Circular-Functions-An-Overview Lesson.pptx

This lesson will discuss about the circular functions and finding the exact values using reference angle.

1. Circular Functions: An Overview Circular functions are essential tools in mathematics, physics, and engineering. They describe the relationships between angles and the sides of right triangles. They are used in various applications, from modeling periodic phenomena to solving trigonometric equations. by Sherra Mae Bagood

2. Circular Function can be defined in terms of an arc length and the coordinates of the terminal point of the arc on the unit circle. Let ɵ be an angle in the standard position and let the point ( x,y ) be a point on the terminal side of ɵ.

3. The Unit Circle   NOTE: The radius of unit circle is equal to 1

4. Sine, Cosine, and Tangent Functions 1 Sine Function The sine function (sin θ) is defined as the y-coordinate of a point on the unit circle corresponding to an angle θ. 2 Cosine Function The cosine function (cos θ) is defined as the x-coordinate of a point on the unit circle corresponding to an angle θ. 3 Tangent Function The tangent function (tan θ) is defined as the ratio of the sine function to the cosine function: tan θ = sin θ/cos θ.

5. TRIGONOMETRIC FUNCTIONS

6. Sine, Cosine, and Tangent Functions known as the basic circular function . Cosecant, Secant, and Cotangent Functions known as the reciprocal function.

7. Example 1: Let be the terminal points of an arc length s on the unit circle. Give the values of six circular functions of s.  

8. Let be the terminal points of an arc length s on the unit circle. Give the values of six circular functions of s.  

9. Example 2: Give the values of six circular functions of s. Let be the terminal points of an arc length s on the unit circle. Let be the terminal points of an arc length s on the unit circle.  

10. Finding Exact Values Using Reference Angles Trigonometric Functions sometimes called circular functions. This is because the two fundamental trigonometric functions – the sine and the cosine – are defined as the coordinates of a point P travelling around on the unit circle.

11. Finding Exact Values Using Reference Angles Reference Angle The reference angle of an angle θ is the acute angle formed between the terminal side of θ and the x-axis. Using reference angles, we can find the exact values of trigonometric functions for any angle.

12. Unit Circle        

13. Reference angle = or  

14. Reference angle = or  

15. Reference angle = or  

16. Reference angle = or  

17. Reference Angles Q1 = Q2 = Q3 = Q4 =  

18. Therefore, we use the special angles in the first quadrant as reference to find the exact values of the circular functions (trigonometric functions) of the other special angles in the second, third, and fourth quadrants. We just need to determine the quadrant where given angle lies.

19. Example: 1. Find the values of the six circular functions of whose terminal side is at .  

20. Example: Find the values of the six circular functions of whose terminal side is at .  

21. TRIGONOMETRIC RATIOS

22. Example: 2. 3. 4.  

23. Applications of Circular Functions Modeling Periodic Phenomena Circular functions are used to model periodic phenomena like sound waves, light waves, and alternating currents. Navigation Circular functions are used in navigation systems to calculate distances, bearings, and positions. Engineering Circular functions are used in engineering to design structures, machines, and other systems.

24. Inverse Circular Functions Function Inverse Function Domain Range sin x arcsin x or sin⁻¹ x -1 ≤ x ≤ 1 -π/2 ≤ y ≤ π/2 cos x arccos x or cos⁻¹ x -1 ≤ x ≤ 1 0 ≤ y ≤ π tan x arctan x or tan⁻¹ x -∞ ≤ x ≤ ∞ -π/2 < y < π/2

25. Graphing Circular Functions 1 Amplitude The amplitude of a circular function is the distance from the midline to the maximum or minimum value. 2 Period The period of a circular function is the horizontal distance between two consecutive maximum or minimum values. 3 Phase Shift The phase shift of a circular function is the horizontal displacement of the graph.

26. Solving Problems with Circular Functions Trigonometric Equations Circular functions are used to solve trigonometric equations, which are equations that involve trigonometric functions. Right Triangle Problems Circular functions are used to solve right triangle problems, where we are given some information about the triangle and need to find other information.