What is Cos Square theta Formula?

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 pythagoras  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.