7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine 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 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

x 0 P(x,y) r y x y Which of the following represents r in the figure below? (Click on the blue.) 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!

x 0 P(x,y) r y x y Which of the following represents sin  in the figure below? (Click on the blue.) Sorry. Does SohCahToa ring a bell? x/r represents cos . CORRECT! Well done. Sorry. Does Some Old Hippy Caught Another Hippy Tripping on Acid sound familiar? y/x represents tan .

x 0 P(x,y) r y x y Which of the following represents cos  in the figure below? (Click on the blue.) CORRECT! Yeah! Sorry. Wrong ratio.Oops! Try something else.

x 0 P(x,y) r y x y Which of the following represents tan  in the figure below? (Click on the blue.) CORRECT! Yeah! Try again.

x 0 P(x,y) r y x y In your notes, please copy down the general ratios but keep in mind that for a unit circle r = 1. In General The Unit Circle 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.

0 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 x 2 + y 2 = r 2 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 x 2 + y 2 = 1. The trigonometric ratios still apply no matter what quadrant, but you will need to pay attention to the +/– sign of each.

Example: If the terminal ray of an angle  in standard position passes through (–3, 2), find sin  and cos . You try this one in your notebook: If the terminal ray of an angle  in standard position passes through (–3, –4), find sin  and cos . (–3,2) r –3 2 Check Answer

Example: If  is a fourth-quadrant angle and sin  = –5/13, find cos . 13 –5 x Since  is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.

Example: If  is a second quadrant angle and cos  = –7/25, find sin . Check Answer

x 0 P(–x,y) r y 0 P(–x, –y) r x y P(x,y) 0 r x y 0 P(x, –y) r x y Determine the signs of sin , cos , and tan  according to quadrant. Quadrant II is completed for you. Repeat the process for quadrants I, III, and IV. Hint: r is always positive; look at the red P coordinate to determine the sign of x and y.

y x AllSine TangentCosine Check your answers according to the chart below: All are positive in I. Only sine is positive in II. Only tangent is positive in III. Only cosine is positive in IV.

y x AllStudents TakeCalculus A handy pneumonic to help you remember! Write it in your notes!

x 0 P(–x,y) r y 0 P(–x, –y) r x y P(x,y) 0 r x y 0 P(x, –y) r x y Let  be an angle in standard position. The reference angle  associated with  is the acute angle formed by the terminal side of  and the x-axis. 1.If necessary, find a coterminal angle between 0  and 360  or 0 and 2π. 2.Find the reference angle. 3.Determine the sign by noting the quadrant. 4.Evaluate and apply the sign.

Example: Find the reference angle for  = 135 . You try it: Find the reference angle for  = 5  /3. You try it: Find the reference angle for  = 870 . Check Answer

Give each of the following in terms of the cosine of a reference angle: Example: cos 160  The angle  =160  is in Quadrant II; cosine is negative in Quadrant II (refer back to All Students Take Calculus pneumonic). The reference angle in Quadrant II is as follows:  =180 –  or  =180 – 160 = 20. Therefore: cos 160  = –cos 20  You try some: cos 182  cos (–100  ) cos 365  Check Answer

Try some sine problems now: Give each of the following in terms of the sine of a reference angle: sin 170  sin 330  sin (–15  ) sin 400  Check Answer

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

Check your work!!!!!! Write this table in your notes!

Give the exact value in simplest radical form. Example: sin 225  Determine the sign: This angle is in Quadrant III where sine is negative. Find the reference angle for an angle in Quadrant III:  =  – 180  or  = 225  – 180  = 45 . Therefore:

You try some: Give the exact value in simplest radical form: sin 45  sin 135  sin 225  cos (–30  ) cos 330  sin 7  /6 cos  /4 Check Answer

Homework: Page , #1, 3, 11, 13, 15, 17