Trigonometry Question: Find the max and min.

In summary, to find the maximum and minimum values of 4cos\theta-3sin\theta, you can rewrite it as 5[(4/5)cos\theta - (3/5)sin\theta] and use the identity sinAcosB - cosAsinB = sin(A-B) to find an angle \alpha such that sin(\theta) = 4/5 and cos(\theta) = 3/5. This allows you to solve for the maximum and minimum values, which are equal to C and -C respectively.
  • #1
Cuisine123
38
0

Homework Statement


Find the maximum and minimum values of 4cos[tex]\theta[/tex]-3sin[tex]\theta[/tex].


Homework Equations


I have no idea.


The Attempt at a Solution


I have no idea how to do this question

Please help me!
 
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  • #2
Try to draw a plot - even if it will not give you an exact answer, it may give you some hints.
 
  • #3
The trick to these kinds of problems is to write Asin(x) - Bcos(x) as Csin(x - [itex]\theta[/itex]). It can then be seen that the maximum value is C and the minimum value is -C (assuming that C > 0).
4cos[itex]\theta[/itex] - 3sin[itex]\theta[/itex]
= 5[(4/5)cos[itex]\theta[/itex] - (3/5)sin[itex]\theta[/itex]]

Now what you need to do is find an angle [itex]\alpha[/itex] such that sin([itex]\theta[/itex]) = 4/5 and cos([itex]\theta[/itex]) = 3/5. Then you can use the identity sinAcosB - cosAsinB = sin(A-B).
 
  • #4
= 5[(4/5)cos - (3/5)sin]

I'm just curious, can you go into more detail how you generated this from,

4cos-3sin
 
  • #5
Do you mean why it's true, or why Mark wrote it like that? It's easy to prove why it's true: just expand and you get 4cos x - 3 sin x. As for why it's useful, you want an equation of the form cos(a)cos(b)-sin(a)sin(b), because such an equation is equal to cos(a+b). Since cos(a) can't be 4 and sin(a) can't be 3, Mark decided to factor out a five. You can just as well factor out a ten, or a 100.
 

Related to Trigonometry Question: Find the max and min.

1. What is Trigonometry?

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems involving right triangles and can also be applied to other shapes and objects in space.

2. What is the purpose of finding the maximum and minimum values in Trigonometry?

Finding the maximum and minimum values in Trigonometry is important because it helps us understand the behavior of a function or equation. It allows us to determine the highest and lowest points on a graph and can be used to optimize real-world problems.

3. How do you find the maximum and minimum values in Trigonometry?

To find the maximum and minimum values in Trigonometry, you must first find the derivative of the function or equation. Then, set the derivative equal to zero and solve for the variable. The resulting value will be the x-coordinate of the maximum or minimum point. Plug this value back into the original function to find the y-coordinate.

4. Can you give an example of finding the maximum and minimum values in Trigonometry?

Sure, let's say we have the function f(x) = 2sin(x). To find the maximum and minimum values, we first find the derivative, which is f'(x) = 2cos(x). We set this equal to zero and solve, giving us x = π/2 as the critical point. Plugging this back into the original function, we get the maximum value of f(π/2) = 2. To find the minimum value, we can use the first derivative test and check the values on either side of the critical point. Since f'(0) = 2 and f'(π) = -2, we can conclude that the minimum value is f(π) = -2.

5. What are some real-world applications of finding the maximum and minimum values in Trigonometry?

Finding the maximum and minimum values in Trigonometry can be applied to various fields such as engineering, physics, and economics. For example, it can be used to optimize the trajectory of a projectile or to find the most efficient way to pack items in a container. In economics, it can be used to determine the highest and lowest points of a profit or cost function.

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