Exploring Trigonometric Values on the Unit Circle
Learning Objective
At the end of the lesson you will be able to identify and use key values of sine and cosine for standard angles on the unit circle.
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Learning Objective
At the end of the lesson you will be able to identify and use key values of sine and cosine for standard angles on the unit circle.
What do you already know about unit circle values?
What is the Unit Circle?
The unit circle is a circle with radius one, centred at the origin of a coordinate plane. It's fundamental in trigonometry for defining values of sine, cosine, and tangent for all angles.
Key Angles on the Unit Circle
Standard angles such as 0°, 30°, 45°, 60°, 90°, and their multiples appear frequently on the unit circle. Each corresponds to a specific point with coordinates (cos θ, sin θ).
Special Trigonometric Values
sin 0° = 0 | sin 30° = 1/2 | sin 45° = √2/2 | sin 60° = √3/2 | sin 90° = 1
cos 0° = 1 | cos 30° = √3/2 | cos 45° = √2/2 | cos 60° = 1/2 | cos 90° = 0
Sine Values
Cosine Values
Signs in Each Quadrant
Quadrant I: All positive. Quadrant II: Sine positive. Quadrant III: Tangent positive. Quadrant IV: Cosine positive. Remember "All Students Take Calculus" for the signs.
Patterns and Symmetry
Sine and cosine values repeat in a predictable cycle around the unit circle, showing symmetry at 180° and 360°, and relating negative and positive values.
Applying Unit Circle Values
With these special values and sign patterns, you can solve trigonometric equations and understand periodic functions in mathematics and science.
Reflect and Connect
Think of examples in real life where circular motion or cycles occur. How do you think the unit circle helps us understand repeating patterns?
Write down 3 things you learned in this lesson.
Write down 2 things you want to know more about.
Ask 1 question about something you haven't quite understood yet.
