4.3A 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) r2 = x2 + y2 r r = √x2 + y2 y q x The three reciprocal ratios are defined as follows: Math 30-1

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? No, the radius of a unit circle is 1. P(-2, 3) Determine the exact value of the six trigonometric ratios for angle q. 3 q -2 r2 = x2 + y2 r2 = (-2)2 + (3)2 r2 = 4 + 9 r2 = 13 r = √ 13 Math 30-1

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

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

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

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

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

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

4.3B 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

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

Using Technology sin 30º= Given sin 30º Given sin θº = ½ 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

Deteriming the Measure of an Angle Given the Ratio Determine the measure of θ. 0 ≤ θ < 3600 cos θ = -0.6691 trig function angle trig ratio 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. θ = 1320 or 2280 nSolve(cos(x)=0.6691,x) Alternate Method Math 30-1

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 Point A is in quadrant IV Reference Reference Math 30-1

Determine Angle q , Given an Exact Trigonometric Ratio Determine the value of angle q. 0 ≤ q < 2p y is positive in Q1 or Q2 RA = x is negative in Q2 or Q3 y/x is negative in Q2 or Q4 RA = RA = Math 30-1

Find the Measure of an Angle q , Given an Exact Trigonometric Ratio 0 ≤ q < 2p x is neg in Q2 & Q3 y/x is pos in Q1 & Q3 RA = RA = y is pos in Q1 & Q2 Quadrantal x value is 0 RA = Math 30-1

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

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