For 180∘< θ< 270∘, which of the primary trigonometric functions may have negative values?

tan⁡θ and sin⁡θ

tan⁡θ and cos⁡θ

cos⁡θ and sin⁡θ

cos⁡θ only

The answer is: The third option, the functions cos(θ) and sin(θ) will have negative values for 180°<θ<270°.

To answer the question we must remember where the trigonometric functions have positive and negative values. We can remember it by considerating where the coordinates of any point are positive or negative along the coordinate plane (x and y).

The primary trigonometric functions are:

Where,

Also, we need to remember the quadrants of the coordinate plane.

First quadrant: I, 0°<θ<90°

We can find the first quadrant between 0° and 90° , taking the values from 0 to the positive numbers for the x-axis and the y-axis, the points located on this quadrant, will always have positive coordinates, meaning that the functions sine, cosine and tangent will always have positive values.

Second quadrant: II, 90°<θ<180°

We can find the second quadrant between 90° and 180°, taking the values from 0 to the negative numbers for the a-axis, and from 0 to the positive numbers, the points located on this quadrant, will have negative coordinates along the x-axis and positive coordinates along the y-axis, meaning that the function cosine and tangent will always have negative values, while the sine function will always have positive values.

Third quadrant: III, 180°<θ<270°

We can find the third quadrant between 180° and 270°, taking values from 0 to the negative numbers for both x-axis and y-axis, where the points located on this quadrant, will always have negative coordinates along the x-axis and the y-axis, meaning that both functions sine and cosine will always have negative values, while the tangent function will have positive values.

Fourth quadrant: IV, 270°<θ<360°

We can find the fourth quadrant between 270° and 360°, taking values from 0 to the positive numbers for the x-axis, and from 0 to the negative numbers for the y-axis, the points located at this quadrant will always have positive coordinates along the x-axis and negative coordinates along the y-axis, meaning that the sine and tangent function will always have negative values, while the cosine function will always have positive values.

Hence, the answer to the question is the third option, the functions cos(θ) and sin(θ) will have negative values for 180°<θ<270°.

Have a nice day!

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