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# Sine line cosine line tangent line example4

Question:

Do not using calculator, compare the value of sin (-53^{o}), cos (-53^{o}) and tan (-53^{o}). Writing them in ascending order.

Solution:

The circle is a unit circle. The radius of the circle is one. That is, r = 1.

The angle a = -53^{o}. Now we are going to determine the terminal side of the angle a.

The initial side of the angle is the positive x-axis. The vertex of the angle is the origin. Because the angle a = -53^{o}, so we clockwise rotate to 53^{o} to get
the terminal side of the angle. The terminal side of the angle is in Quadrant IV. So, the angle a is the angle of Quadrant IV.

The terminal side of the angle intersects the unit circle at point P. The point P has the coordinate (x, y). Now we determine the (x, y).

Draw a line passing the point P and perpendicular to the positive x-axis, this line intersects the positive x-axis at the point M. M is the perpendicular point. So, x = OM and OM > 0. y = MP and MP < 0.

From definition, sin a = y/r = y = MP < 0. (MP is the sine line.)

cos a = x/r = x = OM > 0. (OM is the cosine line.) So, sin (-53^{o}) < cos (-53^{o})

Now we are going to find tan a. From definition, tan a = y/x

The unit circle intersects the positive x-axis at the point A. We draw a line passing the point A and tangent to the circle, this line intersects the terminal side of the angle at point T. Point T has the coordinate (x, y). Now we are going to find the x-coordinate of the point T and y-coordinate of the point T.

tan a = y/x = AT/OA = AT/r = AT. (AT is the tangent line.)

Because both AT and MP are perpendicular to the positive x-axis, so, AT // MP

- so, |AT| > |MP|
- - |AT| < - |MP|
- so, AT < MP

Therefore, tan (-53^{o}) < sin (-53^{o}) < cos (-53^{o}). Watch the video for more details.