- A Unit Circle is a type of circle whose center is at and whose radius length is equal to unit.
- This circle is modeled by the equation . This equation is derived using the distance formula.
- The circumference of the unit circle is equal to units or approximately equal to units.
- To inspect whether a point is on the unit circle, we check the coordinates if they satisfy the equation .
- The standard arc on the circle starts at the point and travels counterclockwise if it is positive and clockwise if it is negative.
- Since the circumference of the unit circle is , then the arc lengths are rational multiples of .
- Consider the unit circle being divided by the coordinate axes into four congruent arcs.
- As point X moves from point A counterclockwise and terminates at point B, the length of arc is of or .
- The length of arc is one-half of the circumference, which is .
- The length of arc is of the circumference, which is .
- If we consider as the length of the arc, then we have the following:
Properties of the Unit Circle and the Special Angles
- As shown in the figure, each point on the unit circle satisfies the equation .
- Each point on the circle has coordinates corresponding to a special angle or multiple of special angles, such as and .
- Study the tables below.
- The domain of the unit circle is . The range is .
- Every real number (arc length) is associated uniquely with the central angle that subtends the standard arc on the unit circle.
- The relationship , for any integer is called the Periodic Theorem. Typical points on the unit circle range from to . Once the angle of the circular function exceeds , then this multiple of has to be subtracted from the given angle. The difference is then equivalent to one common value between to .
Trigonometric Functions and the Unit Circle
- Let us consider a real number and let be a point on the unit circle corresponding to the angle .
- For each arc length on the unit circle, with the starting point and terminal point , the values of and are real numbers, then we have and .
- Study the figure below.
- Point is equal to . This indicates that and . This is based on and
- Using the y and x coordinates, we can define the other trigonometric functions of .
- , and .
- Trigonometric Functions are also called Circular Functions or Periodic Functions because they behave in a cyclic or repetitive manner.
- To illustrate, we have and as these functions have a period of .
Trigonometric Functions of Any Angle
- If we let be any angle in standard position, it indicates that its vertex is at the origin and the initial side is the positive x-axis.
- The six trigonometric functions are described in the tables below, and their respective signs in the four quadrants.
- For any angle , its reference angle is the acute angle that the terminal side of makes with the positive x-axis or negative x-axis.
- If , then the reference angle is equal to .
- In Quadrant II, the reference angle is .
- In Quadrant III, the reference angle is .
- In Quadrant IV, the reference angle is .
In general, the angle and where have the same terminal side. These angles are called co-terminal angles.
Example 1. Find the exact value of using the standard unit circle.
terminates in Quadrant I, which forms in the positive x-axis.
Consider the angle and choose the point , on the unit circle.
This shows that
Note that the sign is positive since in Quadrant I.
Example 2. Given that and , find the other trigonometric functions of .
If the tangent function is positive and sine function is negative, then is Quadrant III.
Solve for the radius:
- Trigonometric functions of different special angles are summarized in the tables below.
- The cosine function and secant functions are even. To illustrate, we have and .
- The sine, cosecant, tangent, and cotangent functions are odd.
- Always be mindful of the sign of each trigonometric function in the four quadrants.
- Reference angles can be used to evaluate trigonometric functions.
- Suppose that and we want to find the other trigonometric values of , we solve for the radius. The radius is always positive.
- The angles associated with trigonometry (in the unit circle approach) are directed angles. Angles obtained from the counterclockwise direction are positive while angles obtained from the clockwise direction are negative.
- Keith Madrilejos
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