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4.3A Trigonometric Ratios

Discover how to find exact and approximate values for trigonometric ratios using the unit circle, angles in standard position, and special angles. Learn to determine ratios, angles, and points in various quadrants with math formulas and technology.

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4.3A Trigonometric Ratios

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

  2. Finding the Trig Ratios of an Angle in Standard Position The point P(-2, 3) is on the terminal arm of qin standard position. Does point P(-2, 3) lie on the unit circle? No, the radius of a unit circle is 1. Determine the exact value of the six trigonometric ratios for angle q. P(-2, 3) 3 q -2 r2 = x2 + y2 r2 = (-2)2 + (3)2 r2 = 4 + 9 r2 = 13 r = √ 13 Math 30-1

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

  4. 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

  5. (0, 1) (1, 0) (-1, 0) (0, -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. Math 30-1

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

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

  8. Approximate Values for Trig Ratios (four decimal places) 1. sin 250 = 2. cos 3750 = 3. 4. csc 1.57= 5. 6. cot 2700= The mode of calculator must match the domain of the angle, degrees or radians. Math 30-1

  9. 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

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

  11. 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

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

  13. 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

  14. 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

  15. 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

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

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

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