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Trigonometric Identities for Physics Help

By — McGraw-Hill Professional
Updated on Sep 17, 2011

Introduction

The following paragraphs depict common trigonometric identities for the circular functions. Unless otherwise specified, these formulas apply to angles θ and ϕ in the standard range as follows:

0 ≤ θ < 2π (in radians)

0 ≤ θ < 360 (in degrees)

0 ≤ ϕ < 2π (in radians)

0 ≤ ϕ < 360 (in degrees)

Angles outside the standard range usually are converted to values within the standard range by adding or subtracting the appropriate multiple of 2π radians (360°). You occasionally may hear of an angle with negative measure—that is, measured clockwise rather than counterclockwise—but this can always be converted to some angle with positive measure that is at least zero but less than 360°. The same is true for “angles” greater than 360°. Sometimes physicists will use strange angular expressions (for example, talking about reversed or multiple rotations or revolutions), but it is usually best to reduce angles to values in the standard range. Some of these formulas deal with negative angles, but in these cases, the intent is to let you determine an equivalent value for a trigonometric function of some angle within the standard range.

Pythagorean Theorem For Sine And Cosine

The sum of the squares of the sine and cosine of an angle is always equal to 1. The following formula holds:

sin 2 θ + cos 2 θ = 1

The expression sin 2 θ refers to the sine of the angle squared (not the sine of the square of the angle). That is to say:

sin 2 θ = (sin θ) 2

The same holds true for the cosine, tangent, cosecant, secant, cotangent, and all other similar expressions you will see in the rest of this chapter and in physics.

Pythagorean Theorem For Secant And Tangent

The difference between the squares of the secant and tangent of an angle is always equal to either 1 or −1. The following formulas apply for all angles except θ = π/2 radians (90°) and θ = 3π/2 radians (270°):

sec 2 θ − tan 2 θ = 1

tan 2 θ − sec 2 θ = −1

Sine Of Negative Angle

The sine of the negative of an angle (an angle measured in the direction opposite to the normal direction) is equal to the negative (additive inverse) of the sine of the angle. The following formula holds:

sin −θ = −sin θ

Cosine Of Negative Angle

The cosine of the negative of an angle is equal to the cosine of the angle. The following formula holds:

cos −θ = cos θ

Tangent Of Negative Angle

The tangent of the negative of an angle is equal to the negative (additive inverse) of the tangent of the angle. The following formula applies for all angles except θ = π/2 radians (90°) and θ = π/2 radians (270°):

tan −θ = −tan θ

Cosecant Of Negative Angle

The cosecant of the negative of an angle is equal to the negative (additive inverse) of the cosecant of the angle. The following formula applies for all angles except θ = 0 radians (0°) and θ = π radians (180°):

cos −θ = −cos θ

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