1. TRIGONOMETRY – Functions 1 We will now place the angle in the x–y plane. The initial side of the angle will always be placed on the (+) side of the x – axis and the terminal side will move either clockwise or counterclockwise with the vertex at the origin. +x+x - x +y+y - y θ

2. TRIGONOMETRY – Functions 1 We will now place the angle in the x–y plane. The initial side of the angle will always be placed on the (+) side of the x – axis and the terminal side will move either clockwise or counterclockwise with the vertex at the origin. This angle is said to be in “standard position”. +x+x - x +y+y - y θ

3. TRIGONOMETRY – Functions 1 We will now place the angle in the x–y plane. The initial side of the angle will always be placed on the (+) side of the x – axis and the terminal side will move either clockwise or counterclockwise with the vertex at the origin. This angle is said to be in “standard position”. +x+x - x +y+y - y θ As the terminal side creates an angle, it represents a point located in the coordinate plane.

4. TRIGONOMETRY – Functions 1 We will now place the angle in the x–y plane. The initial side of the angle will always be placed on the (+) side of the x – axis and the terminal side will move either clockwise or counterclockwise with the vertex at the origin. This angle is said to be in “standard position”. +x+x - x +y+y - y θ As the terminal side creates an angle, it represents a point located in the coordinate plane. Recall from Algebra 1 the signs of each quadrant. I II III IV

5. TRIGONOMETRY – Functions 1 We will now place the angle in the x–y plane. The initial side of the angle will always be placed on the (+) side of the x – axis and the terminal side will move either clockwise or counterclockwise with the vertex at the origin. This angle is said to be in “standard position”. +x+x - x +y+y - y θ As the terminal side creates an angle, it represents a point located in the coordinate plane. Recall from Algebra 1 the signs of each quadrant. I II III IV The coordinate point created by the angle will have these signs.

6. TRIGONOMETRY – Functions 1 +x+x - x +y+y - y θ Trigonometry was developed using a right triangle and the relationships between the sides of that triangle.

7. TRIGONOMETRY – Functions 1 Trigonometry was developed using a right triangle and the relationships between the sides of that triangle. Let’s create a right triangle with a hypotenuse of 1 from the given point. +x+x - x +y+y - y θ 1

8. TRIGONOMETRY – Functions 1 Trigonometry was developed using a right triangle and the relationships between the sides of that triangle. Let’s create a right triangle with a hypotenuse of 1 from the given point. +x+x - x +y+y - y θ 1 We need to label the sides of this triangle. The angle is still created with the vertex at the origin.

9. TRIGONOMETRY – Functions 1 Trigonometry was developed using a right triangle and the relationships between the sides of that triangle. Let’s create a right triangle with a hypotenuse of 1 from the given point. +x+x - x +y+y - y θ 1 We need to label the sides of this triangle. The angle is still created with the vertex at the origin. The side opposite the right angle will still be called the hypotenuse. Hypotenuse

10. TRIGONOMETRY – Functions 1 Trigonometry was developed using a right triangle and the relationships between the sides of that triangle. Let’s create a right triangle with a hypotenuse of 1 from the given point. +x+x - x +y+y - y θ 1 We need to label the sides of this triangle. The angle is still created with the vertex at the origin. The side opposite the right angle will still be called the hypotenuse. The side opposite the given angle is called the opposite side Hypotenuse Opposite

11. TRIGONOMETRY – Functions 1 Trigonometry was developed using a right triangle and the relationships between the sides of that triangle. Let’s create a right triangle with a hypotenuse of 1 from the given point. +x+x - x +y+y - y θ 1 We need to label the sides of this triangle. The angle is still created with the vertex at the origin. The side opposite the right angle will still be called the hypotenuse. The side opposite the given angle is called the opposite side The side “next to” the angle is called the adjacent side. Hypotenuse Opposite Adjacent

12. TRIGONOMETRY – Functions 1 The 3 major trigonometric functions are ratios of these sides +x+x - x +y+y - y θ 1 Hypotenuse Opposite Adjacent

13. TRIGONOMETRY – Functions 1 The 3 major trigonometric functions are ratios of these sides +x+x - x +y+y - y θ 1 Hypotenuse Opposite Adjacent

14. TRIGONOMETRY – Functions 1 The 3 major trigonometric functions are ratios of these sides +x+x - x +y+y - y θ 1 Hypotenuse Opposite Adjacent

15. TRIGONOMETRY – Functions 1 The 3 major trigonometric functions are ratios of these sides +x+x - x +y+y - y θ 1 Hypotenuse Opposite Adjacent

16. TRIGONOMETRY – Functions 1 The 3 major trigonometric functions are ratios of these sides +x+x - x +y+y - y θ 1 Hypotenuse Opposite Adjacent

17. TRIGONOMETRY – Functions 1 Let’s try a few examples… +x+x - x +y+y - y θ 1 Hypotenuse Opposite Adjacent

18. TRIGONOMETRY – Functions 1 Let’s try a few examples… +x+x - x +y+y - y θ 5 Hypotenuse Opposite Adjacent Find the sin, cos, and tangent ratios for the given triangle. 3 4

