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Sin and Sec

100

Sin 60° 

Sin = y/1 

The y coordinate of 60° on the unit circle is √ 3/2

200

Sin(255°) 

Sin = y/1 

The y coordinate of 225° on the unit circle is -√2/2

300

Sin 5π/3

Sin = y/1 

The y coordinate of 5π/3 on the unit circle is -√3/2. 

400

Sec 150° 

Sec = 1/x

The x coordinate of 150° is -√3/2. 1 over (-√3/2) is (-2√3/3)

500

Sec 4π/3

Sec = 1/x

The x coordinate of 4π/3 is -1/2. 1 over -1/2 is -2

Cos and Csc

100

Cos60° 

Cos = x/1

The x-coordinate of 60° on the unit circle is 1/2

200

Cos(-2π /3)

Cos = x/1

The x-coordinate of -2π /3 on the unit circle is -1/2

300

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. 

400

Csc 7π/4

Csc = 1/y

The y coordinate of 7π/4 is -√2/2. 1 over -√2/2 is -√2.

500

Csc (3π/4)

Csc = 1/y

The y coordinate of 3π/4 is √2/2. 1 over √2/2 is √2.

Tan and Cot

100

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

200

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.

300

Cot 0 

Cot = x/y

However, any sign of 0 is always undefined.

400

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.

500

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.

Random

100

Sin 30° 

Sin = y/1 

The y coordinate of 30° on the unit circle is 1/2.

200

Sin 7π/4

Sin = y/1 

The y coordinate of 7π/4 on the unit circle is -√2/2.

300

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

400

Csc π/4

Csc = 1/y

The y coordinate of π/3 is √2/2. 1 over √2/2 is √2.

500

Sec 5π/6

Sec = 1/x

The x coordinate of 300° is -√3/2. 1 over -√3/2 is 2√ 3.

Random 2

100

Cos 135° 

Cos = x/1

The x-coordinate of 135° on the unit circle is -√2/2.

200

Cos π/6

Cos = x/1

The x-coordinate of Cos π/6 on the unit circle is √3/2

300

Sec 300° 

Sec = 1/x

The x coordinate of 300° is 1/2 1 over 1/2 is 2

400

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.

500

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.