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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 = cos^{2} α - sin^{2} α

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 = cos^{2} α - sin^{2} α

= 2 cos^{2} α – 1

= 1 - sin^{2} α

sin 2α = 2 sin α cos α

tan 2α = 2 tan α/1- tan^{2} 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

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:

sin^{2} θ = 1 – cos (2θ) / 2

sin^{2} 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.

cos^{2} θ = 1 + cos (2θ) / 2

cos^{2} 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

tan^{2} θ = 1 – cos (2θ) / 1 + cos (2θ)

tan^{2} (α/2) = 1 – cos (2 * α/2) / 1 + cos (2 * α/2)

tan (α/2) = ± √1 – cos α / 1 + cos α

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} = sin^{2} θ + 2 sin θ cos θ + cos^{2} θ

= (sin^{2} θ + cos^{2} 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} = a^{2} ± 2ab ± b^{2}

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.

**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

sin^{2} x + sin x cos x + sin x cos x + cos^{2} x = 1 + sin 2x

sin^{2} x + 2sin x cos x + cos^{2} x = 1 + sin 2x (combine like terms)

sin^{2} x + sin 2x + cos^{2} x = 1 + sin 2x (substitution: double-angle identity)

sin^{2} x + cos^{2} 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

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