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Sine line cosine line tangent line example3

Question:

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

Solution:

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

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

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 125o 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.)

how to compare the values of sin (-125 degrees), cos (-125 degrees) and tan (-125 degrees)?

The angle a = -125o = the angle AOP

Because MP is parallel to the negative y-axis, so, angle MPO = angle AOP - 900 = 125o - 90o = 35o

In right triangle OMP, the angle MOP = 90o - angle MPO = 90o - 35o = 55o. MP opposite the angle MOP which is 55o. OM opposite the angle MPO which is 35o. 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 125o < cos 125o

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 (-125o) < cos (-125o) < tan (-125o). Watch the video for more details.