Find All Solutions to the Equation Cos X

Trigonometry

Trigonometric Equations

Basic Trigonometric Equations

The equation cos x = a

Trigonometric equations

An equation that involves one or more trigonometric functions, of an unknown arc, angle or number, is called trigonometric equation.

Basic trigonometric equations

The equation cos x = a , -1 < a < 1

The solutions of the equation are arcs x whose function's value of cosine equals a .

Infinite many arcs whose cosine value equals a end in points, P

and P ′ , that are

x = a ^rad + k � 2 p and x′ = - a ^rad + k � 2 p , k � Z.

This is the set of the general solutions of the given equation.

For k = 0 follows the basic solutions of the equation

x ₀ = a ^rad and x ₀ ′ = - a ^rad .

Therefore, if cos x = a, - 1 < a < 1 then, x = + a ^rad + k � 2 p = + arccos a , k � Z.

For example if, a = - 1 , then, cos x = - 1 => x = p + k � 2 p , k � Z ,

a = 0 cos x = 0 => x = p /2 + k � p , k � Z ,

or a = 1 cos x = 1 => x = k � 2 p , k � Z.

Since cosine function passes through all values from range - 1 to 1 while arc x increases from 0 to p , one of the arcs from this interval must satisfy the equation cos x = a .

This arc, denoted x ₀ , we call the basic solution .

Thus, the basic solution of the equation cos x = a, - 1 < a < 1 is the value of inverse cosine function, x ₀ = arccos a or x ₀ = cos ^{- 1} a , that is, an arc or angle (whose cosine equals a ) between 0 and p which is called the principal value .

Scientific calculators are equipped with the arccos (or cos ^{- 1} ) function which, for a given argument between - 1 and 1 , outputs arc (in radians) or angle (in degrees) from the range x ₀ � [0, p ] .

Example: Solve the equation, cos x = - 0.5.

Solution: In the unit circle in the below figure shown are the two arcs, of which cosine value equals - 0.5 , that represent the basic solutions of the given equation

x ₀ = 120 � or x ₀ ′ = - 120 �

while the abscissas of the intersection points of the line y = - 0.5 with the graph of cosine function represent the set of the general solution

x = + 120 � + k � 360 � or x = + 2 p /3 + k � 2 p , k � Z.

The same results we obtain by using calculator if we set DEG then input

- 0.5 INV cos (or cos ^{- 1} ) =>x ₀ = 120 � and x ₀ ′ = - 120 � that are the basic solutions.

Or we input the same while calculator is set in RAD mode to get the arc in radians that is

x ₀ = 2.094395102 ^rad = 2 p /3 ^rad .

Trigonometry contents B

Find All Solutions to the Equation Cos X

Source: http://www.nabla.hr/TF-TrigFunEqu6.htm

Weston Ancingingen

Find All Solutions to the Equation Cos X

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