4-2 Trigonometric Functions: The Unit Circle Identify a unit circle and describe its relationship to real numbers Evaluate Trigonometric functions using the unit circle Use domain and period to evaluate sine and cosine functions Use a calculator to evaluate trigonometric functions sin is an abbreviation for sine cos is an abbreviation for cosine tan is an abbreviation for tangent csc is an abbreviation for cosecant Sec is an abbreviation for secant Cot is an abbreviation for cotangent

y 𝑥 2 + 𝑦 2 =1 Unit Circle (0, 1) (𝑥, 𝑦) 𝑟=1 y (−1, 0) 𝜃 (1, 0) x x 𝑐𝑠𝑐𝜃= 1 𝑦 , y≠0 𝑠𝑖𝑛𝜃=𝑦 (0, −1) 𝑐𝑜𝑠𝜃=𝑥 s𝑒𝑐𝜃= 1 𝑥 , x≠0 𝑡𝑎𝑛𝜃= 𝑦 𝑥 , 𝑥≠0 𝑐𝑜𝑡𝜃= 𝑥 𝑦 , y≠0

Which of the following represents r in the figure below Which of the following represents r in the figure below? (Click on the blue.) x P(x,y) r y Close. The Pythagorean Theorem would be a good beginning but you will still need to “get r alone.” You’re kidding right? (xy)/2 represents the area of the triangle! CORRECT!

Which of the following represents sin  in the figure below Which of the following represents sin  in the figure below? (Click on the blue.) x P(x,y) r y Sorry. Does Some Old Hippy Caught Another Hippy Tripping on Acid sound familiar? y/x represents tan. Sorry. Does SohCahToa ring a bell? x/r represents cos. CORRECT! Well done.

Which of the following represents cos  in the figure below Which of the following represents cos  in the figure below? (Click on the blue.) x P(x,y) r y CORRECT! Yeah! Oops! Try something else. Sorry. Wrong ratio.

Which of the following represents tan  in the figure below Which of the following represents tan  in the figure below? (Click on the blue.) x P(x,y) r y CORRECT! Yeah! Try again. Try again.

In General The Unit Circle In your notes, please copy down the general ratios but keep in mind that for a unit circle r = 1. x P(x,y) r y Note that csc, sec, and cot are reciprocals of sin, cos, and tan. Also note that tan and sec are undefined when x = 0 and csc and cot are undefined when y = 0. In General The Unit Circle 𝑐𝑠𝑐𝜃= 𝑟 𝑦 , y≠0 𝑐𝑠𝑐𝜃= 1 𝑦 , y≠0 𝑠𝑖𝑛𝜃=𝑦 s𝑒𝑐𝜃= 𝑟 𝑥 , x≠0 𝑐𝑜𝑠𝜃=𝑥 s𝑒𝑐𝜃= 1 𝑥 , x≠0 𝑡𝑎𝑛𝜃= 𝑦 𝑥 , 𝑥≠0 𝑐𝑜𝑡𝜃= 𝑥 𝑦 , y≠0 𝑡𝑎𝑛𝜃= 𝑦 𝑥 , 𝑥≠0 𝑐𝑜𝑡𝜃= 𝑥 𝑦 , y≠0

A few key points to write in your notebook: P(x,y) r x y A few key points to write in your notebook: P(x,y) can lie in any quadrant. Since the hypotenuse r, represents distance, the value of r is always positive. The equation x2 + y2 = r2 represents the equation of a circle with its center at the origin and a radius of length r. Hence, the equation of a unit circle is written x2 + y2 = 1. The trigonometric ratios still apply no matter what quadrant, but you will need to pay attention to the +/– sign of each.

Can you complete this chart? 60 30 45 60 30 45

Check your work!!!!!! Write this table in your notes! See any patterns?

So now what? Use your knowledge to evaluate the six trigonometric functions at a given . Example: Evaluate the six trigonometric functions when θ= 2𝜋 3 . θ= 2𝜋 3 corresponds with the point − 1 2 , 3 2 which means 𝑥=− 1 2 , 𝑦= 3 2 , and 𝑟=1. 𝑠𝑖𝑛𝜃=𝑦 3 2 𝑐𝑠𝑐𝜃= 1 𝑦 , y≠0 2 3 = 2 3 3 𝑐𝑜𝑠𝜃=𝑥 − 1 2 s𝑒𝑐𝜃= 1 𝑥 , x≠0 − 2 𝑡𝑎𝑛𝜃= 𝑦 𝑥 , 𝑥≠0 3 2 − 1 2 3 2 ∙− 2 1 =− 3 𝑐𝑜𝑡𝜃= 𝑥 𝑦 , y≠0 −1 3 =− 3 3

YOU TRY IT! Evaluate the six trigonometric functions when θ= 4𝜋 3 . Then again when θ=2π. And one more time when θ= 𝜋 2

HOMEWORK: Page 274, #31-36. Don’t want to bring the book home? Evaluate the six trigonometric functions when 31.) θ= 3𝜋 4 32.) θ= 5𝜋 6 33.) θ= 𝜋 2 34.) θ= 3𝜋 2 35.) θ=− 2𝜋 3 36.) θ=− 7𝜋 4