Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
LESSON 3: TRIGONOMETRIC RATIOS Learning Outcomes: To relate the trigonometric ratios to the coordinates of points in the unit circle To determine exact values for trigonometric ratios To identify the measures of angles that generate specific trigonometric values To solve problems using trigonometric ratios Recall: Suppose θ is any angle in standard position, and P(x, y) is any point on its terminal arm, at a distance r from the origin. If given the values (x, y) of the two smaller legs of a right angled triangle, what formula would you use to find the r value? Pythagorean Theorem, 𝑟 = √𝑥 2 + 𝑦 2 You can use a reference triangle to determine the three primary trigonometric ratios in terms of x, y and r. Knowing that: Express the sin, cos and tan ratios in terms of x, y and r: With a unit circle with radius of 1: 𝑥 = 𝑥 , 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 1 𝑦 sin 𝜃 = = 𝑦 , 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 1 cos 𝜃 = Therefore, you can describe the coordinates of any point 𝑃(𝜃) as (cos 𝜃, sin 𝜃). This is true for any point 𝑃(𝜃) at the intersection of the terminal arm of an angle θ and the unit circle. Reciprocal Trigonometric Ratios: 1 cosecant ratio csc sin cotangent ratio cot secant ratio sec 1 cos 1 tan These are reciprocals of the primary trigonometric ratios. Ex. The point (15, 8) lies on the terminal arm of as shown. Calculate the value of r and the exact values of the primary and reciprocal ratios. 1 2√2 Ex. The point B(− 3 , 3 ) lies at the intersection of the unit circle and the terminal arm of an angle θ in standard position. a. In what quadrant does point B lie? b. Determine the values of the six trigonometric ratios for θ. Express your answers in lowest terms. Exact values for the trigonometric ratios can be determined using special triangles. With the knowledge of the special triangles, the CAST rule can be used to tell us where each trigonometric ratio is positive. CAST Rule: Sine ratio All ratios The reciprocal trigonometric ratios positive positive follow the same framework as their corresponding primary ratio. Tangent Cosine ratio ratio positive positive The trigonometric ratios of any angle can be written as the same function of a positive acute angle called the reference angle with the sign of the ratio being determined by the CAST rule. Ex. Determine the exact value for each. a. 210 b. cos 5 3 𝜋 c. 𝑡𝑎𝑛 2 d. sec 60° e. sin(-300˚) Assignment: pg. 201-205 #1, 3-6, 8