Trigonometric Functions of a Right Triangle

Trigonometric Functions of a Right Triangle

Trigonometry by definition is the "measurement of the triangle" or "measuring triangles".This page focused on plane trigonometric functions which deals with the plane angles and triangles. When solving parts of a triangle's measurements, we should determine some given initials from the given triangle or the "five special triangles" might needed. This topic also deals with special functions of angles. In the provided picture above, it shows that angle FAG, made perpendiculars such as BAC, and DAE. These are called similar triangles. This also means that;

$\frac{AB}{AC}=\frac{AD}{AE}=\frac{AF}{AD}$

and

$\frac{AC}{AB}=\frac{AF}{AD}=\frac{AG}{AF}$

also

$\Delta ABC=\Delta ADE=\Delta AFG$

and so on...

Special note: The six trigonometric functions applies only to right triangle. Triangle with one side measure 90 degrees.

The six trigonometric functions

Relationships between an acute angle of a right triangle and its sides can be determined by its ratios,

see figure below;

Figure:

From the figure above, the six trigonometric functions are:

$sine {\color{Red} \theta} =\frac{opposite}{hypotenuse}$

$cosine {\color{Red} \theta} =\frac{adjacent}{hypotenuse}$

$tangent{\color{Red}\theta} =\frac{opposite}{adjacent}$

$cotangent{\color{Red}\theta} =\frac{adjacent}{opposite}$

$secant{\color{Red}\theta} =\frac{hypotenuse}{adjacent}$

$cosecant{\color{Red}\theta} =\frac{hypotenuse}{opposite}$

This does not mean "in order". These six trigonometric functions are permanent to its relationships. We wanna need to find the location of the "reference angle" as our basis for the "function". In the example here, we used theta (𝛉).

Illustrative Examples

1. Find the six trigonometric functions of the given triangle below.

Here, we use alpha (⍺) as our reference angle.It faces at letter "a", so as our opposite side. So the six trigonometric functions of this right triangle are found to be;

$sin{\color{Red} \alpha} =\frac{opp}{hyp}=\frac{a}{c}$ $cot{\color{Red}\alpha} =\frac{adj}{opp}=\frac{b}{a}$

$cos{\color{Red}\alpha} =\frac{adj}{opp}=\frac{b}{c}$ $sec{\color{Red}\alpha}=\frac{hyp}{adj}=\frac{c}{b}$

$tan{\color{Red}\alpha} =\frac{adj}{hyp}=\frac{a}{b}$ $csc{\color{Red}\alpha}=\frac{hyp}{opp}=\frac{c}{a}$

2. In the figure below, find the six trigonometric functions of both angle theta(𝜽) and alpha(α).

For angle theta(𝜽).

$sin{\color{Blue} \theta }=\frac{opp}{hyp}=\frac{4}{5}$ $cot{\color{Blue}\theta}=\frac{adj}{opp}=\frac{3}{4}$

$cos{\color{Blue} \theta}=\frac{adj}{hyp}=\frac{3}{5}$ $sec{\color{Blue}\theta} =\frac{hyp}{adj}=\frac{5}{3}$

$tan{\color{Blue}\theta}=\frac{opp}{adj}=\frac{4}{3}$ $csc{\color{Blue}\theta} =\frac{hyp}{opp}=\frac{5}{4}$

For angle alpha(α).

$sin{\color{Red}\alpha} = \frac{opp}{hyp}=\frac{3}{5}$ $cot{\color{Red}\alpha} = \frac{adj}{opp}=\frac{4}{3}$

$cos{\color{Red}\alpha} = \frac{adj}{hyp}=\frac{4}{5}$ $sec{\color{Red}\alpha} = \frac{hyp}{adj}=\frac{5}{4}$

$tan{\color{Red}\alpha} = \frac{opp}{adj}=\frac{3}{4}$ $csc{\color{Red}\alpha} = \frac{hyp}{opp}=\frac{5}{3}$

Notice that the relationships of two angles are co functions, they are sinα = cosθ.

Example problems:

1. In the figure below, find the six trigonometric functions of the given triangle.

Figure i

Answer: The figure is not a right triangle, so the six trigonometric functions cannot be applied.

2.) In figure ii below, find the six trigonometric functions of each angle of a triangle.

Figure ii

Answer:

$sin{\color{Red}\alpha} =\frac{3}{5}$ $cot{\color{Blue}\beta} =\frac{3}{4}$

$cos{\color{Red}\alpha} =\frac{4}{5}$ $sec{\color{Blue}\beta}=\frac{5}{3}$

$tan{\color{Red}\alpha}=\frac{3}{4}$ $csc{\color{Blue}\beta}=\frac{5}{4}$

$cot{\color{Red}\alpha}=$ answer $sin{\color{Blue}\beta}=$   answer

$sec{\color{Red}\alpha}=$ answer    $cos{\color{Blue}\beta}=$ answer

$csc{\color{Red}\alpha}=$ answer    $tan{\color{Blue}\beta}=$   answer

Given the figure iii below, find the sine, cosine, and tangent functions of the following right triangles from their given angles.

3.) Angle 𝛉

4.) Angle 𝞪

5.) Angle 𝞫

figure iii

Solution for # 3,Angle 𝛉

$sin{\color{Red} \theta}=\frac{6}{6.7}$
$cos{\color{Red} \theta}=\frac{3}{6.7}$

$tan{\color{Red} \theta}=\frac{6}{3}=2$

$cot{\color{Red} \theta} =$ answer

$sec{\color{Red} \theta} =$ answer

$csc{\color{Red} \theta} =$ answer

Solution for # 4,Angle 𝞪

$sin{\color{Blue} \alpha} =\frac{3}{5}$

$cos{\color{Blue} \alpha} =\frac{4}{5}$

$tan{\color{Blue} \alpha} =\frac{3}{4}$

$cot{\color{Blue} \alpha }=$ answer

$sec{\color{Blue} \alpha }=$ answer

$csc{\color{Blue} \alpha }=$ answer

Solution for # 3,Angle 𝞫

$sin{\color{DarkGreen} \beta} =\frac{3}{6.7}$

$cos{\color{DarkGreen} \beta} =\frac{6}{6.7}$

$tan{\color{DarkGreen} \beta} =\frac{3}{6}=\frac{1}{2}$

$cot{\color{DarkGreen} \beta} =$ answer

$sec{\color{DarkGreen} \beta} =$ answer

$csc{\color{DarkGreen}\beta} =$ answer

Practice Test

Given the triangle below,

Given triangle