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Chapter 5: Problem 67

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
The statement is false. Whether a trigonometric function is even or odd does not directly determine if its value is positive or negative.

Step by step solution

01

Understanding Even and Odd Trigonometric Functions

Even functions are those for which \(f(-x) = f(x)\). For trigonometric functions, cosine (\(cos\)) is an even function since \(cos(-x)=cos(x)\). Odd functions are those for which \(f(-x) = - f(x)\). For trigonometric functions, sine (\(sin\)) and tangent (\(tan\)) are both odd since \(sin(-x)= - sin(x)\) and \(tan(-x)= - tan(x)\).
02

Evaluating the statement

The given statement is that 'Even and odd trigonometric identities help determine whether a trigonometric function's value is positive or negative'. The evenness and oddness of a function can help determine the symmetry of a function around the y-axis or the origin. However, it doesn't provide explicit indication about whether the value of a trigonometric function is positive or negative. The sign of trigonometric functions is more related to which quadrant of the unit circle the angle lies in.
03

Conclusion

Based on the above understanding, we can say that the statement is false because knowing whether the trigonometric function is even or odd does not directly help in determining if the function's value is positive or negative.

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