Double Angle Formulas

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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.

sin2A=2sinAcosA

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.

cos2A=cos2A−sin2A

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:

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

Double‐Angle and Half‐Angle Identities

Special cases of the sum and difference formulas for sine and cosine yields what is known as the double‐angle identities and the half‐angle identities. First, using the sum identity for the sine,

sin 2α = sin (α + α)

sin 2α = sin α cos α + cos α sin α

sin 2α = 2 sin α cos α

Similarly for the cosine,

Using the Pythagorean identity, sin 2 α+cos 2α=1, two additional cosine identities can be derived.

and 

The half‐angle identities for the sine and cosine are derived from two of the cosine identities described earlier.

The sign of the two preceding functions depends on the quadrant in which the resulting angle is located.

Example 1: Find the exact value for sin 105° using the half‐angle identity.

In the following verification, remember that 105° is in the second quadrant, and sine functions in the second quadrant are positive. Also, 210° is in the third quadrant, and cosine functions in the third quadrant are negative. From Figure  1, the reference triangle of 210° in the third quadrant is a 30°–60°–90° triangle. Therefore, cos 210° = −cos 30°.

Figure  1  
                 Drawing for Example 1.

Using the half‐angle identity for sine,

Example 2: Find the exact value for cos 165° using the half‐angle identity.

In the following verification, remember that 165° is in the second quadrant, and cosine functions in the second quadrant are negative. Also, 330° is in the fourth quadrant, and cosine functions in the fourth quadrant are positive. From Figure 2 , the reference triangle of 330° in the fourth quadrant is a 30°–60°–90° triangle. Therefore, cos 330° = cos 30°.

Figure  2
                 Drawing for Example 2.

Using the half‐angle identity for the cosine,

Example 4: Verify the identity 1 − cos 2 x = tan x sin 2 x.

What is a Double Angle?

Double angle identities and formula are useful for solving certain integration problems where a double formula may make things much simpler to solve. Thus in math as well as in physics, these formulae are useful to derive many important identities. Notice that there are many listings for the double angle for sine, cosine, and tangent.

The Double Angle Formula

We may start by recalling the addition formulae of trigonometry ratios with two angles A and B. Students, are already knowing these.

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