Solutions of Right Triangles
Solutions of Right Triangles
Triangles are made up of three line segments. They meet to form three angles. The
sizes of the angles and the lengths of the sides are related to one another.
If you know the size (length) of three out of the six parts of the triangle (at least one
side must be included), you can find the sizes of the remaining sides and angles.
If the triangle is a right triangle, you can use simple trigonometric ratios to find the
In a general triangle (acute or obtuse), you need to use other techniques, including
the law of cosines and the law of sines. You can also find the area of triangles by
using trigonometric ratios.
All triangles are made up of three sides and three angles. If the three angles of the
triangle are labeled ∠ A, ∠ B and ∠ C, then the three sides of the triangle should be
labeled as a, b, and c.
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Figure 1 illustrates how lowercase letters are used to name the sides of the triangle
that are opposite the angles named with corresponding uppercase letters.
If any three of these six measurements are known (other than knowing the measures
of the three angles), then you can calculate the values of the other three
The process of finding the missing measurements is known as solving the triangle. If
the triangle is a right triangle, then one of the angles is 90°.
Therefore, you can solve the right triangle if you are given the measures of two of the
three sides or if you are given the measure of one side and one of the other two
Example 1: Solve the right triangle shown in Figure 1 (b) if ∠ B = 22°
Because the three angles of a triangle must add up to 180°, ∠ A = 90 ∠ B thus ∠ A
Example 2: Solve the right triangle shown in Figure 1 (b) if b = 8 and a = 13.
You can use the Pythagorean theorem to find the missing side, but trigonometric
relationships are used instead. The two missing angle measurements will be found
first and then the missing side.
In many applications, certain angles are referred to by special names. Two of these
special names are angle of elevation and angle of depression. The examples shown
in Figure 2 make use of these terms.
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Example 3: A large airplane (plane A) flying at 26,000 feet sights a smaller plane
(plane B) traveling at an altitude of 24,000 feet. The angle of depression is 40°. What
is the line of sight distance ( x) between the two planes?
Example 4: A ladder must reach the top of a building. The base of the ladder will be
25′ from the base of the building. The angle of elevation from the base of the ladder to
the top of the building is 64°. Find the height of the building (h) and the length of the
ladder ( m).
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