Angles in the four quadrants (Unit circle)
- Author:
- Cliff Packman, Nick Kochis, talljerome
- Topic:
- Angles, Circle, Unit Circle
Keywords
Inquiry questions
Moving into the other quadrants, we need to use the symmetry of the circle
2. Sine Waves: As we sail from 0° to 90°, the sine value climbs the rigging. How high does it reach at 90°? Now, plunge down to 270°—what happens to the sine value?
3. Reflective Waters: The applet shows that cos(160°) is about -0.94. If we reflect across the y-axis to 200°, does the cosine value change? What does this tell us about symmetry in the unit circle?
3. Reflective Waters: The applet shows that cos(160°) is about -0.94. If we reflect across the y-axis to 200°, does the cosine value change? What does this tell us about symmetry in the unit circle?
4. Quadrant Quest: Each quadrant holds its secrets. In which quadrants will you find positive sine values? And where will cosine lead us to positive shores?
5. Angle Amplification: Suppose we amplify our angle from 20° to 200°, passing through the stormy sea of the third quadrant. How does the sign of cosine and sine change?
6. Trigonometric Treasure: If cos(20°) is approximately 0.94, can we predict cos(160°) without a compass? Use the unit circle to verify your guess and find the value.
7. Full Circle: As we complete our 360° journey, can you predict the sine and cosine values at the cardinal points of 0°, 90°, 180°, and 270°? Confirm your predictions using the unit circle.
8. Opposite Odyssey: We know angles have opposite pairs on the unit circle. If the cosine of 20° is 0.94, what's the cosine of its opposite angle? Use the unit circle to find out!
9. Sine and Cosine Saga: Create your own angle and predict its sine and cosine values. Then, embark on a quest to find an angle in a different quadrant with the same sine value.
10. Circle of Life: The unit circle is never-ending. If you keep adding 360° to an angle, what happens to the sine and cosine values? Do they change or remain steadfast like the North Star?
Part 2 - The unit circle exact values
Activity 1: Finding sin(60°), cos(60°), and tan(60°) 1. **Draw an equilateral triangle** with side length 2 (for simplicity). Label the vertices A, B, and C. 2. **Draw the altitude** from vertex A to the midpoint of the base BC, creating two 30°-60°-90° right triangles. Let's call the midpoint D. 3. **Determine lengths** of AD, BD, and AB. Since AB = 2, and BD = 1 (half of AB), use Pythagoras' theorem to find AD. 4. **Calculate trigonometric ratios** for 60° using the right triangle ABD.
Activity 2: Finding sin(30°), cos(30°), and tan(30°) 1. Using the same triangle, **apply the trigonometric functions** to angle BAD (30°). 2. **Calculate** the sine, cosine, and tangent of 30° using the lengths found in Activity 1.
Activity 3: Finding sin(45°), cos(45°), and tan(45°) 1. Draw a right isosceles triangle** with the two legs of equal length. For simplicity, let each leg have a length of 1. 2. Label the hypotenuse** and calculate its length using Pythagoras' theorem. 3. Calculate trigonometric ratios** for 45° using the lengths of the legs and the hypotenuse.