4 3 A Trigonometric Ratios Point Px y

4. 3 A Trigonometric Ratios Point P(x, y) is the point on the terminal arm of angle , an angle in standard position, that intersects a circle. P(x, y) r 2 = x 2 + y 2 r= √x 2 + y 2 r y q x The three reciprocal ratios are defined as follows: Math 30 -1 1

Finding the Trig Ratios of an Angle in Standard Position The point P(-2, 3) is on the terminal arm of q in standard position. Does point P(-2, 3) lie on the unit circle? Determine the exact value of the six trigonometric ratios for angle q. P(-2, 3) 3 No, the radius of a unit circle is 1. q -2 r 2 = x 2 + y 2 r 2 = (-2)2 + (3)2 r 2 = 4 + 9 r 2 = 13 r = √ 13 Math 30 -1 2

Review of Exact Trig Ratios is a point on the terminal arm of angle θ that intersects the unit circle Math 30 -1 3

The point lies at the intersection of the unit circle and the terminal arm of an angle θ in standard position. Diagram q r=1 Math 30 -1 4

The Unit Circle (0, 1) (1, 0) (-1, 0) Exact Values for the trigonometric ratios can be determined using multiples of special angles for points P(θ) on the unit circle. (0, -1) Math 30 -1 5

Exact Values For Trigonometric Ratios 1. sin 3300 = RA = 300 quadrant IV 3. tan RA = quadrant III 2. cos RA = quadrant II 4. cos = RA = quadrant IV Math 30 -1 6

Using the unit circle Determine the exact value of: Math 30 -1 7

Approximate Values for Trig Ratios (four decimal places) The mode of calculator must match the domain of the angle, degrees or radians. 1. sin 250 = 2. cos 3750 = 3. 4. csc 1. 57 = 5. 6. cot 2700= Math 30 -1 8

4. 3 B Trigonometric Ratios Does point P lie on the unit circle? Yes If Point P is the point on the terminal arm of angle that intersects the unit circle, in which quadrant does P lie? III Determine the exact values of the 6 trigonometric ratios. Math 30 -1 9

Determine the exact value of the trig ratios given Must be in Quad IV x q -5 7 Math 30 -1 10

Using Technology sin 30º= trig function angle trig ratio Given sin 30º Given sin θº = ½ Asking for the sine ratio value when angle θ is 30° Asking for the value of angle θ when the sine ratio is ½ Enter sin 30°, the answer is a ratio Use the inverse of the sine ratio which gives the angle. Enter sin-1 (½), the answer is an angle. When a positive ratio is used, the calculator will display the reference angle. The mode of calculator must match the domain of the angle, degrees or radians. Math 30 -1 11

Deteriming the Measure of an Angle Given the Ratio Determine the measure of θ. 0 ≤ θ < 3600 cos θ = -0. 6691 trig angle trig ratio function The trig ratio is negative, indicating that the x-coordinate is negative, therefore, the angle θ would be found in Quadrants II or III. The trig ratio is not an exact value: use inverse cosine. Domain is in degrees. The reference angle is 480. Alternate Method n. Solve(cos(x)=0. 6691, x) θ = 1320 or 2280 Math 30 -1 12

The point lies at the intersection of the unit circle and the terminal arm of an angle θ in standard position. Determine the measure of angle θ for each domain. Degrees Radians Point A is in quadrant IV Reference Math 30 -1 13

Determine Angle q , Given an Exact Trigonometric Ratio Determine the value of angle q. 0 ≤ q < 2 p y is positive in Q 1 or Q 2 RA = x is negative in Q 2 or Q 3 y/x is negative in Q 2 or Q 4 RA = Math 30 -1 14

Find the Measure of an Angle q , Given an Exact Trigonometric Ratio 0 ≤ q < 2 p x is neg in Q 2 & Q 3 y/x is pos in Q 1 & Q 3 RA = y is pos in Q 1 & Q 2 Quadrantal x value is 0 RA = Math 30 -1 15

Jackie stated that Is she correct? What is the relationship between the angles? Complete each equality. Math 30 -1 16

Page 201 1, 2, 3, 5 a, b, 6, 7, 8, 10, 11 a, c, 12, 13, 14, 16, 19 Math 30 -1 17