Question

How do you determine whether the sign of a trigonometric function is positive or negative when dealing with half angles?

Answer 1

To determine the sign of a trigonometric function for half angles, identify which quadrant the half angle lies in and use the rules for trigonometric function signs in that quadrant.

Step-by-step explanation:

To determine whether the sign of a trigonometric function is positive or negative for half angles, first consider which quadrant the angle lies in after you calculate the half angle. Each quadrant has a specific rule for the signs of trigonometric functions.

In the first quadrant (0 to 90 degrees), all trigonometric functions are positive. In the second quadrant (90 to 180 degrees), only sine and its reciprocal function cosecant are positive. In the third quadrant (180 to 270 degrees), only tangent and its reciprocal function cotangent are positive. And in the fourth quadrant (270 to 360 degrees), only cosine and its reciprocal function secant are positive.

Answer 2

Step-by-step explanation:

Consider the quadrant of the original angle and where half that angle would lie.

Half of any positive 1st- or 2nd-quadrant angle will lie in the first quadrant.

Half of any positive 3rd- or 4th-quadrant angle will lie in the second quadrant.

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Half of any negative 1st- or 2nd-quadrant angle will lie in the third quadrant.

Half of any negative 3rd- or 4th-quadrant angle will lie in the fourth quadrant.

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Here, we are using the wording "x-quadrant angle" to mean an angle whose terminal ray lies in the x-quadrant. The measure of that angle may be positive or negative. (Half of -5π/4 radians will have different trig function values than half of +3π/4 radians, even though they both have their terminal ray in the 2nd quadrant.)

We are assuming the angle does not exceed 360° (or 2π radians). If it does, you need to adjust the half-angle quadrant accordingly.