**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.

**Negative Angles**

**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*

**Practice 1**

Why does the above formula not work when *θ* = 90° (π/2 rad) or *θ* = 270° (3π/2 rad)?

**Solution 1**

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.

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