7-3 Sine and Cosine (and Tangent) Functions 7-4 Evaluating Sine and Cosine 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 Homework: Page 279-280, #1, 3, 11, 13, 15, 17
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.
(–3,2) r –3 2 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 .
Example: 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 –5 13 Since is in quadrant IV, the coordinate signs will be (+x, –y), therefore x = +12.
Determine 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) r y r x y P(x,y) x P(–x, –y) r x y P(x, –y) r x y
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 All Sine Tangent Cosine
Find 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. y y P(–x,y) P(x,y) r r If 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. x x y P(x, –y) r x y x r P(–x, –y)
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.
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 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
Can you complete this chart? 60 30 45 60 30 45
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