The Unit Circle

Sine t = y/rCosine t = x/rTangent t = y/xCotangent t = x/ySecant t = r/xCosecant t = r/y

The variable r is the Radius. This will always equal 1 on the unit circle.

Here at 30° x = √3/2Also at 30° y = ½45° both x & y = √2/2At 60° x = ½Also at 60° y = √3/2

By using the Pythagorean Theorem the sides of a 45°, 45°, 90° can be found.

You will use these on to find all Six Trigonometric Functions.

The Hypotenuse must equal1 therefore the sides opposite45° is √2/2.

Use the Pythagorean Theorem to find the sides of a 30°, 60°, 90° Triangle.

Remember the Hypotenuse must equal 1.

Therefore, the side opposite 30° = ½.

The side opposite 60° = √3/2.

Here you can see Sine t = y/r.Cosine t = x/r.Notice that r = 1.

Sin 30° = 1/2 Cos 30° = √3/2Tan 30° = 3√3Cot 30° = √3Sec 30° = 2√3/3Csc 30° = 2

Sin 45° = √2/2Cos 45° = √2/2Tan 45° = 1Cot 45° = 1Sec 45° = √2Csc 45° = √2

Sin 60° = √3/2Cos 60° = 1/2Tan 60° = √3Cot 60° = 3√3Sec 60° = 2Csc 60° = 2√3/3

Notice that at 90° the (x,y) is (0,1).Sin 90° = 1Cos 90° = 0Tan 90° = undefinedCot 90° = 0Sec 90° = undefinedCsc 90° = 1

All y values stay the same as in Quadrant I.

All x values are negative here. The Six Trig Functions are reflected

across the y-axis here. (-cos t, sin t)

All x values are negative here.All y values are negative here. The Six Trig Functions are reflected

across the origin from Quadrant I. (-cos t, -sin t)

All x values stay the same as in Quadrant I.

All y values are negative here. The Six Trig Functions are reflected

across the x-axis here. (cos t, -sin t)

Sin 135° = √2/2Cos 135° = - √2/2Tan 135° = -1Cot 135° = -1 Sec 135° = -√2Csc 135° = √2

Sin 150° = -√3/2Cos 150° = 1/2Tan 150° = -√3Cot 150° = -3√3Sec 150° = 2Csc 150° = -2√3/3

Sin 225° = -√2/2Cos 225° = -√2/2Tan 225° = 1Cot 225° = 1Sec 225° = -√2Csc 225° = -√2

Sin 300° = -√3/2Cos 300° = 1/2Tan 300° = -√3Cot 300° = -3√3Sec 300° = 2Csc 300° = -2√3/3

Sin 315° = √2/2 Cos 315° = - √2/2 Tan 315° = -1Cot 315° = -1Sec 315° = -√2Csc 315° = √2

All the Trig Functions in the second Quadrant have the same values as Quad I except the x values are now negative.

The same thing happens in Quad III both the x and y values are negative here.

In Quad IV the x values are positive and the y values here are negative.