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.
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