19. TRIGONOMETRY – Functions 1 Let’s try a few examples… +x+x - x +y+y - y θ 5 Hypotenuse Opposite Adjacent Find the sin, cos, and tangent ratios for the given triangle. 3 4

20. TRIGONOMETRY – Functions 1 Let’s try a few examples… +x+x - x +y+y - y θ 5 Hypotenuse Opposite Adjacent Find the sin, cos, and tangent ratios for the given triangle. 3 4

21. TRIGONOMETRY – Functions 1 Example # 2 : Find the sin, cos, and tangent ratios for the point ( 5, 7 ) +x+x - x +y+y - y θ

22. TRIGONOMETRY – Functions 1 Example # 2 : Find the sin, cos, and tangent ratios for the point ( 5, 7 ) +x+x - x +y+y - y θ ? 7 5 First : label what you know.

23. TRIGONOMETRY – Functions 1 Example # 2 : Find the sin, cos, and tangent ratios for the point ( 5, 7 ) +x+x - x +y+y - y θ 7 5 First : label what you know. Second : find the hypotenuse using Pythagorean theorem

24. TRIGONOMETRY – Functions 1 Example # 2 : Find the sin, cos, and tangent ratios for the point ( 5, 7 ) +x+x - x +y+y - y θ 7 5 First : label what you know. Second : find the hypotenuse using Pythagorean theorem Third : find your ratios

25. TRIGONOMETRY – Functions 1 As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

26. TRIGONOMETRY – Functions 1 Example : Let’s find the decimal equivalent of 30 ⁰ for sin, cos, and tan. As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

27. TRIGONOMETRY – Functions 1 For example : Let’s find the decimal equivalent of 30 ⁰ for sin, cos, and tan. ** if you are using a graphing calculator, make sure it is in degree mode. As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

28. TRIGONOMETRY – Functions 1 For example : Let’s find the decimal equivalent of 30 ⁰ for sin, cos, and tan. ** if you are using a graphing calculator, make sure it is in degree mode. On the calculator type … sin,30,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

29. TRIGONOMETRY – Functions 1 For example : Let’s find the decimal equivalent of 30 ⁰ for sin, cos, and tan. ** if you are using a graphing calculator, make sure it is in degree mode. On the calculator type … sin,30,enter On the calculator type … cos,30,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

30. TRIGONOMETRY – Functions 1 For example : Let’s find the decimal equivalent of 30 ⁰ for sin, cos, and tan. ** if you are using a graphing calculator, make sure it is in degree mode. On the calculator type … sin,30,enter On the calculator type … cos,30,enter On the calculator type … tan,30,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

31. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

32. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

33. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. On the calculator type … 2 nd,cos,0.5,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

34. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. On the calculator type … 2 nd,cos,0.5,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

35. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. On the calculator type … 2 nd,cos,0.5,enter On the calculator type … 2 nd,tan,0.8391,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

36. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. On the calculator type … 2 nd,cos,0.5,enter On the calculator type … 2 nd,tan,0.8391,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

37. TRIGONOMETRY – Functions 1 Example # 2: Find the angle for each decimal value. On the calculator type … 2 nd,cos,0.5,enter On the calculator type … 2 nd,tan,0.8391,enter On the calculator type … 2 nd,sin,0.7071,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

38. TRIGONOMETRY – Functions 1 Example #3: Find the decimal equivalent of 100 ⁰ for sin, cos, and tan. As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

39. TRIGONOMETRY – Functions 1 Example #3: Find the decimal equivalent of 100 ⁰ for sin, cos, and tan. On the calculator type … sin,100,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

40. TRIGONOMETRY – Functions 1 Example #3: Find the decimal equivalent of 100 ⁰ for sin, cos, and tan. On the calculator type … sin,100,enter On the calculator type … cos,100,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

41. TRIGONOMETRY – Functions 1 Example #3: Find the decimal equivalent of 100 ⁰ for sin, cos, and tan. On the calculator type … sin,100,enter On the calculator type … cos,100,enter This value is negative because 100° is in Quadrant 2 where cos ( which is the x distance ) is negative. As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

42. TRIGONOMETRY – Functions 1 Example #3: Find the decimal equivalent of 100 ⁰ for sin, cos, and tan. On the calculator type … sin,100,enter On the calculator type … cos,100,enter This value is negative because 100° is in Quadrant 2 where cos ( which is the x distance ) is negative. Tangent is negative in Quadrant 2 On the calculator type … tan,100,enter As you can see, the ratios become decimal values. These decimal values can be turned into angle measure by the use of a table or calculator. You can also use the tables or calculator to change an angle into its decimal equivalent. We will use the calculator for these calculations. ** round to 4 decimal places

43. TRIGONOMETRY – Functions 1 +x+x - x +y+y - y I II III IV