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Trigonometric Identities Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

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° (0 rad) and θ = 180° ( π rad):

csc – θ = –csc θ

Secant Of Negative Angle

The secant of the negative of an angle is equal to the secant of the angle. The following formula applies for all angles except θ = 90° ( π/2 rad) and θ = 270° (3π/2 rad):

sec – θ = sec θ

Cotangent Of Negative Angle

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

cot – θ = –cot θ

Double Angles

Sine Of Double Angle

The sine of twice any given angle is equal to twice the sine of the original angle times the cosine of the original angle:

sin = 2 sin θ cos θ

Cosine Of Double Angle

The cosine of twice any given angle can be found according to either of the following:

cos 2 θ = 1 – (2 sin 2 θ )

cos 2 θ = (2 cos 2 θ ) – 1

Angular Sum and Difference

Sine Of Angular Sum

The sine of the sum of two angles θ and ø can be found using this formula:

sin ( θ + ø) = (sin θ )(cos ø) + (cos θ ) (sin ø)

Cosine Of Angular Sum

The cosine of the sum of two angles θ and ø can be found using this formula:

cos ( θ + ø) = (cos θ )(cos ø) – (sin θ )(sin ø)

Sine Of Angular Difference

The sine of the difference between two angles θ and ø can be found using this formula:

sin ( θ – ø) = (sin θ )(cos ø) – (cos θ )(sin ø)

Cosine Of Angular Difference

The cosine of the difference between two angles θ and ø can be found using this formula:

cos ( θ – ø) = (cos θ )(cos ø) + (sin θ )(sin ø)

That’s enough fact-stating for now. Some of these expressions look messy, but they involve nothing more than addition, subtraction, multiplication, division, squaring, and taking the square roots of numbers you work out on a calculator.

Precedence Of Operations

When various operations and functions appear in an expression that you want to solve or simplify, there is a well-defined protocol to follow. If you have trouble comprehending the sequence in which operations should be performed, use a pencil and scratch paper to write down the numbers derived by performing functions on variables; then add, subtract, multiply, divide, or whatever, according to the following rules of precedence.

  • Simplify all expressions within parentheses from the inside out
  • Perform all exponential operations, proceeding from left to right
  • Perform all products and quotients, proceeding from left to right
  • Perform all sums and differences, proceeding from left to right

Here are a couple of examples of this process, in which the order of the numerals and operations is the same in each case, but the groupings differ.

A Flurry of Facts Identities Precedence Of Operations

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