The value of the double angle. Trigonometry formulas

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Double angle formulas are used to express the sines, cosines, tangents, cotangents of an angle with a value of 2 α using the trigonometric functions of the angle α . This article will introduce all double angle formulas with proofs. Examples of application of formulas will be considered. In the final part, the formulas for the triple, quadruple angles will be shown.

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List of double angle formulas

To convert double angle formulas, remember that angles in trigonometry have the form n α notation, where n is natural number, the value of the expression is written without brackets. Thus, sin n α is considered to have the same meaning as sin (n α) . With the notation sin n α we have a similar notation (sin α) n . The usage of notation is applicable for all trigonometric functions with powers of n.

The following are the double angle formulas:

sin 2 α = 2 sin α cos α cos 2 α = cos 2 α - sin 2 α , cos 2 α = 1 - 2 sin 2 α , cos 2 α = 2 cos 2 α - 1 tg 2 α = 2 tg α 1 - tg 2 α ctg 2 α - ctg 2 α - 1 2 ctg α

Note that these sin and cos formulas are applicable with any value of the angle α. The formula for the tangent of a double angle is valid for any value of α, where t g 2 α makes sense, that is, α ≠ π 4 + π 2 · z, z is any integer. The cotangent of a double angle exists for any α , where c t g 2 α is defined on α ≠ π 2 · z .

The cosine of a double angle has a triple notation of a double angle. All of them are applicable.

Proof of double angle formulas

The proof of the formulas originates from the addition formulas. We apply the formulas for the sine of the sum:

sin (α + β) = sin α cos β + cos α sin β and the cosine of the sum cos (α + β) = cos α cos β - sin α sin β. Suppose that β = α , then we get that

sin (α + α) = sin α cos α + cos α sin α = 2 sin α cos α and cos (α + α) = cos α cos α - sin α sin α = cos 2 α - sin2α

Thus, the formulas for the sine and cosine of the double angle sin 2 α \u003d 2 sin α cos α and cos 2 α \u003d cos 2 α - sin 2 α are proved.

The remaining formulas cos 2 α \u003d 1 - 2 sin 2 α and cos 2 α \u003d 2 cos 2 α - 1 lead to the form cos 2 α \u003d cos 2 α \u003d cos 2 α - sin 2 α, when replacing 1 with the sum of squares by the basic identity sin 2 α + cos 2 α = 1 . We get that sin 2 α + cos 2 α = 1. So 1 - 2 sin 2 α \u003d sin 2 α + cos 2 α - 2 sin 2 α \u003d cos 2 α - sin 2 α and 2 cos 2 α - 1 \u003d 2 cos 2 α - (sin 2 α + cos 2 α) \u003d cos 2 α - sin 2 α.

To prove the formulas for the double angle of tangent and cotangent, we apply the equalities t g 2 α \u003d sin 2 α cos 2 α and c t g 2 α \u003d cos 2 α sin 2 α. After the transformation, we get that tan 2 α \u003d sin 2 α cos 2 α \u003d 2 sin α cos α cos 2 α - sin 2 α and ctg 2 α \u003d cos 2 α sin 2 α \u003d cos 2 α - sin 2 α 2 · sin α · cos α . Divide the expression by cos 2 α where cos 2 α ≠ 0 with any value of α when t g α is defined. Divide another expression by sin 2 α , where sin 2 α ≠ 0 with any values ​​of α , when c t g 2 α makes sense. To prove the double angle formula for tangent and cotangent, we substitute and get: