Find the square of cosine function based on the identity 4cosx-sin x=0.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have the identity 4*cos x - sin x = 0

4*cos x - sin x = 0

=> 4*cos x = sin x

square both the sides

=> 16*(cos x)^2 = (sin x)^2

=> 16*(cos x)^2 = 1 - (cos x)^2

=> 17*(cos x)^2 = 1

=> (cos x)^2 = 1/17

The value of square of the cosine is 1/17

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll apply one of the 3 forms of fundamental formula of trigonometry:

(tan x)^2 + 1 = 1/(cos x)^2

(cos x)^2 = 1/[(tan x)^2 + 1] (1)

Now, we'll find out tan x from the given constraint:

4cos x - sin x = 0

We'll isolate cos x to the left side:

4 cos x = sin x

We'll divide by cos x both sides to create tangent function:

sin x/cos x = 4

tan x = 4 (2)

We'll substitute (2) in (1):

(cos x)^2 = 1/[(4)^2 + 1]

(cos x)^2 = 1/(16 + 1)

(cos x)^2 = 1/17

The requested square of cosine is: (cos x)^2 = 1/17.

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