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# Sine line cosine line tangent line example3

Question:

Do not use a calculator, compare the values of sin (-125^{o}), cos (-125^{o}) and tan (-125^{o}). Writing them in ascending order.

Solution:

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

The angle a = -125^{o} = -90^{o} + (-35^{o})

Now we determine the terminal side of the angle a. The initial side of the angle is in the positive x-axis. The vertex of the angle is in the origin. Because the angle is negative, so, we
rotate the angle from the initial side clockwise to 125^{o} to get the terminal side of the angle. The terminal side of the angle lies in Quadrant III. So, the angle a is an angle of Quadrant III.

The terminal side of the angle intersects the unit circle at the point P. Pint P has the coordinate (x, y). Now we are going to determine the (x, y). Draw a line from the point P and perpendicular to the negative x-axis. This line intersects the negative x-axis at point M. M is the perpendicular point. So, x = OM and OM < 0. y = MP and MP < 0.

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

cos a = x/r = OM/r = OM < 0 (OM is cosine line.)

The angle a = -125^{o} = the angle AOP

Because MP is parallel to the negative y-axis, so, angle MPO = angle AOP - 90^{0} = 125^{o} - 90^{o} = 35^{o}

In right triangle OMP, the angle MOP = 90^{o} - angle MPO = 90^{o} - 35^{o} = 55^{o}. MP opposite the angle MOP which is 55^{o}. OM opposite the
angle MPO which is 35^{o}. In a triangle, the side opposite the large angle is longer than the side opposite the small angle. So, |MP| > |OM|. Then -|MP| < -|OM|. Because MP is
negative and OM is negative, so, MP < OM. That is, sin 125^{o} < cos 125^{o}

Now we are going to find tan a. We extended the terminal side of the angle a in the opposite direction. Now the extended terminal side of the angle goes to the Quadrant I. 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 the point T. The point T has the coordinate (x, y). Now we are going to determine the (x, y).

By definition, tan a = y/x = TA/OA = TA/r = TA > 0 (TA is the tangent line.)

Because MP < OM < TA, So, sin (-125^{o}) < cos (-125^{o}) < tan (-125^{o}). Watch the video for more details.

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