File Name: circular functions and trigonometry .zip
- Module 2 - Circular Functions and Trigonometry.pdf
- 10.3: The Six Circular Functions and Fundamental Identities
As such, these functions earn the moniker circular functions. Not only do these observations help explain the names of these functions, they serve as the basis for a fundamental inequality needed for Calculus which we'll explore in the Exercises. Of the six circular functions, only cosine and sine are defined for all angles. However, when solving for tangent or cotangent, we usually stick with what we're dealt.
What this module is about This module is about determining the coordinates of angles in standard position in a unit circle; the six circular functions and finding the six circular functions of special angles; As you go over the discussion, examples and exercises, you will understand what circular functions are all about. Anytime you feel you are at a loss, do not hesitate to go back to the discussion and examples.
The x-coordinate of an angle in the along the unit circle is. If the terminal side of 4 the angle is located in the fourth quadrant, what is its y-coordinate? An angle measuring 30o is in standard position along the unit circle. What are its coordinates? What are the coordinates of the point of intersection of the terminal side of a o angle and the unit circle? Circular function c.
Cosine function b. Sine function d. Tangent function. In the previous module you have learned about the measures of arcs on a unit circle. Now, let us find the coordinates of the point where the terminal side of an angle in standard position lies. Consider a circle whose center is at 0,0. The circle of radius one with center at origin is called the unit circle. Y It intersects at the points: B. Knowing this equation, the other coordinate of a point of intersection of the unit circle and the terminal side of an angle in standard position can be obtained when one of its coordinates is given.
Example 1: 1 If the x-coordinate of an angle in standard position is , what is the y- 2 coordinate? Substitute the given value of x in the equation of the unit circle to obtain the value of y. Example 2: 1 The y-coordinate of an angle in standard position is. If the terminal side 7 of the angle lies between 90o and o, what is its x-coordinate? Substitute the given y- coordinate to find the x-coordinate. Try this out A. Let B be a point on the first quadrant of the unit circle.
The x-coordinate of 1 the point of intersection of the unit circle and the terminal side at B is. Name the angle in standard position. What special kind of triangle is formed out of points O, B, and F? How will you find the y-coordinate of B? What is the sign of the y-coordinate of B? What then are the coordinates of B? What are the coordinates of point C which is a reflection of point B in the second quadrant? In what quadrant is the abscissa or x-coordinate of the terminal point of an angle positive?
What about the y-coordinate or ordinate? In what quadrant is it positive? One of the coordinates of the point of intersection of the unit circle and the terminal side of an angle in standard position is given. Find the other coordinate.
Find the missing coordinate of the point of intersection of the unit circle and the terminal ray of an angle in standard position. In geometry, you have learned that angles whose measures are 30o, 45o, and 60o are called special angles. Now, let these special angles be amount of rotations on the unit circle as you can see in the figures below. Figure 1 Figure 2 Figure 3. Let P x, y be the terminal point of the angle. Drop a perpendicular from P to the x-axis and call the point of intersection M.
Notice that triangle OPM is a 30ooo triangle. Y Recall that in a 30ooo triangle, the 1 length of the leg opposite the 30o angle is the 2 P x,y 1 1 length of the hypotenuse. Consider this time figure 2. Notice that triangle OPM is a 45ooo triangle and that a 45o angle in standard position has its terminal side in the first quadrant. Y Recall that in a 45ooo, the lengths of P x, y the two legs are equal. Now, consider figure 3.
This can be done by determining the location of the terminal point of the given angle and the reference angle, A. Determine the coordinates of the point of intersection of the terminal side of a o angle on the unit circle.
Its 4 reference angle can be determined using the form on — A. Hence, its coordinates can be obtained using the coordinates of 30o but following the signs of the ordered pairs in the third quadrant.
It can be written 3 in the form on - A. X Observe that o is a reflection of 60o on the fourth quadrant. Hence, the coordinates of the terminal point can be obtained from the P x, y coordinates of 60o but following the signs of coordinates in the fourth quadrant. Determine the coordinates of the point of intersection of the terminal side of a - o angle on the unit circle. Thus, the reference angle of o is 30o. The coordinates of the terminal point can be obtained from the coordinates of the points of intersection of the terminal side of 30o and the unit circle.
The signs of the coordinates depend upon the position of the terminal side of the angle. Try this out Find the reference angle for each of the following angles on a unit circle and determine the coordinates of its terminal point. These coordinates of points have special names. The relation between the angle and the y-coordinate is called the sine function while the relation between the angle and the x-coordinate is called the cosine function. Evaluate sin 90o and cos 90o.
Its terminal side intersects the unit circle at the point 0, 1. Note that the sine and cosine functions of angles which are integral multiples of 90o can be easily evaluated. To evaluate angles that are not multiples of 90o, use the concept of reference angles as in the previous lesson.
Evaluate sin 30o and cos 30o. Evaluate sin o and cos o. Solution: Y The figure at the left shows o in standard position. Its reference angle is 30o and the terminal side intersects the. The unit circle below is divided into 8 congruent arcs. Complete the table. These are the tangent function, cotangent function, secant function and cosecant function.
The third basic function is the tangent function abbreviated as tan. This function is defined in terms of sine and cosine functions. Did you notice that tangent and cotangent functions are reciprocal functions? The same is true for secant and cosecant functions. The value of the tangent, cotangent, secant and cosecant of special angles can be obtained using their x and y coordinates. Evaluate tan 30o, cot 30o, sec 30o and csc 30o. Determine tan 45o, cot 45o, sec 45o and csc 45o.
Notice that the tan 45o and cot 45o are equal and that sec 45o and csc 45o are also equal. It is because the right triangle formed by a 45ooo is an isosceles right triangle.
Find tan 60o, cot 60o, sec 60o and csc 60o. Why is it so? It is because 30o and 60o are complementary angles. Hence, their functions are also complementary. Evaluate tan 0o, cot 0o, sec 0o, and csc 0o. Evaluate tan 90o. The circular functions of angles which are not multiples of the quadrantal angles and special angles can be obtained using a scientific calculator or a table of trigonometric functions.
In this module, the use of a scientific calculator is encouraged for you to make use of the technology. Caution is given that before you use a scientific calculator, that is, you have to familiarize yourself with the model of the scientific calculator you are going to use.
This module will not prescribe a particular scientific calculator. Answers in the examples are given up to six decimal places. Find the six circular functions of the following angles using the definition of the functions. Sine function is the relation between an angle and the y-coordinate while cosine function is the relation between an angle and the x-coordinate.
Tangent function is the ratio of y to x while cotangent function is the ratio of x to y, where x and y are the coordinates of the point of intersection of the terminal side of an angle in standard position and the unit circle. Secant function is the reciprocal of the cosine function while cosecant function is the reciprocal of the sine function.
The y-coordinate of an angle in the along the unit circle is -. If the terminal side of 5 the angle is located in the third quadrant, what is its x-coordinate?
Module 2 - Circular Functions and Trigonometry.pdf
In this module, we will revise the basics of triangle trigonometry, including the sine and cosine rules, and angles of any magnitude. We will then look at.
10.3: The Six Circular Functions and Fundamental Identities
To use trigonometric functions, we first must understand how to measure the angles. The radian measure of an angle is defined as follows. We say the angle corresponding to the arc of length 1 has radian measure 1.
In this geometry activity, students identify missing sides and angles using the unit circle. Radian Measure Technical Definition: An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of 1 radian. Use the information provided to write the standard form equation of each circle.