (since tan(52°) > 1). 57.6 m is larger than 45 m, consistent.
Common modeling choices in elevation problems
- If the measurement is taken from eye level, the computed “opposite” is the height above eye level. Add eye height to get total height.
- Make sure the angle is measured from horizontal, not from vertical. If an angle is given from vertical, convert it: angle-from-horizontal = 90° − angle-from-vertical.
Context 3: Simple Navigation (Bearings to Components)
Navigation problems often ask you to break a movement into north/south and east/west components. This creates a right triangle where the path is the hypotenuse and the components are perpendicular legs.
Understanding bearings in a right-triangle model
- A bearing like N 30° E means: start facing north, then rotate 30° toward east.
- In that case, the north component is adjacent to the 30° angle (because the angle is measured from north), and the east component is opposite.
Example E: Components from a bearing
A boat travels 12.0 km on a bearing of N 25° E. Find how far north and how far east it traveled.
- Diagram: right triangle with hypotenuse = 12.0 km, angle at the north axis = 25°.
- Unknowns: north component (adjacent), east component (opposite).
- Choose ratios: cosine for adjacent, sine for opposite (relative to the 25° angle).
north = 12.0 * cos(25°) ≈ 12.0 * 0.906308 ≈ 10.88 km => 10.9 km (rounded)
east = 12.0 * sin(25°) ≈ 12.0 * 0.422618 ≈ 5.07 km => 5.1 km (rounded)Reasonableness check: 25° is closer to north than east, so the north component should be larger than the east component. It is.
Example F: Bearing from components
A hiker walks 3.0 km east and 7.0 km north. Find the straight-line distance and the bearing (as N θ E).
- Diagram: legs: east = 3.0, north = 7.0, hypotenuse = distance.
- Distance: use the Pythagorean theorem.
- Bearing angle: angle measured from north toward east, so use tangent with opposite = east and adjacent = north.
distance = sqrt(7.0^2 + 3.0^2) = sqrt(58) ≈ 7.62 km
tan(θ) = east / north = 3.0 / 7.0 => θ = arctan(3/7) ≈ 23.20°Rounded: distance 7.6 km, bearing N 23° E (to the nearest degree).
Reasonableness check: since north is more than twice east, the direction should be mostly north with a small tilt east; 23° fits.
Rounding and Reasonableness: Quick Rules You Can Apply Every Time
Rounding
- Keep intermediate values unrounded (or at least 4–5 significant digits) until the final step.
- Match the precision of the given data unless told otherwise (e.g., if distances are given to the nearest tenth, round your answer to the nearest tenth).
Reasonableness checks tied to angle size
- If the angle is very small (near 0°): opposite should be much smaller than adjacent; hypotenuse should be close to adjacent.
- If the angle is very large (near 90°): opposite should be much larger than adjacent; hypotenuse should be close to opposite.
- Hypotenuse must be the longest side; if your “hypotenuse” comes out shorter than a leg, re-check the ratio and labeling.
Scenario Mini-Quizzes (Choose the Ratio and Justify the Setup)
Mini-Quiz 1: Ramp length
A ramp must rise 0.90 m and is built at 5° above horizontal. You need the ramp’s surface length.
- Which sides (relative to 5°) are known and unknown?
- Which ratio will you use (sin, cos, or tan), and why?
- Write the equation you would solve for the ramp length.
Mini-Quiz 2: Tree height from shadow
A tree casts a 14.5 m shadow when the sun’s angle of elevation is 41°.
- Identify opposite/adjacent relative to the 41° angle.
- Select the correct ratio and justify it.
- Write the equation for the tree’s height (do not compute yet).
Mini-Quiz 3: Distance to a lighthouse
From a boat, the angle of elevation to the top of a lighthouse is 8°. The lighthouse is 30 m tall (ignore eye height). Find the horizontal distance from the boat to the lighthouse base.
- Which side is 30 m relative to the 8° angle?
- Which ratio matches the unknown horizontal distance?
- Write the equation you would solve.
Mini-Quiz 4: Bearing components
A drone flies 2.5 km on a bearing of S 60° E.
- In your diagram, which axis is the angle measured from?
- Which component is adjacent and which is opposite?
- Choose the ratios to compute the south and east components and write both equations.
Mini-Quiz 5: Choose the correct setup (no calculation)
Match each scenario to the correct equation form.
- A) Height from shadow: height = (shadow) · tan(angle)
- B) Ramp length from rise: length = (rise) / sin(angle)
- C) Horizontal distance from height and elevation: distance = (height) / tan(angle)
- D) North component from N θ E: north = (distance) · cos(θ)
Scenarios:
- 1) A 1.5 m step is climbed by a ramp at 7°. Find ramp length.
- 2) A building casts a 22 m shadow at 33°. Find building height.
- 3) A cliff is 80 m tall; angle of elevation to the top is 12°. Find how far you are from the base.
- 4) A ship travels 40 km on bearing N 15° E. Find how far north it traveled.
