Right Triangle Trigonometry: Apply It

  • Use right triangles to evaluate trigonometric functions.
  • Use cofunctions of complementary angles.
  • Use the deļ¬nitions of trigonometric functions of any angle.
  • Use right triangle trigonometry to solve applied problems.

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye.

angle of elevation or depression

Diagram of a radio tower with line segments extending from the top and base of the tower to a point on the ground some distance away. The two lines and the tower form a right triangle. The angle near the top of the tower is the angle of depression. The angle on the ground at a distance from the tower is the angle of elevation.

The angle of depression and the angle of elevation have the same measure thanks to alternate interior angles formed by parallel lines.
How To: Given a tall object, measure its height indirectly.

  1. Make a sketch of the problem situation to keep track of known and unknown information.
  2. Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
  3. At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
  4. Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
  5. Solve the equation for the unknown height.
To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of [latex]57^\circ[/latex] between a line of sight to the top of the tree and the ground. Find the height of the tree.A tree with angle of 57 degrees from vantage point. Vantage point is 30 feet from tree.

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of [latex]\frac{5\pi }{12}[/latex] with the ground? Round to the nearest foot.