Introduction to trigonometry
Angles and measurement
Angle
An angle is formed by either two line segments or rays meeting at a common point. An angle is measured in degrees.
The following graphic contains several angles and their measurements in degrees.
Right triangle
A triangle which has an angle measuring 90^\circ is called a right triangle.
Parts of a right triangle
Hypotenuse of a right triangle
The hypotenuse of a right triangle is the side of the triangle which is opposite of the right angle.
Adjacent and opposite sides of a right triangle
Suppose \theta is an angle of a right triangle which is not a right angle. Then there are two sides of the triangle which form the angle \theta.
- The side of the triangle which forms the angle \theta but is not the hypotenuse of the triangle is called the adjacent side of the triangle relative to \theta.
- The side of the triangle which is not the hypotenuse nor the adjacent side relative to \theta is called the opposite side of the triangle relative to \theta.
The following graphic contains a right triangle with an angle labeled \theta. The adjacent side of the triangle relative to \theta is labeled as A.
The Pythagorean theorem and its converse
The pythagorean theorem
Pythagorean Theorem
Consider a right triangle with side measurements a, b and c (c being the length of the hypotenuse of the triangle). Then:
a^2+b^2=c^2
Consider the following right triangle:
Use the Pythagorean Theorem to solve for c :
c^2=3^2+4^2=25
c^2=25
c=5
The converse of the pythagorean theorem
Converse of the pythagorean theorem
Suppose a,b,c are the lengths of the three sides of a triangle. If a^2+b^2=c^2, then the triangle is a right triangle and the hypotenuse has length c.
We can use the converse of the Pythagorean Theorem to prove the following triangle is a right triangle.
Observe that:
5^2+12^2=25+144=169
And:
13^2=169
Therefore:
5^2+12^2=13^2
The triangle in the diagram is a right triangle with a hypotenuse of length 13.
Trigonometric ratios and properties
In this section we will introduce various functions, namely \cos\left(x\right), \sin\left(x\right), and \tan\left(x\right), which relate the sides of a right triangle with its hypotenuse.
Cos, sin and tan
Cosine and right triangles
Consider the right triangle in the following graphic:
Then:
\cos\left(\theta\right)=\dfrac{a}{c}
Consider the right triangle in the following graphic:
Then:
\cos\left(\theta\right)=\dfrac{4}{5}
Sine and right triangles
Consider the right triangle in the following graphic:
Then:
\sin\left(\theta\right)=\dfrac{b}{c}
Consider the right triangle in the following graphic:
Then:
\sin\left(\theta\right)=\dfrac{3}{5}
Tangent and right triangles
Consider the right triangle in the following graphic:
Then:
\tan\left(\theta\right)=\dfrac{b}{a}
Consider the right triangle in the following graphic:
Then:
\tan\left(\theta\right)=\dfrac{3}{4}
Cosines and sines can be used to solve for missing pieces of information in a right triangle.
To solve for the missing value of c we use:
\cos\left(30^\circ\right)=\dfrac{2}{c}
The calculator gives:
\cos\left(30°\right)=\dfrac{\sqrt{2}}{2}
Therefore:
\dfrac{\sqrt{2}}{2}=\dfrac{2}{c}
c=\dfrac{4}{\sqrt{2}}=2\sqrt{2}
Special right triangles and special measures
45^\circ - 45^\circ - 90^\circ triangle
A 45^\circ - 45^\circ - 90^\circ triangle is a right triangle whose three angles measure 45^\circ, 45^\circ, and 90^\circ.
The following graphic contains a 45^\circ - 45^\circ - 90^\circ triangle.
The two sides of a 45^\circ - 45^\circ - 90^\circ which are not the hypotenuse have the same length. If c is the length of the hypotenuse of a 45^\circ - 45^\circ - 90^\circ triangle and a is the measurement of one of the other sides of the triangle, then:
2a^2=c^2
Consider the following 45^\circ - 45^\circ - 90^\circ triangle:
Solve for the length of the hypotenuse:
c^2=2\left(2\right)^2=8
Since c is a positive number, we have:
c=2\sqrt{2}
30^\circ - 60^\circ - 90^\circ triangle
A 30^\circ - 60^\circ - 90^\circ triangle is a triangle whose angles have measurements of 30^\circ, 60^\circ, and 90^\circ.
Measurements of a 30^\circ - 60^\circ - 90^\circ triangle
Suppose we are given a 30^\circ - 60^\circ - 90^\circ right triangle as in the following graphic:
Then we have the following relations among the sides of the triangle:
- \dfrac{b}{c}=\dfrac{1}{2}
- \dfrac{a}{c}=\dfrac{\sqrt{3}}{2}
- \dfrac{b}{a}=\dfrac{\sqrt{3}}{3}
- \dfrac{a}{b}=\sqrt{3}
Consider the following 30^\circ - 60^\circ - 90^\circ triangle:
Solve for the length of the side adjacent to the 30^\circ angle:
\dfrac{a}{2}=\sqrt{3}\\a=2\sqrt{3}
Solving a right triangle
To solve a right triangle means to solve for all missing angles and missing side lengths of the right triangle. To solve a right triangle, we can use all of the above theorems, including the Pythagorean Theorem and the fact that the sum of all the angles in any triangle is equal to 180^\circ.
When solving a right triangle we often need to compute the values of \cos\left(\theta\right), \sin\left(\theta\right), and \tan\left(\theta\right) where \theta is an angle in degrees. All of these computations can be done with a calculator.
Consider the following triangle:
To solve for the right triangle we need to solve for the following missing values:
- The angle \beta
- The side length a
- The hypotenuse length c
To solve for \beta we use the fact that the sum of the three angles of a triangle are 180^\circ :
\beta +30^\circ +90^\circ=180^\circ\\\beta=60^\circ
To solve for the side length a we can use the measurements of a 30^\circ - 60^\circ - 90^\circ triangle and find:
\dfrac{a}{5}=\sqrt{3}\\a=5\sqrt{3}
To solve for the hypotenuse we can use Pythagorean Theorem:
c^2=\left(5\sqrt{3}\right)^2+5^2\\c^2=75+25\\c^2=100\\c=10
Therefore, we have solved the right triangle:
