How to identify tan6 is in 4th quadrant?
Answers
Step-by-step explanation:
It means: In the first quadrant (I), all ratios are positive. In the second quadrant (II), sine (and cosec) are positive. In the third quadrant (III), tan (and cotan) are positive. In the fourth quadrant (IV), cos (and sec) are positive.
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Answer:
The four quadrants
The coordinate axes divide the plane into four quadrants, labelled first, second, third and fourth as shown. Angles in the third quadrant, for example, lie between 180∘ and 270∘.
Illustration of coordinate axes with quadrants labelled one, two, three, four.
By considering the x- and y-coordinates of the point P as it lies in each of the four quadrants, we can identify the sign of each of the trigonometric ratios in a given quadrant. These are summarised in the following diagrams.
Two illustrations of coordinate axes. Four quadrants identified with the sign of each of the trigonometric ratios in a given quadrant.
Detailed description of diagram
Related angles
In the module Further trigonometry (Year 10), we saw that we could relate the sine and cosine of an angle in the second, third or fourth quadrant to that of a related angle in the first quadrant. The method is very similar to that outlined in the previous section for angles in the second quadrant.
We will find the trigonometric ratios for the angle 210∘, which lies in the third quadrant. In this quadrant, the sine and cosine ratios are negative and the tangent ratio is positive.
To find the sine and cosine of 210∘, we locate the corresponding point P in the third quadrant. The coordinates of P are (cos210∘,sin210∘). The angle POQ is 30∘ and is called the related angle for 210∘.
Circle with radius of 1, centre of circle at origin O. Point marked on circle in quadrant 1 as (cos 30 degrees, sin 30 degrees).
Detailed description of diagram
So,
cos210∘=−cos30∘=−3–√2andsin210∘=−sin30∘=−12.
Hence
tan210∘=tan30∘=13–√.
In general, if θ lies in the third quadrant, then the acute angle θ−180∘ is called the related angle for θ.
Exercise 2
Use the method illustrated above to find the trigonometric ratios of 330∘.
Write down the related angle for θ, if θ lies in the fourth quadrant.
The basic principle for finding the related angle for a given angle θ is to subtract 180∘ from θ or to subtract θ from 180∘ or 360∘, in order to obtain an acute angle. In the case when the related angle is one of the special angles 30∘, 45∘ or 60∘, we can simply write down the exact values for the trigonometric ratios.
In summary, to find the trigonometric ratio of an angle between 0∘ and 360∘: