Vocabulary
Unit Circle- The circle x² + y² = 1, which has center (0,0) and radius 1. For an angle θ in standard position, the terminal side of θ intersects the unit circle at the point (cos θ, sin θ).
Quadrantal Angle- An angle in standard position whose terminal side lies on an axis.
Reference Angle- If θ is an angle in standard position, its reference angle is the acute angle θ′ formed by the terminal side of θ and the x-axis.
Quadrantal Angle- An angle in standard position whose terminal side lies on an axis.
Reference Angle- If θ is an angle in standard position, its reference angle is the acute angle θ′ formed by the terminal side of θ and the x-axis.
Essential Information
The definition of right triangles of trigonometric functions can be generalized so that they apply to any angle in standard position.
General Definitions of Trigonometric Functions
Let θ be an angle in standard position, and let (x,y) be the point where the terminal side of θ intersects the circle x² + y² = r².
General Definitions of Trigonometric Functions
Let θ be an angle in standard position, and let (x,y) be the point where the terminal side of θ intersects the circle x² + y² = r².
These functions are sometimes called circular functions.
It is easy to use the unit circle to find trigonometric functions of quadrantal angles. A quadrantal angle is an angle in standard position whose terminal side lies on an axis. The measure of a quadrantal angle is always a multiple of 90⁰ or π/2 radians.
Reference Angle Relationships
Let θ be an angle in standard position. The reference angle for θ is the acute angle θ′ formed by the terminal side θ and the x-axis. The relationship between θ and θ′ is shown for nonquadrantal angles θ such that 90⁰< θ <360⁰ (π/2 < θ <2π).
Let θ be an angle in standard position. The reference angle for θ is the acute angle θ′ formed by the terminal side θ and the x-axis. The relationship between θ and θ′ is shown for nonquadrantal angles θ such that 90⁰< θ <360⁰ (π/2 < θ <2π).
Evaluating Trigonometric Functions
Reference angles allow for evaluation of a trigonometric function for any angle θ. The sign of the trigonometric function values depends on the quadrant in which θ lies.
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