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What is Cos Square theta Formula?

  • Last Updated : 07 Jan, 2022

The equations that relate the different trigonometric functions for any variable are known as trigonometric identities. These trigonometric identities help us to relate various trigonometric formulas and relationships with different angles. They are sine, cosine, tangent, cotangent, sec, and cosec. Here, we will look at the cos square theta formula.

According to the trigonometric identities, the cos square theta formula is given by

cos2θ + sin2θ = 1

where θ is an acute angle of a right-angled triangle.

Proof:

The trigonometric functions for any right angled triangle is defined as:

cosθ = base/hypotenuse

sinθ = altitude/hypotenuse

So, we can write

cos2θ + sin2θ = base2/hypotenuse2 + altitude2/hypotenuse2

Thus, cos2θ + sin2θ = (base2 + altitude2)/hypotenuse2

Applying pyhogorus theorem for right angled triangle, we get

base2 + altitude2 = hypotenuse2

Thus, we get

cos2θ + sin2θ = 1

Other than this, there are some generalized formulas derived using this property:

  • cos2θ = 1 – sin2θ
  • cos2θ = cos2θ – sin2θ
  • cos2θ = 2cos2θ – 1

Sample Problems

Question 1. Find the value of cosθ, given the value of sinθ, is 3/5.

Solution:

Given, the value of sinθ = 3/5

Using cos square formula, we get

cos2θ + sin2θ = 1

cos2θ = 1 – sin2θ = 1 – (3/5)2 = 1 – 9/25

cos2θ = 16/25

cosθ = √(16/25) = ± 4/5

Thus, the value of cosθ is ± 4/5.

Question 2. Find the value of cosθ, given the value of cosθ – sinθ = 1

Solution:

Given, cosθ – sinθ = 1.

or, cosθ = 1 + sinθ        —- (i)

Using cos square formula, we get

cos2θ + sin2θ = 1

cos2θ = 1 – sin2θ = (1 + sinθ)(1 – sinθ)

cos2θ = cosθ (1 – sinθ)

cosθ (cosθ – 1 + sinθ) = 0

So, we get two cases,

cosθ = 0

else, cosθ – 1 + sinθ = 0

or, cosθ = 1 – sinθ     —- (ii)

From eq.(i) and eq.(ii), we get

1 – sinθ = 1 + sinθ

2sinθ = 0

sinθ = 0

From eq.(i), we get cosθ = 1 + sinθ = 1 + 0 = 1

Thus, cosθ = 1

So, we get two possibilities. The value of cosθ is 0 or 1.

Question 3. If cosθ = 3/5, find the value of sin2θ – cos2θ.

Solution:

Given, the value of cosθ = 3/5

Now, using cos square formula, we can write

sin2θ – cos2θ = (1 – cos2θ) – cos2θ = 1 – 2cos2θ

Putting the value of cosθ = 3/5, we get

sin2θ – cos2θ = 1 – 2cos2θ = 1 – 2 × (3/5)2 = 1 – 2 × 9/25 =1 – 18/25 = 7/25

So, the answer is 7/25 .

Question 4. Find the value of cos2θ, given the value of cosθ = 1/2.

Solution:

Using the generalized formula,

cos2θ = 2cos2θ – 1

we can find the value of cos2θ, by substituting cosθ = 1/2

cos2θ = 2 × (1/2)2 – 1 = 2/4 – 1 = 1/2 – 1 = – 1/2

Thus, cos2θ = – 1/2.

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