Unit Circle The values for sine and cosine create a nice pattern. If we let cos θ be the x value and sin θ be the y value, the plot looks like an arc. There are some values of for which we can find the exact values of cos θ and sin θ; these are the values found on the special right triangles.

Special Right Triangles s s s 2s2s s√2s√2 s√3s√3

The Unit Circle: Quadrant I

Trigonometric Functions: θ>90° Even though we can’t construct a right triangle with non-acute angles, we can find the sine and cosine of angles larger than 90°. These values make sense for application problems that we will look at later. To find sin θ, cos θ, etc., when θ>90°, we need to calculate θ’s reference angle.

Reference Angle The reference angle of θ is the acute angle formed by the terminal ray of θ and the x-axis. Ex.: θ=132°, θ ref =48° ; θ=275°, θ ref =85° Reference angles allow us to construct the remainder of the circle.

The Unit Circle: Quadrants II–IV The trigonometric functions of θ are the same magnitude (numerical size) as those of θ ref. The difference may be the size, which is determined by the quadrant in which the angle lies.

Sine and Cosine Because sine is the y value on the unit circle, it is positive for angles in quadrants I and II and negative for angles in quadrants III and IV. Because cosine is the x value on the unit circle, it is positive for angles in quadrants I and IV and negative for angles in quadrants II and III.

The Other Trigonometric Functions Once we know sine and cosine, we can find the other trigonometric functions using identities. An identity is a mathematical statement that is always true (as opposed to an equation, which is only true for some values).

The Six Trigonometric Functions SineCosecant CosineSecant TangentCotangent

Positive or Negative? The mnemonic “All Students Take Calculus” can be used to determine in which quadrants sine, cosine, and tangent are positive.  I: All  II: Sine +, Cosine and Tangent –  III: Tangent +, Sine and Cosine –  IV: Cosine –, Sine and Tangent + (Note: reciprocals always have the same sign... so secant is positive wherever cosine is, etc.)

You Should Be Able To… Use the unit circle to find the sine and cosine for the angles that come from the special right triangles. Determine reference angles and use the unit circle to find the sine and cosine for non-acute angles. Find the other functions (tangent, etc.) based on the ratios of sine and cosine.

Quick Check Find the reference angle and the exact values of the six trigonometric functions of: θ=7π/6.