General Principle of Transforming Functions of the Form α + a/bπ to a Simpler Form

To reduce the angle of a trigonometric function, the argument of the function must be presented in the form:

General Principle of Transforming Functions of the Form α + a/bπ to a Simpler Form

After this, you can use the rules and formulas listed below.

Rule 1: If a = 2, b = 1 and z is a natural number, the name of the function does not change

Formulas for Reducing the Argument of a Trigonometric Function for the Angle 2π for sin, cos, tg

• sin(α + 2π) = sin(α)
• cos(α + 2π) = cos(α)
• tan(α + 2π) = tan(α)
• cot(α + 2π) = cot(α)

These equalities follow from the properties of the trigonometric circle.

Example: Simplify the expression cos(390).

Considering that π = 180 degrees, then: cos(390) = cos(30 + 2π) Using the reduction formula (2) above: cos(30 + 2π) = cos(30)

Rule 2: Subtracting the Angle α from 2π

Formulas for Reducing the Angle Argument of the Function sin, cos, tan 2π - α

• sin(2π - α) = -sin(α)
• cos(2π - α) = cos(α)
• tan(2π - α) = -tan(α)
• cot(2π - α) = -cot(α)

These properties follow from the properties of the trigonometric circle, considering the parity of functions.

Example: Simplify the expression sin(345). Considering that π = 180 degrees, then: sin(345) = sin(2π - 15) Using the reduction formula (1) above:
sin(2π - 15) = -sin(15)

Rule 3: Adding the Angle π/2

If the argument of a trigonometric function contains the expression π/2 or n*π/2, where n is a natural odd number, the function changes according
to the following rules:

Formulas for Reducing the Argument of a Trigonometric Function of the Form α + π/2 for sin, cos, tan, cot

• sin(α + π/2) = cos(α)
• cos(α + π/2) = -cos(α)
• tan(α + π/2) = -cot(α)
• cot(α + π/2) = -tan(α)

From the properties of the trigonometric circle, it follows that:

• sin(α + π/2 + 2π) = cos(α)
• cos(α + π/2 + 2π) = -cos(α)
• tan(α + π/2 + 2π) = -cot(α)
• cot(α + π/2 + 2π) = -tan(α)

Example: Simplify the expression sin(α + 5π/2). sin(α + 5π/2) = sin(α + π/2 + 2π1) sin(α + π/2 + 2π1) = cos(α)

Rule 4: Subtracting the Angle π/2 (π/2 - α)

If the argument of a trigonometric function involves subtracting the angle π/2, for example, cos(π/2 - α), you can use the following transformation formulas:

Formulas for Reducing Trigonometric Functions When Subtracting the Angle π/2 - α for sin, cos, tg, cot

From the properties of a right triangle, it follows that:

• sin(π/2 - α) = cos(α)
• cos(π/2 - α) = sin(α)
• tan(π/2 - α) = cot(α)
• cot(π/2 - α) = tan(α)

From the properties of the trigonometric circle, it follows that:

• sin(π/2 - α + 2π) = cos(α)
• cos(π/2 - α + 2π) = sin(α)
• tan(π/2 - α + 2π) = cot(α)
• cot(π/2 - α + 2π) = tan(α)

Example: Simplify the expression cos(5π/2 - α). cos(5π/2 - α) = cos(π/2 - α + 2π1) cos(π/2 - α + 2π1) = sin(α)

Rule 5: Trigonometric Formulas for Reducing the Angle α + π

If the argument of a trigonometric function can be presented as the sum of angles of the form α + π + 2πz, such a function
can be reduced to a simpler form using the following formulas:

Trigonometric Formulas for Reducing the Angle α + π to a Simpler Form for sin, cos, tg, cot (α + π)

• sin(α + π) = -sin(α)
• cos(α + π) = -cos(α)
• tan(α + π) = tan(α)
• cot(α + π) = cot(α)

Also:

• sin(α + π + 2π) = -sin(α)
• cos(α + π + 2π) = -cos(α)

Example: Simplify the expression sin(7π + α). sin(7π + α) = sin(π + α + 2π3) sin(π + α + 2π3) = -sin(α)

Rule 6: Trigonometric Formulas for Reducing the Angle (π - α)

If the argument of a trigonometric function involves subtracting the angle π, i.e., simplifying the argument of the form (π - α), you can use the following formulas:

Formulas for Reducing the Argument of the Function π - α for sin, cos, tg, cot (π - α)

Example: Simplify the expression sin(5π - α). sin(5π - α) = sin(π - α + 2π2) sin(π - α + 2π2) = sin(α)

Rule 7: Trigonometric Formulas for Reducing Functions for Angles (3π/2 + α)

Formulas for Reducing Trigonometric Functions of the Form sin(α + 3π/2), cos(α + 3π/2), tg(α + 3π/2) to a Simpler Form

Example: Simplify the expression cos(α + 7π/2). cos(α + 7π/2) = cos(α + 3π/2 + 2π1) cos(α + 3π/2 + 2π1) = sin(α)

Rule 8: Trigonometric Formulas for Reducing the Argument for Cases of Subtracting the Angle α from 3π/2

Formulas for Reducing Trigonometric Functions with the Argument of the Angle of the Form (3π/2 - α) for sin(3π/2 - α),
cos(3π/2 - α), tg(3π/2 - α), cot(3π/2 - α)

Example: Simplify the expression sin(7π/2 - α). sin(7π/2 - α) = sin(3π/2 - α + 2π1) sin(3π/2 - α + 2π1) = -cos(α)

Proof of the Correctness of the Formulas for Transforming Trigonometric Functions of the Form (α + a/bπ)

Values of Sine, Cosine, and Tangent of Acute Angles

The values of trigonometric functions of acute angles are derived from the ratios of the sides of a right triangle.

THEOREM: For any acute angle α:

• sin(90° - α) = cos(α)

• cos(90° - α) = sin(α)

• tg(90° - α) = ctg(α)

Consider a right triangle with an acute angle α. By the Pythagorean theorem:

AC² = AB² + BC².

Definition of Sine, Cosine, and Tangent for Angles 90 - α for a Right Triangle

By the definition of trigonometric functions:

• sin(α) = AB/AC and cos(α) = BC/AC (1)

Since ∠A + ∠C = 90°, then: ∠A = 90° - ∠C ∠A = 90° - α,

• sin(90° - α) = BC/AC and cos(90° - α) = AB/AC (2)

Comparing equalities (1) and (2), we write the identical equalities:

• sin(90° - α) = cos(α)

• cos(90° - α) = sin(α) (3)

Calculate tg(90° - α), knowing that the tangent of an angle is the ratio of the sine and cosine of the same angle and using equality (3):

tg (90°-α) = sin(90°-α) / cos(90°-α) =cos α / sin α = ctg α.