Mục lục bài viết
Double Angle Formulas
Double angle formulas are used to express the trigonometric ratios of double angles (2θ) in terms of trigonometric ratios of single angle (θ). The double angle formulas are the special cases of (and hence are derived from) the sum formulas of trigonometry and some alternative formulas are derived by using the Pythagorean identities. Let us recall the sum formulas of trigonometry.
- 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)
What Are Double Angle Formulas?
We will derive the double angle formulas of sin, cos, and tan by substituting A = B in each of the above sum formulas. Also, we will derive some alternative formulas are derived using the Pythagorean identities. Here are the double angle formulas followed by the derivation of each formula.
Double Angle Formulas
The double angle formulas of sin, cos, and tan are,
- sin 2A = 2 sin A cos A (or) (2 tan A) / (1 + tan2A)
- cos 2A = cos2A – sin2A (or) 2cos2A – 1 (or) 1 – 2sin2A (or) (1 – tan2A) / (1 + tan2A)
- tan 2A = (2 tan A) / (1 – tan2A)
Double Angle Formulas Derivation
Let us derive the double angle formula(s) of each of sin, cos, and tan one by one.
Double Angle Formulas of Sin
The sum formula of sine function is,
sin (A + B) = sin A cos B + cos A sin B
When A = B, the above formula becomes,
sin (A + A) = sin A cos A + cos A sin A
sin 2A = 2 sin A cos A
Let us derive an alternate formula for sin 2A in terms of tan using the Pythagorean identity sec2A = 1 + tan2A.
Thus, the double angle formulas of sine function are
sin 2A = 2 sin A cos A (or) (2 tan A) / (1 + tan2A)
Double Angle Formulas of Cos
The sum formula of cosine function is,
cos (A + B) = cos A cos B – sin A sin B
When A = B, the above formula becomes,
cos (A + A) = cos A cos A – sin A sin A
cos 2A = cos2A – sin2A
Let us use this as a base formula to derive two other formulas of cos 2A using the Pythagorean identity sin2A + cos2A = 1.
(i) cos 2A = cos2A − (1 − cos2A) = 2cos2A – 1
(ii) cos 2A = (1- sin2A) – sin2A = 1 – 2sin2A
Now, we will derive the formula of cos 2A in terms of tan using the base formula.
Thus, the double angle formulas of the cosine function are:
cos 2A = cos2A – sin2A (or) 2cos2A – 1 (or) 1 – 2sin2A (or) (1 – tan2A) / (1 + tan2A)
Double Angle Formulas of Tan
The sum formula of tangent function is,
tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
When A = B, the above formula becomes,
tan (A + A) = (tan A + tan A) / (1 – tan A tan A) =(2 tan A) / (1 – tan2A)
Thus, the double angle formula of tan function is,
tan 2A = (2 tan A) / (1 – tan2A)
Let us see the applications of the double angle formulas in the section below.
Examples Using Double Angle Formulas
Example 1: If tan A = 3 / 4, find the values of sin 2A, cos 2A, and tan 2A.
Solution:
Since the value of tan A is given, we use the double angle formulas for finding each of sin 2A, cos 2A, and tan 2A in terms of tan.
Solution:
The double angle formula of cos is,
cos 2A = 2cos2A – 1 (or) 1 – 2sin2A
We will use these formulas to prove the given identity.
Answer: The given identity is proved.
Example 3: Use the double angle formulas to derive the formula for sin 3x.
Solution:
sin (3x) = sin (2x + x)
= sin 2x cos x + cos 2x sin x (using sin (A + B) = sin A cos B + cos A sin B)
= (2 sin x cos x) cos x + (1 – 2 sin2x) sin x (using the double angle formulas)
= 2 sin x cos2x + sin x – 2 sin3x
= 2 sin x (1 – sin2x) + sin x – 2 sin3x
= 2 sin x – 2 sin3x + sin x – 2 sin3x
= 3 sin x – 4 sin3x
Answer: sin 3x = 3 sin x – 4 sin3x.
FAQs on Double Angle Formulas
What Are Double Angle Formulas?
The double angle formulas are formulas in trigonometry that deals with the double angles of trigonometric functions. Some important double angle formulas are:
- sin 2A = 2 sin A cos A
- cos 2A = cos2A – sin2A
- tan 2A = (2 tan A) / (1 – tan2A)
How To Derive Double Angle Formulas?
We substitute A = B in the sum formulas of sin, cos, and tan functions to derive the double angle formulas. For a more and clear explanation, pls refer to the “Double Angle Formulas Derivation” section of this page.
What Are Applications of Double Angle Formulas?
The double angle formulas are used to find the values of double angles of trigonometric functions using their single angle values. For example, the value of cos 30o can be used to find the value of cos 60o. Also, the double angle formulas can be used to derive the triple angle formulas.
How To Use Double Angle Formulas in Integration?
We use the double integration formulas in integration. To understand this, consider the example below.
∫ sin x cos x dx = (1/2) ∫ (2 sin x cos x) dx
= (1/2) ∫ sin 2x dx
= (1/2) (- cos 2x / 2) + C (by using the substitution 2x = u)
= -1/4 cos 2x + C
Related articles
✅ Cos Inverse Formula ⭐️⭐️⭐️⭐️⭐
✅ Cos Theta Formula ⭐️⭐️⭐️⭐️⭐
✅ Cosec Cot Formula ⭐️⭐️⭐️⭐️⭐
✅ Cosine Formula ⭐️⭐️⭐️⭐️⭐
✅ Law of Cosines Formula ⭐️⭐️⭐️⭐️⭐