Trigonometric Word Problems

Objective: Relate vertical rise and horizontal distance with tangent.

Prerequisite skills: right-triangle trigonometry and units. For a level sightline reference, tangent of the elevation angle equals vertical rise divided by horizontal distance. The rise is measured from the observer’s eye or instrument, not automatically from the ground. Common error: using the slanted sightline distance as the horizontal side. Restrictions and interpretation: Assume level ground and vertical objects only when stated; add observer height separately from the triangle’s vertical rise.

Objective: Add instrument height after finding the vertical rise.

If the instrument is h₀ above level ground, the triangle’s opposite leg measures only the rise from that instrument to the target. Add h₀ to obtain the total object height. Common error: forgetting the instrument height or adding it inside the tangent. Restrictions and interpretation: Assume level ground and vertical objects only when stated; add observer height separately from the triangle’s vertical rise.

Objective: Use sine when the known distance is a hypotenuse.

A ramp or line-of-sight distance forms the hypotenuse when paired with horizontal and vertical directions. The vertical rise is hypotenuse times sine of the angle with horizontal. Use cosine for horizontal reach. Common error: using tangent with a sloped distance as the adjacent side. Restrictions and interpretation: Assume level ground and vertical objects only when stated; add observer height separately from the triangle’s vertical rise. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.