Trigonometry: Angles with sine, cosine and tangent
Triangles
A triangle is determined by three points in the plane that we connect with line segments. The points are called the vertices, and the line segments are the sides of the triangle.
- The vertices are denoted by upper case letters, for example #\blue A#, #\green B# and #\orange C#.
- The length of the side #BC#, the line segment between vertex #B# and vertex #C#, is denoted by lower case letters, such as #\blue a#, #\green b# and #\orange c#.
- We indicate the size of the angles with Greek letters, such as #\blue \alpha#, #\green \beta#, #\orange \gamma#.
The size of an angle #\blue A# is the corresponding letter in the Greek alphabet #\blue \alpha#. The side opposite of angle #\blue A# gets a lowercase #\blue a#.
A triangle with a right angle is called a right-angled triangle.
The sum of the three angles of a triangle is equal to #180^\circ#:
\[\blue \alpha + \green \beta + \orange \gamma = 180 ^\circ \]
This means that if we know the size of two angles of a triangle, we can calculate the third angle.
Let's say we know the angles #\green \beta# and #\orange \gamma#. We can calculate #\blue \alpha# with the formula:
\[\blue \alpha=180^\circ-\green \beta-\orange \gamma\]
What is the measure of angle #\gamma#?
The sum of the three angles of a triangle is equal to #180^\circ#.
Therefore:
\[\gamma=180^\circ-\alpha-\beta=180^\circ-44^\circ-85^\circ=51^\circ\]
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