The Unit Circle & Trig Ratios

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Presentation transcript:

The Unit Circle & Trig Ratios B. Afghani Algebra PP2.ppt The Unit Circle & Trig Ratios LBUSD Math Office, 2002

Today’s Learning Questions: How do I create a unit circle? What are the “special angles” and how do I graph their arcs & triangles? How do I derive the side lengths for isosceles triangles whose hypotenuse = 1?

Today’s Learning Questions: How do I derive the side lengths for 30- 60-90 triangles whose hypotenuse = 1? What 5 values and their opposites are the sine and cosine values for each special angle between 0° and 360°? How may I use the unit circle graph to find the sine of a special angle? The cosine of an angle?

Today’s Learning Bonus Question: Why is the ratio of sine θ to cosine θ called tangent θ?

Angles Mathematicians Like Solve for a. 45° 1 a 45° a

Angles Mathematicians Like Solve for a. a 1 1 b 60° a a 60° 1

Angles Mathematicians Like Solve for b. a 1 1 b 60° a a 60° 1

Put in order from least to greatest:

Please label these items on your coordinate grid: The origin The x-axis The y-axis

What makes a circle a “unit circle”? y-axis r = 1 (0, 0) the origin What makes a circle a “unit circle”? x-axis

(0, 1) What are the coordinates for the graphed points? y (1, 0) (-1, 0) x (0, -1)

y 1 y  x x

Trigonometric Ratios

90° y 90° 120° 60° 45° 135° 150° 45° 30° a 1 0°, 360° 0°, 360° 180° 180° 0° 0° 45° x 210° 330° 225° 315° 240° 270° 270° 300°

y 0, x

90° y 90° 120° 60° 45° 135° 150° 45° 30° a 1 0°, 360° 0°, 360° 180° 180° 0° 0° 45° x 210° 330° 225° 315° 240° 270° 270° 300°

Trig Ratios

y 90° 120° 60° 150° 30° 180° 0°, 360° 0° x 210° 330° 240° 270° 300°

y 0, x

Trig Ratios 60° 300°

Today’s Learning Questions: How do I create a unit circle? What are the “special angles” and how do I graph their arcs & triangles? How do I derive the side lengths for isosceles triangles whose hypotenuse = 1?

Today’s Learning Questions: How do I derive the side lengths for 30- 60-90 triangles whose hypotenuse = 1? What 5 values and their opposites are the sine and cosine values for each special angle between 0° and 360°? How may I use the unit circle graph to find the sine of a special angle? The cosine of an angle?

Today’s Learning Bonus Question: Why is the ratio of sine θ to cosine θ called tangent θ?

Bonus Question