Trigonometry: Angles with sine, cosine and tangent
Triangles
A triangle is determined by three points in the plane that we connect to line segments. The points are called the vertices and the line segments 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 lower case #\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:
\[\gamma=180^\circ-\alpha-\beta=180^\circ-15^\circ-45^\circ=120^\circ\]
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