Trigonometric Identities Help (page 2)
Introduction to Trigonometric Identities
The following paragraphs depict common trigonometric identities for the circular functions. Unless otherwise specified, these formulas apply to angles θ and ø in the standard range, as follows:
0 rad ≤ θ < 2π rad
0° ≤ θ < 360°
0 rad ≤ ø < 2π rad
≤ ø < 360°
Angles outside the standard range are converted to values within the standard range by adding or subtracting the appropriate multiple of 360° (2 π rad). You might occasionally hear of an angle with negative measure or with a measure of more than 360° ( 2π rad), but this can always be converted to some angle with positive measure that is at least zero but less than 360° ( 2π rad).
An Encouraging Word
When you look at the next few paragraphs and see one equation after another, peppered with Greek symbols, exponents, and parentheses, don’t let them intimidate you. All you have to do when working with them is substitute numbers for the angles, and work through the formulas with a calculator. You are not expected to memorize these formulas. They are here for your reference. If you ever need one of these identities, you can refer back to this chapter and look it up!
Trigonometric identities can be useful in solving complicated angle/distance problems in the real world, because they allow the substitution of “clean” expressions for “messy” ones. It’s a lot like computer programming. There are many ways to get a computer to perform a specific task, but one scheme is always more efficient than any of the others. Trigonometric identities are intended to help scientists and engineers minimize the number of calculations necessary to get a desired result. This in turn minimizes the opportunity for errors in the calculations. As any scientist knows, the chance that a mistake will be made goes up in proportion to the number of arithmetic computations required to solve a problem.
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 = 90° (π/2 rad) and θ = 270° (3π/2 rad):
tan – θ = –tan θ
Why does the above formula not work when θ = 90° (π/2 rad) or θ = 270° (3π/2 rad)?
The value of the tangent function is not defined for those angles. Remember that the tangent of any angle is equal to the sine divided by the cosine. The cosine of 90° ( π /2 rad) and the cosine of 270° (3π/2 rad) are both equal to zero. When a quotient has zero in the denominator, that quotient is not defined. This is also the reason for the restrictions on the angle measures in some of the equations that follow.
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
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 2θ = 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.
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