Presentation on theme: "Identify a unit circle and describe its relationship to real numbers"— Presentation transcript:
17-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine Identify a unit circle and describe its relationship to real numbersEvaluate Trigonometric functions using the unit circleUse domain and period to evaluate sine and cosine functionsUse a calculator to evaluate trigonometric functionssin is an abbreviation for sinecos is an abbreviation for cosinetan is an abbreviation for tangentcsc is an abbreviation for cosecantSec is an abbreviation for secantCot is an abbreviation for cotangentHomework: Page , #1, 3, 11, 13, 15, 17
2In General The Unit Circle In your notes, please copy down the general ratios but keep in mind that for a unit circle r = 1.xP(x,y)ryNote 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 GeneralThe Unit Circle𝑐𝑠𝑐𝜃= 𝑟 𝑦 , y≠0𝑐𝑠𝑐𝜃= 1 𝑦 , y≠0𝑠𝑖𝑛𝜃=𝑦s𝑒𝑐𝜃= 𝑟 𝑥 , x≠0𝑐𝑜𝑠𝜃=𝑥s𝑒𝑐𝜃= 1 𝑥 , x≠0𝑡𝑎𝑛𝜃= 𝑦 𝑥 , 𝑥≠0𝑐𝑜𝑡𝜃= 𝑥 𝑦 , y≠0𝑡𝑎𝑛𝜃= 𝑦 𝑥 , 𝑥≠0𝑐𝑜𝑡𝜃= 𝑥 𝑦 , y≠0
3A few key points to write in your notebook: P(x,y)rxyA 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.
4(–3,2)r–32Example: 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 .
5Example: If is a fourth-quadrant angle and sin = –5/13, find cos . Example: If is a second quadrant angle and cos = –7/25, find sin .x–513Since is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.
6Determine the signs of sin , cos , and tan according to quadrant 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.P(–x,y)ryrxyP(x,y)xP(–x, –y)rxyP(x, –y)rxy
7Check 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.yxAllSineTangentCosine
8Find the reference angle. 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.yyP(–x,y)P(x,y)rrIf necessary, find a coterminal angle between 0 and 360 or 0 and 2π.Find the reference angle.Determine the sign by noting the quadrant.Evaluate and apply the sign.xxyP(x, –y)rxyxrP(–x, –y)
9Example: 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.
10Give each of the following in terms of the cosine of a reference angle: Example: cos 160The 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 20You try some:cos 182cos (–100)cos 365Try some sine problems now: Give each of the following in terms of the sine of a reference angle:sin 170sin 330sin (–15)sin 400
12Give 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 45sin 135sin 225cos (–30)cos 330sin 7/6cos /4