Sin and Sec
Sin 60°
Sin = y/1
The y coordinate of 60° on the unit circle is √ 3/2
Cos60°
Cos = x/1
The x-coordinate of 60° on the unit circle is 1/2
Tan (-π /6)
Tan = y/x
The y coordinate of -π /6 is 1/2 and the x coordinate of -π /6 is √3/2. 1/2 over √3/2 is equal to -√3/3
Sin 30°
Sin = y/1
The y coordinate of 30° on the unit circle is 1/2.
Cos 135°
Cos = x/1
The x-coordinate of 135° on the unit circle is -√2/2.
Sin(255°)
Sin = y/1
The y coordinate of 225° on the unit circle is -√2/2
Cos(2π /3)
Cos = x/1
The x-coordinate of -2π /3 on the unit circle is -1/2
Tan (-11π /6)
Tan = y/x
The y coordinate of -11π /6 is -1/2 and the x coordinate of -11π /6 is √3/2. -1/2 over √3/2 is equal to √3/3.
Sin 7π/4
Sin = y/1
The y coordinate of 7π/4 on the unit circle is -√2/2.
Cos π/6
Cos = x/1
The x-coordinate of Cos π/6 on the unit circle is √3/2
Sin 5π/3
Sin = y/1
The y coordinate of 5π/3 on the unit circle is -√3/2.
Csc -π
Csc = 1/y
Cosecant is the reciprocal of sine, so the cosecant of any angle x for which sin x = 0 must be undefined, since it would have a denominator equal to 0.
So, csc -π is undefined.
Cot 0
Cot = x/y
However, any sign of 0 is always undefined.
Tan 210°
Tan = y/x
The y coordinate of 210° is -1/2 and the x coordinate of 210° is -√ 3/2. -1/2 over -√3/2 is equal to √3/3
Sec 300°
Sec = 1/x
The x coordinate of 300° is 1/2 1 over 1/2 is 2
Sec 150°
Sec = 1/x
The x coordinate of 150° is -√3/2. 1 over (-√3/2) is (-2√3/3)
Csc 7π/4
Csc = 1/y
The y coordinate of 7π/4 is -√2/2. 1 over -√2/2 is -√2.
Cot 315°
Cot = x/y
The y coordinate of Cot 315° is -√2/2 and the x coordinate √2/2. √2/2 over -√2/2 is -1.
Csc π/4
Csc = 1/y
The y coordinate of π/3 is √2/2. 1 over √2/2 is √2.
Tan 4π/3
Tan = y/x
The y coordinate of 4π/3 is -√3/2 and the x coordinate of 4π/3 is -1/2. -√3/2 over -1/2 is equal to √3.
Sec 4π/3
Sec = 1/x
The x coordinate of 4π/3 is -1/2. 1 over -1/2 is -2
Csc (3π/4)
Csc = 1/y
The y coordinate of 3π/4 is √2/2. 1 over √2/2 is √2.
Cot 5π/4
Cot = x/y
The y coordinate of 5π/4 is -√2/2 and the x coordinate is also -√2/2. -√2/2/-√2/2 is 1.
Sec 5π/6
Sec = 1/x
The x coordinate of 300° is -√3/2. 1 over -√3/2 is 2√ 3.
Cot 2π/3
Cot = x/y
The y coordinate of Cot 2π/3 is √3/2 and the x coordinate -√1/2. -√1/2 over √3/2 is -√3/3.