Double Angle Formulas
Double Angle Formulas are formulas in trigonometry to solve trigonometric functions where their angle is in the multiple of 2, i.e. in the form of (2θ). Double angle formulas are special cases of trigonometric formulas and are used to solve various types of trigonometric problems.
In this article, we explore double-angle identities, double-angle identity definitions, and double-angle identity formulas by deriving all double-angle formulas, providing insight into their importance and uses in trigonometry.
Table of Content
- What is Double Angle Formula?
- Double Angle Formulas
- Double Angle Identities Formulas
- Double Angle Formulas Derivation
- Examples Using Double Angle Formulas
- Practice Problems on Double Angle Formulas
- FAQs on Double Angle Formulas
What is Double Angle Formula?
Trigonometric formulae known as “double angle identities” define the trigonometric functions of double angles in terms of the trigonometric functions of the original angles. Numerous mathematical and engineering applications benefit from these identities. The identities for the sum and difference of angles lead to the identities of double angles.
Double Angle Formulas Definition
Double angle identities are trigonometric formulae that represent the angle (θ) sine, cosine, and tangent in terms of the angle (θ) sine, cosine, and tangent.
Double Angle Formulas
The table with double angle formulas is added below:
Double Angle Identities Trigonometry |
|
---|---|
Sine Double Angle Identities |
sin(2θ) = 2sin(θ)cos(θ) sin(2θ) = 2tan(θ) / [1 + tan2(θ)] |
Cosine Double Angle Identities |
cos(2θ)=cos2(θ)-sin2(θ) cos(2θ)=2cos2(θ)-1 cos(2θ)=1-2sin2(θ) cos(2θ) = [1 – tan2(θ)] / [1 + tan2(θ)] |
Tangent Double Angle Identities |
tan(2θ) = 2tan(θ) / [1-tan2(θ)] |
Double Angle Identities Formulas
For sine, cosine, and tangent, the primary double angle identities are as follows:
Double Angle Formulas of Sin
sin(2θ) = 2sin(θ)cos(θ)
sin(2θ) = 2tan(θ) / [1 + tan2(θ)]
Double Angle Formulas of Cos
cos(2θ) = cos2(θ) – sin2(θ)
cos(2θ) = 2cos2(θ) – 1
cos(2θ) = 1 – 2sin2(θ)
cos(2θ) = [1 – tan2(θ)] / [1 + tan2(θ)]
Double Angle Formulas of Tan
tan(2θ) = 2tan(θ) / [1 – tan2(θ)]
These equations define the trigonometric functions of double angles (2θ) in terms of the original angles’ (θ) trigonometric functions. In a variety of mathematical and engineering situations, they are helpful in decomposing trigonometric formulas and resolving issues with double angles.
Double Angle Formulas Derivation
Trigonometric formulae known as the “double angle identities” define the trigonometric functions of twice an angle in terms of the trigonometric functions of the angle itself. I’ll be obtaining the sine, cosine, and tangent double angle identities here.
Derivation of Sine Double Angle Formula
Sine Double Angle Identity:
sin(2θ) = 2 sinθ cosθ
Start with the sum-to-product identity for sine:
sin (A + B) =sin A cos B + cos A sin B
Let A = θ and B = θ
sin(2θ) = sin(θ+θ)
sin(2θ) = sinθ cosθ + cosθ sinθ
sin(2θ) = 2sinθ cosθ
Derivation of Cos Double Angle Formula
Cosine Double Angle Identity:
cos(2θ) = cos2(θ) – sin2(θ)
Start with the sum-to-product identity for cosine:
cos (A + B) = cos A cos B – sin A sin B
Let A = θ and B = θ
cos(2θ) = cos(θ+θ)
cos(2θ) = cosθcosθ – sinθ sinθ
cos(2θ) = cos2θ – sin2θ
Derivation of Tan Double Angle Formula
Tangent Double Angle Identity:
tan(2θ) = 2tanθ / [1 – tan2θ]
Use the quotient identity for tangent:
tan(A+B) = [tan A + tan B] / [1 – tan A tan B]
Let A=θ and B=θ
tan(2θ) = tan(θ+θ)
tan(2θ) = [tan(θ) + tan(θ)] / [1 – tan(θ)tan(θ)]
tan(2θ) = 2tan(θ) / [1 – tan2(θ)]
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Examples Using Double Angle Formulas
Example 1: Solve sin(2θ) = cos(θ) for θ
Solution:
sin(2θ) = cos(θ)
Using double angle identity for sine sin(2θ)=2sin(θ)cos(θ)), substitute:
2sin(θ)cos(θ) = cos(θ)
Now, divide both sides by cos(θ) (assuming cos(θ) ≠ 0
2sin(θ) = 1
Finally, solve for θ
sin(θ) = 1 /2
This implies θ = 30° or θ = 150°.
Example 2: Express tan(2x) in terms of tan(x):
Solution:
Using double angle identity for tangent
tan(2x) = 2tan(x) / {1 – tan2(x)}
This expression provides the tangent of twice the angle x in terms of the tangent of x.
Example 3: Use double angle identities to find the exact value of sin(120°)
Solution:
sin(2θ) = sin (240°)
Using, sin (180°+ θ) = – sin(θ)
We can rewrite expression,
-sin (60°) = – √3/2
Example 4: Prove the double angle identity for sine: sin(2θ) = 2sinθcosθ.
Solution:
Starting with (LHS)
sin(2θ) = sin(θ+θ)
sin(2θ) = sinθ cosθ + cosθ sinθ
Using trigonometric identity:
sin (a + b) = sin(a)cos(b) + cos(a)sin(b)
we get:
sin (θ + θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ)
Thus, LHS is equal to the right-hand side (RHS), and double angle identity for the sine is proved.
Practice Problems on Double Angle Formulas
Q1. Solve for sin(2θ) if sinθ = 3/5.
Q2. Express cos(2α) in terms of cos(α) if cos(α) = -4/7.
Q3. If tan(β) = 125, find the value of tan(2β).
Q4. Given that sin(ϕ) = 1/2 and ϕ is acute, determine cos(2ϕ).
Q5. Evaluate cot(2θ) if cotθ = -3/4.
FAQs on Double Angle Formulas
What is Double Angle Formula?
The identity which related with double angles of trigonometric functions is called double angle identity.
What is Double Angle Formula in Trigonometry?
Double angle formula of all trigonometry is given by:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos2(θ) -sin2(θ)
- cos(2θ) = 2cos2(θ) – 1
- cos(2θ) = 1 – 2sin2(θ)
- tan(2θ) = 2tan(θ) / [1 – tan2(θ)]
What is Sum Formula for a Double Angle?
Sum formula for a double angle is given by:
- sin (A + B) =sin A cos B + cos A sin B
- cos (A + B) = cos A cos B – sin A sin B
- tan(A+B) = [tan A + tan B] / [1 – tan A tan B]
How to Prove Double Angle Identities?
We can prove double angle identities using the sum-product identities of trigonometric functions.
What is the formula for the double angle of sin?
Sin 2x double angle formula is,
- sin 2x = 2 sin x cos x
- sin 2x = (2tan x)/(1 + tan2x)
What is the double angle formula for cos 2A?
There are multiple Cos2x Formulas that are:
- cos2x = 2cos2x – 1
- cos2x = cos2x – sin2x
- cos2x = 2cos2x – 1