7.4 Inverse Trig Functions

Learn about the diverse applications of inverse trigonometric functions in various fields such as actuarial science, aerospace engineering, mechanics, economics, and more. Understand the criteria for a function to have an inverse and explore the inverse sine, cosine, and tangent functions along with their practical uses. Delve into finding exact values without a calculator and evaluating trigonometric functions to three decimal places. Enhance your understanding of how these functions are utilized by professionals in different industries.

• Trigonometry
• Inverse Functions
• Applications
• Engineering
• Mathematics

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1. 7.4 Inverse Trig Functions Who uses this? Actuaries Aerospace Engineers Mechanical Engineers Nuclear Engineers Economist Boilermakers

2. For a function to have an inverse it must be one-to-one. One-to-one functions have to pass the horizontal line test. Each y-value can be paired with no more than one x-value. And each x-value can be paired with no more than one y- value. Are the graphs of the trig functions one-to-one? Inverse Sine Function We can make the sine function one-to-one by restricting the domain. For y = sinx the domain is all real numbers and the range is [-1, 1]. To find the inverse we are going to reflect the restricted sine graph over the line y = x.

4. y = sin-1x "the angle between - and with a sine of x." 2 2 Domain: 1,1 Range: , 2 2

5. Ex sin-1 What angle between and has a sine that is ? 2 2 6 Range of the inverse is restricted to right side of unit circle.

6. Find the exact values without a calculator! 1 2 1 1 und sin sin 3 6 2 3 arcsin arcsin 4 3 2 2 1 2 sin ( 1) arcsin 2 1 6

7. Use your calculator to evaluate to 3 decimal places. (You must be in radians!!!) 0.961 sin-1(0.82) 0.309 arcsin(-0.3042) 9.145 cot[sin-1(-0.1087)]

8. Inverse Cosine y = cos-1x or y = arccosx 1,1 Domain: Range: 0,

9. y = cos-1x means the angle in the interval between 0 and pi whose cosine is x. Values of the inverse cosine are located in the upper half of the unit circle.

10. Find the exact values without a calculator! 1 2 und 1 1 cos cos ( 2) 3 3 arccos 5 6 3 2 arccos 4 2 2 ( ) 0.0349 1.001 1 1 cos(cos (0.7)) csc cos 0.7

11. Inverse Tangent y = tan-1x or y = arctanx Domain: or (- , ) Range: , 2 2 The vertical asymptotes become horizontal asymptotes when we reflect the graph over the line y = x. The values are on the right side of the unit circle.

13. Find the exact values without a calculator! tan 1 tan 1 1 3 4 3 3 ( ) arctan arctan 6 20 3 1.521 ( ) 0.1308 1.009 1 arctan(tan ) sec tan und 2

14. Domain of Compositions of Trig Functions [ ( )] x = 1 f f x x = If 1 If 1 1, then sin(sin 1, then cos(cos 1 ) x x = x 1 ) x x = 1 If , then tan(tan ) x x x = 1 [ ( )] f x f x = If -2 If 1 , then sin (sin ) x x x 2 , then cos (cos ) = 0 1 x x x = If -2 1 , then tan (tan ) x x x 2

15. Find the exact value, if it exists. 1 2 1 2 1 sin sin 5 1 sin sin 4 4 und arctan tan2 2 arcsin sin 3 3

16. Find the exact value if it exists. ( ) ( ) 2 sin 2 45 ( sin 90 1 sin 2cos 2 ) 1 ( ) + 1 2 os 90 c cos 120 3 ) 0 + 1 1 cos cos 0 sin ( 1 2

17. Find the exact value without a calculator. 2 3 3 5 1 cos sin 2 = cos 3 5 1 2 1 2 = tan 1 tan sin 5 5 1

18. Write as an algebraic expression in x. ( ) sin(cos-1x) = 1 c ) 2 1 os co s x = 1 cos ( cos 2 sin u u = 1 cos u x ) ( 2 = 1 1 cos x = 1 x 2 OR = = 1 co s Let cos x 1 1 x 2 x x

19. Write as an algebraic expression in x. ( ) tan(cos-1x) 2 1 1 cos cos x 2 sin cos 1 cos cos u u u = = = tan u ( ) u 1 cos cos x = 1 cos u x 2 1 x x = OR = = 1 co s Let cos x 1 1 x 2 x 2 1 x x = tan x

20. Write as an algebraic expression in x. ( ) cos(arcsinx) = 1 s 2 1 in si n x = 1 sin 2 cos u u = 1 sin u x = 1 x 2 OR 1 x = = 1 si n x Let sin x 1 x 2

21. 7.4 b Inverse Trig Functions Sec, csc, cot

22. y = sin-1x y = cos-1x Domain: 1,1 1,1 Domain: Range: 0, Range: , 2 2 y = csc-1x y = sec-1x [1, ) Domain: ( , 1] Domain: ( , 1] [1, ) Range: , , 0 y Range: 0, , y 2 2 2 y = tan-1x Domain: ( y = cot-1x , ) Domain: ( Range: 0, , ) ( ) Range: , 2 2

