Question Video: Determining the Quadrant in Which an Angle Lies given Two of Its Trigonometric Ratios Mathematics

Determine the quadrant in which πœƒ lies if cos πœƒ > 0 and sin πœƒ < 0.

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Video Transcript

Determine the quadrant in which πœƒ lies if cos of πœƒ is greater than zero and sin of πœƒ is less than zero.

We’re trying to consider a coordinate grid and find which quadrant an angle would fall in. We’re told that cos of πœƒ is greater than zero, this means it has a positive cosine value, while the sin of πœƒ is less than zero, which means the sine has a negative value. One method we use for identifying the sine and cosine values in different quadrants is the CAST diagram that looks like this.

In the CAST diagram, we know that in the first quadrant, all values are positive. In the second quadrant, only the sine value is positive. In the third quadrant, only the tangent value is positive. And in the fourth quadrant, only cosine is positive. If we have a negative sine value and a positive cosine value, we can eliminate quadrant one as all values must be positive there. We can eliminate quadrant two as sine is positive there. In quadrant three, sine is negative, but so is cosine. And that means quadrant three will not work. In quadrant four, cosine is positive and sine is negative. And that means our angle πœƒ under these conditions must fall in the fourth quadrant.

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