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### Compare sin125^{o}, cos125^{o}, tan125^{o} without using calculator

Question:

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

Solution:

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

The angle a = 125^{o}. The vertex of the angle is in the origin. The initial side of the angle is in the positive x-axis. Now count clockwise rotate 125^{o}. So, the terminal side of the angle lies in Quadrant II.

The terminal side of the angle intersects the unit circle at point P. Draw a line passing 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. The x-coordinate of the point P is OM and OM < 0. The y-coordinate of the point P is PM and PM > 0.

By definition: sin a = y/r. Because the circle is a unit circle, so, r = 1. sin a = y. y is the sine line. So, sin 125^{o} = PM and PM > 0.

By definition cos a = x/r = x. x is the cosine line. cos 125^{o} = OM and OM < 0. So, OM < PM. That is: cos125^{o} < sin125^{o}

Now we will find tangent line. Extended the terminal side of the angle in the opposite direction which goes to Quadrant IV. The unit circle intersects the positive x-axis at point A. Draw a line passing the
point A and tangent to the circle. This line intersects the extended terminal side of the angle at the point T. The x-coordinate of the point T is OA and OA > 0. The y-coordinate of the point T is AT and
AT < 0. By definition tan a = y/x = AT/OA = AT/r = AT/1 = AT. AT is the tangent line. So, tan 125^{o} = AT. Because both AT and OM are negative and AT is longer than OM, so, AT < OM < PM.
Therefore, tan 125^{o} < cos 125^{o} < sin 125^{o}. Watch the video for more details.