Properties of Triangle

Side
Side of a triangle is a line segment that connects two vertices. Triangle has three sides, it is denoted by a, b, and c in the figure below.
 

Vertex
Vertex is the point of intersection of two sides of triangle. The three vertices of the triangle are denoted by A, B, and C in the figure below. Notice that the opposite of vertex A is side a, opposite to vertex B is side B, and opposite to vertex C is side c.
 

Derivation of Cosine Law

COMPLEX Mode - Ditch the COSINE LAW?

The following are the formulas for cosine law for any triangles with sides a, b, c and angles A, B, C, respectively.
 

$a^2 = b^2 + c^2 - 2bc\cos A$

$b^2 = a^2 + c^2 - 2ac\cos B$

$c^2 = a^2 + b^2 - 2ab\cos C$

 

Derivation of Sine Law

For any triangles with vertex angles and corresponding opposite sides are A, B, C and a, b, c, respectively, the sine law is given by the formula...
 

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

 

Derivation of the Half Angle Formulas

Half angle formulas can be derived from the double angle formulas, particularly, the cosine of double angle. For easy reference, the cosines of double angle are listed below:
 

cos 2θ = 1 - 2sin2 θ   →   Equation (1)
cos 2θ = 2cos2 θ - 1   →   Equation (2)

 

Note that the equations above are identities, meaning, the equations are true for any value of the variable θ. The key on the derivation is to substitute θ with ½ θ.
 

Derivation of Basic Identities

The derivation of basic identities can be done easily by using the functions of a right triangle. For easy reference, these trigonometric functions are listed below.

Right triangle with sides a, b, and c and angle thetaa/c = sin θ
b/c = cos θ
a/b = tan θ
c/a = csc θ
c/b = sec θ
b/a = cot θ

 

Functions of a Right Triangle

From the right triangle shown below,
 

Right triangle with sides a, b, and c and angle theta

 

the trigonometric functions of angle θ are defined as follows:

Summary of Trigonometric Identities

Derivation of Trigonometric Identities | Trigonometry

Basic Identities
Click here for the derivation of basic identities.

1. $\sin \theta = \dfrac{1}{\csc \theta}; \,\, \csc \theta = \dfrac{1}{\sin \theta}$
2. $\cos \theta = \dfrac{1}{\sec \theta}; \,\, \sec \theta = \dfrac{1}{\cos \theta}$
3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta}$
4. $\cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$
 

Derivation of Pythagorean Identities

Right triangle with sides a, b, and c and angle thetaIn reference to the right triangle shown and from the functions of a right triangle:
a/c = sin θ
b/c = cos θ
c/b = sec θ
c/a = csc θ
a/b = tan θ
b/a = cot θ
 

05 - 08 Number Problems in Maxima and Minima

Problem 5
The sum of two positive numbers is 2. Find the smallest value possible for the sum of the cube of one number and the square of the other.

Solution 5

 

01 - 04 Number Problems in Maxima and Minima

Problem 1
What number exceeds its square by the maximum amount?

Solution 1

 

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