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How do you tell whether the value of $\sin \left( { - {{45}^ \circ }} \right)$ is positive, negative, zero or undefined?

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Last updated date: 20th Jun 2024
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Answer
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Hint: We will first discuss the rules of trigonometric ratios in all the quadrants about them being positive or negative. Now, after that we will see where – 45 degrees lie and then answer the required.

Complete step-by-step answer:
Let us first of all discuss which quadrant which trigonometric function or ratio takes positive and negative values.
In Quadrant I: All the trigonometric functions take up positive values. Therefore, if the value of angle lies between 0 degrees to 90 degrees, its trigonometric ratio will always be greater than zero.
In Quadrant II: Only the sine and cosecant function takes up positive values. Therefore, if the value of angle lies between 90 degrees to 180 degrees, its sine and cosecant value will always be greater than zero. But all other values will be less than zero.
In Quadrant III: Only the tangent and cotangent function takes up positive values. Therefore, if the value of angle lies between 180 degrees to 270 degrees or – 90 degrees to – 180 degrees (if taken clockwise), its tangent and cotangent value will always be greater than zero. But all other values will be less than zero.
In Quadrant IV: Only the cosine and secant function takes up positive values. Therefore, if the value of angle lies between 270 degrees to 360 degrees or – 0 degrees to – 90 degrees (if taken clockwise), its cosine and secant value will always be greater than zero. But all other values will be less than zero.
Now, we see that the given angle is – 45 degrees which lies in IV quadrant for sure. In IV quadrant as we see from above, sine is negative. Therefore, it will be negative.

Hence, the answer is negative value.

Note:
The students must note that there is a trick to learn the same thing about all the quadrants.
Remember the phrase: “ADD SUGAR TO COFFEE”.
Here we are moving from first quadrant to fourth quadrant with each word.
In the first quadrant we have ADD, take A from it, it stands for all which means all are positive here.
In the second quadrant we have SUGAR, take S from it, it stands for sine which means sine is positive here and thus cosecant as well.
In the third quadrant we have TO, take T from it, it stands for tangent which means tangent is positive here and thus cotangent as well.
In the fourth quadrant we have COFFEE, take C from it, it stands for cosine which means cosine is positive here and thus secant as well.