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Chapter 5: Problem 127

Determine whether the statement is true or false. Justify your answer. The even and odd trigonometric identities are helpful for determining whether the value of a trigonometric function is positive or negative.

Short Answer

Expert verified
False. Even and odd identities of trigonometric functions only provide information about the symmetry of the functions, not whether they are positive or negative. The sign of a trigonometric function depends on the quadrant in which the angle lies.

Step by step solution

01

Understanding Even and Odd Trigonometric Identities

The concept of even and odd functions extends to trigonometric functions as well. A function \( f(x) \) is called an even function if, for all \( x \) in the domain of \( f(x) \), \( f(x) = f(-x) \). We can see this property in the cosine function. A function \( f(x) \) is called an odd function if, for all \( x \) in the domain of \( f(x) \), \( f(-x) = -f(x) \). The sine function is an example of an odd function.
02

Sign Determination by Quadrant

Angles in standard position lie in one of the four quadrants. In quadrant I, all trigonometric functions are positive. In quadrant II, only sine and its reciprocal are positive hence cosine and tangent are negative. In the third quadrant, only tangent and its reciprocal are positive hence sine and cosine are negative. In quadrant IV, only cosine and its reciprocal are positive, making sine and tangent negative. Hence the sign of a trigonometric function depends on the quadrant in which the angle lies not the function鈥檚 even or odd property.
03

Outcome Analysis

Given the relationship between even and odd identities with trigonometric functions as well as the behavior of trigonometric functions in various quadrants, it鈥檚 clear that the value of being positive or negative of trigonometric functions can鈥檛 be determined by their even and odd identities alone.

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