23. Find the exact values without a calculator! 1 2 csc und ( ) 6 1 1 csc 2 3 ( ) 2 3 3 6 2 arcsec 4 arcsec ( ) 3 1 cot ( 1) arccot 0 4 2

24. Use your calculator to evaluate to 3 decimal places. (You must be in radians!!!) 0.582 csc-1(1.82) arcsec(-5.3042) 1.760 0.322 tan[cot-1(-3.1087)]

25. Find the exact value, if it exists. 3 ( ) ( ) 2 1 sin sec 2 5 und 1 sin csc 4 0 arctan cot2 3 3 arccot sin 2 4

26. Find the exact value without a calculator. 3 ( ) ( ) = tan 2 2 1 tan sec 3 2 2 1 5 3 4 3 1 cot csc 4 = cot 3 5

27. 7.5 Trig Equations An equation that contains trig functions is called a trig equation. To solve a trig equation we find ALL VALUES of the variable that make the equation true.

28. Ex 2sinx 1 = 0 + + 1 1 1 x = 2sin 2 2 sin x = 1 2 11 5 = = sin , x 2 6 6 = + 2 x k 6 k Z 5 = + 2 x k 6

29. Ex tan2x - 3 = 0 + + 3 3 3 x = 2 tan tan x = x = tan = 3 2 = 1 , 3 3 3 + x k 3 2 = + x k 3 k Z

30. Ex 2sin2x 13sinx + 6 = 0 (2sin 1)(sin x 2sin 1 0 x = 1 + + 2sin x = 2 2 1 sin 2 5 , 6 = 6) 0 x 1 1 x = + sin sin 6 6 x = und 0 6 6 + x = = x = + 2 x k 6 6 k Z 5 = + 2 x k 6

31. Ex 2cos2x 7cosx + 3 = 0 (2cos 1)(cos x 2cos 1 0 x = 1 + + 2cos x = 2 2 1 cos 2 x = cos = 3) 0 3 3 + x x = 0 3 3 + 1 1 x = und cos x = 5 = + 2 x k , 3 3 3 k Z 5 = + 2 x k 3

32. Ex sinx = cosx cos x cos x sin cos x x= 1 x = tan 1 5 = = 1 tan 1 x , 4 4 = + k Z x k 4

33. Ex 3cosx = 2sin2x 3cos x = 2 2cos x + (2cos x x+ = cos x = und cos 2 0 2 x = x = 2cos cos 1 0 1 2 , 2(1 cos 3cos 1)(cos 2 ) x 2 2) = + 0 x x = 0 5 = x 3 3 = + 2 x k 3 k Z 5 = + 2 x k 3

34. Ex 1 + sinx = 2cos2x 1 sin x + = 2 2sin sin x + (2sin 1)(sin x 2(1 sin x x 2 ) x 0 = x+ = x = = x = sin sin 1 0 2sin 1 0 1 2 , = 1 + 1 x = sin 1) 0 5 x = x 2 6 6 = + 2 x k = + 2 x k 6 2 5 = + 2 x k k Z 6

35. Ex 1 sin2x cosx = 0 2sin cos x cos (2sin x x = cos 0 x = 2sin 1 0 = cos 1) 0 x x x = 0 1 2 3 x = sin = , x 2 2 5 = , x 6 6 = + = + 2 2 x k x k 6 2 3 k Z 5 = + 2 x k = + 2 x k 2 6

36. Ex 2 3tan3x tanx = 0 tan (3tan x x x = = = 2 1) 0 x = 2 tan 0 3tan 1 0 1 0, x x = tan 3 5 = , x 6 6 = + 0 x k = + x k 6 5 k Z = + x k 6

37. Ex 3 secxtanx = 4sinx 1 sin cos cos x x sin 4sin 1 sin x = x 2 4sin 3 0 x = 0, sin x 0 = 4sin x = 3 4 x = 2 sin x x= x 2 3 x = sin = 3 sin 4sin 4sin x x x 2 4 2 5 = 3 = 4sin 3sin 0 x x , , , x 3 3 3 3 x = 2 sin (4sin x 3) 0 = + = + x k 0 x k 3 2 = + x k = + k Z 0 x k 3 3

38. Ex 4 cosx + 1 = sinx 2 cos 2cos x + cos 2cos x + 2cos 2cos x + (2cos )(cos x x+ = cos x = cos 1 0 1 x = 3 , 1 sin + = 1 1 cos + = 0 x = 1) 0 x+ = 2 2cos 0 x x x 2 2 x = x 2 2 = 2 x = + = + x k 2 x k 2 k Z

39. Ex 5 2sin3x 1 = 0 [0,2 ] 1 2 x = sin3 5 = 3 , x 6 6 5 = + = + 3 2 3 2 x k x k 6 6 2 5 18 2 k k = + = + x x 18 3 3 13 25 18 5 18 17 18 29 18 = = , , , , x x 18 18

40. Ex 5 ?tan? ? 1 = 0 3 x= tan2 x 3 7 = , 2 6 6 x = + k 2 6 = + k Z 2 x k 3