Study Notes
The unit circle is a fundamental concept in trigonometry, representing a circle with a radius of one unit centered at the origin. It is used to define the sine, cosine, and tangent functions based on the coordinates of points on the circle.
- Unit Circle — A circle of radius one unit with its center at the origin. Example: For a point (x, y) on the unit circle, x = cosθ and y = sinθ.
- Sine Function — A trigonometric function that represents the y-coordinate of a point on the unit circle. Example: sinθ = 0 when θ = 0°, 180°, 360°; maximum value is 1 at θ = 90°.
- Cosine Function — A trigonometric function that represents the x-coordinate of a point on the unit circle. Example: cosθ = 1 when θ = 0°, 360°; minimum value is -1 at θ = 180°.
- Tangent Function — A trigonometric function defined as the ratio of sine to cosine. Example: tanθ = 0 when θ = 0°, 180°, 360°; undefined at θ = 90°, 270°.
Exam Tips
Key Definitions to Remember
- Unit Circle: A circle with a radius of one unit centered at the origin.
- Sine Function: y-coordinate of a point on the unit circle.
- Cosine Function: x-coordinate of a point on the unit circle.
- Tangent Function: Ratio of sine to cosine.
Common Confusions
- Confusing the maximum and minimum values of sine and cosine.
- Forgetting that tangent is undefined at θ = 90° and 270°.
Typical Exam Questions
- What is the maximum value of sinθ? The maximum value is 1 at θ = 90°.
- When is cosθ equal to -1? Cosθ is -1 at θ = 180°.
- At what angles is tanθ undefined? Tanθ is undefined at θ = 90° and 270°.
What Examiners Usually Test
- Understanding of the unit circle and its properties.
- Ability to identify key values of sine, cosine, and tangent functions.
- Application of trigonometric functions to solve problems.