Question Video: Finding the Value of a Trigonometric Function of an Angle given the Coordinates of the Point of Intersection of the Terminal Side and the Unit Circle

Find tan ๐œƒ, given ๐œƒ is in standard position and its terminal side passes through the point (โˆ’3/5, โˆ’4/5).

02:03

Video Transcript

Find the tangent of ๐œƒ, given ๐œƒ is in standard position and its terminal side passes through the point negative three-fifths, negative four-fifths.

Since this is an ๐‘ฅ-๐‘ฆ point, letโ€™s think about where this will be located. In quadrant number one, ๐‘ฅ and ๐‘ฆ would be positive. In quadrant two, ๐‘ฅ would be negative and ๐‘ฆ would be positive. And in quadrant three thatโ€™s in the bottom left-hand corner, ๐‘ฅ would be negative and ๐‘ฆ would be negative. And in the fourth quadrant, ๐‘ฅ would be positive and ๐‘ฆ would be negative.

So since ๐‘ฅ and ๐‘ฆ are both negative, we will be in quadrant three, because weโ€™re in the negative direction for ๐‘ฅ and in the negative direction for ๐‘ฆ. When we create an angle, thereโ€™s an initial side and a terminal side. The terminal side is when it stops, so weโ€™re gonna stop in quadrant three. And it says that this terminal side passes through the point negative three-fifths, negative four-fifths.

The ๐‘ฅ-coordinate of the point where the terminal side of an angle measuring ๐œƒ in standard position in a rectangular coordinate system intersects the unit circle is cos ๐œƒ, and the ๐‘ฆ-coordinate is sin ๐œƒ. Since the angle is in standard position and its terminal side intersects the unit circle at a point with the coordinate of negative three-fifths, negative four-fifths, sin of ๐œƒ must be equal to negative four-fifths and cos of ๐œƒ must be equal to negative three-fifths.

Now tangent of ๐œƒ will be equal to the sin of ๐œƒ divided by the cos of ๐œƒ, so we have negative four-fifths divided by negative three-fifths. And when we divide fractions, we actually multiply by the reciprocal, so we keep our negative four-fifths, but instead of dividing by the denominator, we multiply by the denominatorโ€™s reciprocal, so we flip it.

And now we multiply. The fives cancel and the two negatives cancel to become a positive, so we get four-thirds. Therefore, the tangent of ๐œƒ is equal to four-thirds.

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