Double And Half-Angle Identities

Introduction to double and half-angle identities

Double-angle and half-angle identities are special instances of the sum and distinct formulas for sine and cosine. First, using the sine's sum individuality,

sin 2α = sin (α + α)

sin 2α = sin α cos α + cos α sin α

sin 2α = 2 sin α cos α

Likewise, consider the cosine.

cos 2α = cos (α + α)

cos 2α = cos α cos α - sin α sin α

cos 2α = cos² α - sin² α

How to use Double-Angle Formulas to Find Exact Values?

In this answer, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles in this concept. The Double-Angle (2a) Formulas will be introduced here.

Double-Angle (2a) Formulas:

cos 2α = cos² α - sin² α

            = 2 cos² α – 1

            = 1 - sin² α

sin 2α = 2 sin α cos α

tan 2α = 2 tan α/1- tan² α

For cos 2α and tan α/2, any formula can be used to resolve for the exact value.

Let us find the exact value of cos π/8.

π/8 is half of π/4 and in the first quadrant.

cos (1/2 * π/4) = √1 + cos π/4 / 2

                        = √1 + √2/2 / 2

                        = √1/2 * 2 + √2/2

                        = √2 + √2 / 2

Now, let us find the exact value of sin 2α if cos α = - 4/5 and 3π/2 ≤ α < 2π.

In order to use the sine double-angle formula, we also need to find sin α, which would be 3/5 because α is in the 4^th quadrant.

sin 2α = 2 sin α cos α

            = 2 * 3/5 * -4/5

            = -24/25

Finally, let us determine the exact value of tan 2α for α from the prior problem.

Use tan α = sin α/ cos α = 3/5 / -4/5 = -3/4 to solve for tan 2α.

tan 2α = 2 * -3/4 / 1 – (-3/4)² = -3/2 / 7/16 = -3/2 * 16/7 = -24/7

How to use half-angle Formulas to Find Exact Values?

In this answer, we will learn how to find the exact values of the trig functions for angles that are half or double of other angles in this concept. The Half-Angle (a/2) Formulas will be introduced here.

Half-Angle (a/2) Formulas:

sin α/2 = ± √1 – cos α / 2

cos α/2 = ± √1 + cos α / 2

tan α/2 = 1 – cos α / sin α

            = sin α / 1 + cos α

The signs of sin α/2 and cos α/2 rely on which quadrant α/2 lies.

The sine half-angle formula is deduced as follows:

sin² θ = 1 – cos (2θ) / 2

sin² α/2 = 1 – (cos 2 * α/2) / 2

            = 1 - cos α / 2

sin α/2 = ± √1 - cos α / 2

We have used the half-angle formula for cosine to generate the formula.

cos² θ = 1 + cos (2θ) / 2

cos² α/2 = 1 + (cos 2 * α/2) / 2

            = 1 + cos α / 2

cos (π/2) = ± √1 + cos α / 2

For the tangent identity, we have

tan² θ = 1 – cos (2θ) / 1 + cos (2θ)

tan² (α/2) = 1 – cos (2 * α/2) / 1 + cos (2 * α/2)

tan (α/2) = ± √1 – cos α / 1 + cos α

How to Use double-angle formulas to verify identities?

The same steps used to derive the sum and difference formulas are used to establish identities using the double-angle formulas. Rewrite the more complex side of the equation until it matches the other side.

Using double-angle formulas, confirm the following identity:

1 + sin(2θ) = (sin θ + cos θ)²

Solution:

We will start with the right side of the equal sign and rewrite the expression until it matches the left.

(sin θ + cos θ)² = sin² θ + 2 sin θ cos θ + cos² θ

                        = (sin² θ + cos² θ) + 2 sin θ cos θ

                        = 1 + sin θ cos θ

                        = 1 + sin (2θ)

Analysis

This procedure is simple if we remember the perfect square formula from algebra:

(a ± b)² = a² ± 2ab ± b²

Where a = sin θ and b = cos θ. Recognizing patterns is an important part of being successful in mathematics. The algebra remains constant even if the terms or symbols change.

Examples of double and half-angle identities

Double-Angle Identities:

Sum identity for sine:

 sin (x + y) = (sin x)(cos y) + (cos x)(sin y)

 sin (x + x) = (sin x)(cos x) + (cos x)(sin x) (replace y with x)

 sin 2x = 2 sin x cos x

Example 1: Verify, (sin x + cos x)² = 1 + sin 2x:

Answer

(sin x + cos x)² = 1 + sin 2x

(sin x + cos x)(sin x + cos x) = 1 + sin 2x

sin² x + sin x cos x + sin x cos x + cos² x = 1 + sin 2x

sin² x + 2sin x cos x + cos² x = 1 + sin 2x (combine like terms)

sin² x + sin 2x + cos² x = 1 + sin 2x (substitution: double-angle identity)

sin² x + cos² x + sin 2x = 1 + sin 2x

1 + sin 2x = 1 + sin 2x (Pythagorean identity)

Thus, 1 + sin 2x = 1 + sin 2x, is verifiable.

Half-Angle Identities:

Example 2: Without using a calculator, find the exact value for tan 30 degrees and use the half-angle identities.

Answer

tan 30 degrees = tan 60 degrees/ 2

 = sin 60/ (1 + cos 60)

 = ( √3 / 2) / (1 + 1/ 2) +

 = ( √3 / 2) / (3 / 2)

 = ( √3 / 2) (2 / 3) ×

 = √3/3