Working with Trigonometric Functions, calculus homework help

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1.) What is the domain and range of y  cos x ? 2.) True or False: For a trigonometric function, 3.) True or False: For a one-to-one function, 4.) True or False: For any function, y  f x y  f x x  f 1  y  , then , then , then x  f 1  y  x  f 1  y  y  f x . Explain your answer. . Explain your answer. . Explain your answer.   3   ,  5.) True or False: The graph of y  sin x is increasing on the interval  2 2  . Explain your answer. 1 6.) Explain the meaning of y  cos x . 7.) True or False: sec 1 0.5 is undefined. Explain your answer. Find the exact value of each expression. Do not use a calculator. 8.) Cos1  0  9.) Sin1 1  3 10.) Tan1   3    Use a calculator to find the value of each expression. Round your answer to the nearest hundredth. 11.) Arcsin 1 5 12.) Arccos  0.34 13.) Arctan0.6 Find the exact value of each expression. Do not use a calculator.  3 14.) cot cot 1  3     15.) Sin1  cos  2  x  16.) tan  sin1  2  True or False? Explain your answer.      17.) csc 1 csc       4  4   18.) sec sec 1   3   3 19.) For questions 19 – 22, solve each equation: 20.) 2cos x  5  4 1  sin x  1 2 21.) 2 22.) 2cos   3cos   1  0, 0    2 23.) An object attached to a coiled spring is pulled down 5 centimeters from its rest position and released. If the motion is simple harmonic in nature, with a period of  seconds, answer the following questions: a.) b.) c.) d.) What is the maximum displacement from equilibrium of the object? What is the time required for one oscillation? What is the frequency? Write an equation to model the motion of the object. 24.) Sketch a graph of the relation: y  cos1 x . Answer: 25.) Evaluate the expression:   b.) Arctan   3  (a) arctan  3 Answer: 26.) Use a calculator to evaluate the expression. Round your answer to the nearest hundredth. sin1 7 8 Answer: 27.) Solve the equation on the interval 0,2 4sin2 x  3  0 Answer: 28.) Solve the equation on the interval 0,2 cos  3x   1 Answer: 29.) Solve the equation for  , giving a general formula for all of the solutions: tan   1 Answer: 30.) Solve the equation for  , giving a general formula for all of the solutions:  1 sin    2 2 Answer: 31.) Solve the equation, giving a general formula for all of the solutions: 2sin2 x  5sin x  3  0 Answer: 32.) Solve the equation, giving a general formula for all of the solutions:  2  cos x    sec x  1  0 2   Answer:
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Explanation & Answer

Attached. I typed as much as I can. Let me know if you have any questions.

1.) What is the domain and range of y  cos x ?

Domain: the set of all real numbers : (-∞, ∞)
Range: the set of all real numbers from -1 to 1, inclusive: [-1, 1]

2.) True or False: For a trigonometric function,

y  f x

, then

x  f 1  y 

. Explain your answer.

False. Trigonometric functions are not one to one. Hence the inverse doesn’t exist. Inverse function
exists only for trigonometric functions with restricted domain.

3.) True or False: For a one-to-one function,

y  f x

, then

x  f 1  y 

. Explain your answer.

True. Since the function is one to one, the inverse exists. By the definition of inverse, y = f(x), x = f-1(y).

4.) True or False: For any function,

x  f 1  y 

, then

y  f x

. Explain your answer.

False, inverse exists only for one to one functions. All functions are not one to one.

  3 
 ,

y

sin
x
5.) True or False: The graph of
is increasing on the interval  2 2  . Explain your answer.

  3 
 ,

False. Sin(π/2) = 1, sin(π) = 0 and sin(3π/2) = -1. Therefore, the function is decreasing on  2 2 

1
6.) Explain the meaning of y  cos x .

y  cos1 x means x = cos(y),

-1 ≤ x ≤ 1.

7.) True or False: sec 1 0.5 is undefined. Explain your answer.

1

True. 𝑠𝑒𝑐 −1 (0.5) = 𝑐𝑜𝑠 −1 ( ) = 𝑐𝑜𝑠 −1 (2), which is undefined since the domain of y = cos-1x is
0.5

[-1, 1].

8.) Find the exact value of each expression. Do not use a calculator.

9.)

Cos1  0 

cos-1(0) = x, then cos(x) = 0,

x = π/2.

Hence cos-1(0) = π/2

0≤x≤π

10.) Sin1 1

Let sin-1(1) = x,

Then sin(x) = 1, -π/2 ≤ x ≤ π

x = π/2

hence sin-1(1) = π/2

 3
11.) Tan1 
 3 



 3
Tan1 
 3 

 = x,
Let

Tan(x) =

𝑥 =

√3
3

−𝜋

,

2

<

𝜋
2

𝜋
6

√3

𝜋

3

6

𝐻𝑒𝑛𝑐𝑒 𝑡𝑎𝑛−1 ( ) =

Use a calculator to find the value of each expression. Round your answer to the nearest hundredth.
12.) Arcsin
...


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