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Compare sin55o, cos55o and tan55o without use a calculator

Question:

Do not use a calculator, compare the values of sin 55o, cos 55o and tan 55o. Writing then in ascending order.

Solution:

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

how to compare the values of sin a, cos a, and tan a withot using a calculator?

The vertex of the angle is in origin. The initial side of the angle is the positive x-axis, the angle a = 55o. The terminal side of the angle is in Quadrant I.

The terminal side of the angle intersects the unit circle at point P. Draw a line passes the point P and perpendicular to the positive x-axis, this line intersects the positive x-axis at point A. Point A is a perpendicular point.

Because PA is perpendicular to the positive x-axis, so, the angle PAO is 90o. So, triangle PAO is a right triangle. In right triangle OPA,

By sine definition, sin 55o = opposite side/Hypotenuse = PA/OP = PA/r = PA/1 = PA. PA is sine line and PA > 0.

By cosine definition, cos 55o = adjacent side/Hypotenuse = OA/OP = OA/r = OA /1 = OA. OA is cosine line and OA > 0.

The unit circle intersects the positive x-axis at point B. Draw a line passes the point B and perpendicular to the positive x-axis. This line intersects the terminal side of the angle at point Q. Because the point B is a perpendicular point, so the triangle OBQ is a right triangle. In right triangle OBQ, tan 55o = (opposite side)/(adjacent side) = QB/OB = QB/r = QB/1 = QB. QB is the tangent line and QB > 0. Look the figure, QB > PA, so tan 55o > sin 55o

In right triangle OAP, because the angle POA = 55o, so the angle OPA = 90o - 55o = 35o. In triangle OPA, the side opposite 35o is less than the side opposite 55o. So, the length of OA is less than the length of PA. Because the length of PA is less than the length of QB, so, OA < PA < QB. Because OA is the cosine line, PA is the sine line and QB is the tangent line, So, cos 55o < sin 55o < tan 55o. Please watch the video for more details.