Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

0.3: Trigonometry

( \newcommand{\kernel}{\mathrm{null}\,}\)

The right triangle

Trigonometric functions

righttriangle2.jpg

Usually in a triangle when there is no chance of ambiguity, the sides opposite to each vertex have length denoted by the corresponding small letter.

righttriangle3.jpg

In the diagram, the triangle ABC has a right angle at B.

sin(A)=length of the opposite side length of the hypotenuse=ab.


cos(A)=length of the adjacent side length of the hypotenuse=cb.

tan(A)=length of the opposite side length of the length of the adjacent side=ac=sin(A)cos(A).

The remanining functions can be considered to be reciprocal functions.
csc(A)=1sin(A),

sec(A)=1cos(A),

cot(A)=1tan(A).

Solving the general triangle

Rules

Consider the triangle ABC in the diagram:

triangle2.jpg

1. The sine law:
asin(A)=bsin(B)=csin(C)


2. The cosine law:
a2=b2+c22bccos(A)
 or equivalently,
cos(A)=b2+c2a22bc

3. The area of the triangle ABC is 12bcsin(A).
4. If the perimeter of the triangle =2s=a+b+c, then sin(A)=2bcs(sa)(sb)(sc)

and the area of the triangle is s(sa)(sb)(sc).

Unit Circle

Definition

Unit circle is circle with center at the origin (0,0) and radius 1.
Unit circle has the equation x2+y2=1.

circle1.jpg

The radian measure of an angle is equal to the numerical value of the length of the arc of the unit circle from the point
(1,0) to the point (x,y).

circle2.jpg

Thus, length of the arc AP=α radian units.
Hence we obtain the conversion from degrees to radians and vice-versa:πradians=180

Definition of the trigonometric functions

There are two ways to define trigonometric functions:

Definition

Using triangles or circles.

1. The definition using triangles is too restrictive in that it treats only angles between 0 degree and 90 degrees.

2. The definition using circles, on the other hand, leads to a much more general definition, since the angle can now take on any real value whatsoever.

circle2.jpg

In the diagram above, A(1,0) is the fixed point from which measurements begin. Angles and arc lengths are measured from A in a counter-clockwise direction.

If P(x,y) is any point on the circumference of the unit circle, then if the arc length of AP=α units, then AOP=α radians, we define sin(α)=x andcos(α)=y.

Further, we define tan(α)=sin(α)cos(α)


cot(α)=cos(α)sin(α),

sec(α)=1cos(α),

csc(α)=1sin(α).

Some important relationships on unit circle

Rules

sin(πα)=sin(α).cos(πα)=cos(α).tan(πα)=tan(α).} Quadrant II

sin(π+α)=sin(α).cos(π+α)=cos(α).tan(π+α)=tan(α).} Quadrant III

sin(2πα))=sin(α)=sin(α).cos(2πα)=cos(α=cos(α).tan(2πα)=tan(α)=tan(α).} Quadrant IV

sin(π/2α)=cos(α).


cos(π/2α)=sin(α).


Basic identities

Basic identities

sin2(A)+cos2(A)=1.1+tan2(A)=sec2(A).1+cot2(A)=csc2(A).cos(A+B)=cos(A)cos(B)sin(A)sin(B).cos(AB)=cos(A)cos(B)+sin(A)sin(B).sin(A+B)=sin(A)cos(B)+cos(A)sin(B).sin(AB)=sin(A)cos(B)cos(A)sin(B).sin(2A)=2sin(A)cos(A).cos(2A)=cos2(A)sin2(A)=2cos2(A)1=12sin2(A).tan(A+B)=tan(A)+tan(B)1tan(A)tan(B).tan(AB)=tan(A)tan(B)1+tan(A)tan(B).tan(2A)=2tan(A)1tan2(A).

An app for trigonometric functions


This page titled 0.3: Trigonometry is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

Support Center

How can we help?