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7.3 The Sine and Cosine Functions

7.3 The Sine and Cosine Functions. Objective To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations . The Definition of Sine and Cosine Functions. Suppose an acute angle A is drawn in

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7.3 The Sine and Cosine Functions

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  1. 7.3 The Sine and Cosine Functions Objective To use the definitions of sine and cosine to find values of these functions and to solve simple trigonometric equations.

  2. The Definition of Sine and Cosine Functions Suppose an acute angle A is drawn in standard position as shown, point B (x, y) is a point on the terminal side. Right-Triangle-Based Definitions of Trigonometric Functions For any acute angle A in standard position, Since the concept of angle has been generalized, we generalize the above “Right-Triangle-Based” definition to formal trigonometric function definition.

  3. The Sine and Cosine Functions Suppose point P (x, y) is on the circle x2 + y2 = r2 and  is an angle in standard position with terminal side OP, as shown at the right. We define the function sine of , denoted sin, as: y P (x, y) r  x We define the function cosine of , denoted cos, as: O This definition is consistent with the prior “Right-Triangle-Based” definition and is more general. From the formal trigonometric definition, we can conclude:

  4. If this question is changed to “Point (-3, 2) is on a Quadrant II angle”, do the values of sin and cos change?

  5. Example 2: If the terminal ray of an angle  in standard position passes through (–4, –3), find sin and cos . y [Solution] We need to find the radius r of the circle, use the equation x2 + y2 = r2with x = –4, and y = –3:  x O r (-4, -3) Thus:

  6. Example 3. If  is a fourth-quadrant angle and , find cos . [Solution] We need to find x from the equation. Since the angle is in the Quadrant IV, taking x= 12. Thus:

  7. Activity Now we have an important question: Will the position of the point on the terminal side affect the values of sin and cos ? If the answer is “Yes”, then the definitions for sin and cosare meaningless. How can we prove that the answer is “No”? Now given a right triangle ABC, DE AB Since both right triangle ABC and ADE are similar, we have: C E A B D Rewrite these proportions to This shows that the position of the point on the terminal side will NOT affect the values of sin and cos .

  8. Via the activity, we proved that the position of the point on the terminal side will NOT affect the values of sin and cos because the ratios of the “opposite side to hypotenuse” remain the same, and so do the ratios of the “adjacent side to hypotenuse”. Thus, we conclude that the value for the radius (like hypotenuse) will not affect the values of sin and cos . In order to keep the simplicity, we then choose r= 1. The circle with r= 1 is called unit circle. Thus, x=r cos = cos , y=r sin= sin The next we use unit circle to discuss the sign of values for sin and cos when  is in all four quadrants. Again, from the above, we have cos=x, sin =y y (x, y) = (cos, sin )  r = 1 x O

  9. The Sign of Sine and Cosine Functions in Each Quadrant Since cos=x, sin =y y y (x, y) = (cos, sin ) (x, y) = (cos, sin ) (cos, sin ) =(+, +) Quadrant I r = 1   r = 1 x x O O (cos, sin ) =(–, +) Quadrant II y y (cos, sin ) =(–, –) Quadrant III   x x O O r = 1 r = 1 (cos, sin ) =(+, –) Quadrant VI (x, y) = (cos, sin ) (x, y) = (cos, sin )

  10. The Sign of Sine and Cosine Functions in Each Quadrant In summary, we list the sign for sin and cos below: Or visually, y y sin = y cos = x + + – + x x – – – +

  11. Challenge QuestionPlease fill in the blank spaces for the value of sin and cos . You need to show the work. 1 + – –1 0 0 –1 1 – + 0 0 y y sin = y cos = x + + – + x x – – – +

  12. Example 4. Find: a. sin90o b. sin450o c. cos(–) = 90o, and = 450o are the two angles of the same terminal side, which is on the + y-axis. Therefore, sin 90o= 1, and sin450o= 1 The terminal side of =– is on the –x-axis. So, cos(–) = –1.

  13. Example 5 Solve sin  = 1 for  in degrees and in radians. Since  = 90º is one solution of the equation sin  = 1, any angle coterminal with 90º also has 1 as its sine value. are all solutions of the equation. They can be written more conveniently as where n is an integer. In radians, the solutions would be written as where n is an integer.

  14. The sine and cosine functions repeat their values every 360º or 2 radians. The sine and cosine functions are periodic They have a fundamental period of 360º or 2 radians. It is the periodic nature of these functions that makes them useful in describing many repetitive phenomena such as tides, sound waves, and the orbital paths of satellites.

  15. Example 6 Find the sin and cos .

  16. Class Exercises c. neg a. pos b. neg d. pos e. pos f. neg g. neg h. pos l. neg j. neg k. neg i. pos

  17. Class Exercises a. dec b. dec c. inc d. inc a. inc b. dec c. dec d. inc

  18. Assignment P. 272 #1 – 4, 5 – 28, 34 – 40

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