Different identities in one equation

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In summary, The question asks for the angle that maximizes the distance traveled by an object propelled at an angle of 45-90 degrees with an initial velocity of V0 m/s on an inclined plane with an angle of 45 degrees to the horizontal, neglecting air resistance. The equation for the distance, R, is given as R = V^2(sqrt 2)/ 32 (2sinthetacostheta - 2cos^2theta. To solve for theta, the equation 2sinthetacostheta + 1 - 2sin^2theta = 0 can be used, and the identities for sin(2θ) and cos(2θ) may be helpful.
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TN17
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1. Homework Statement

Sorry for the long intro:

An object is propelled up at angle theta 45 deg. < theta < 90 deg. to the horiz. with initial vel. of V0 m/s. from the base of a plane that makes an angle of 45 deg. with the horiz.
If air resistance is ingored, the distance, R, traveled by the object up the inclined plane, is
R = V^2(sqrt 2)/ 32 (2sinthetacostheta - 2cos^2theta

Question
You are asked to find the angle that maximizes R by solving equation
2sinthetacostheta + 1 - 2sin^2theta = 0
Solve for theta.

2. Homework Equations
Not really any equations, just solving.

3. The Attempt at a Solution
I tried to continue with this, but I don't know what to do when there are two different identities.

Would I factor?
 
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  • #2
Try using the identities for sin(2θ) and cos(2θ).
 

What is the concept of different identities in one equation?

The concept of different identities in one equation refers to the idea that an equation can have multiple forms or representations, all of which are equivalent. This means that the different identities may have different variables, terms, or structures, but they all yield the same solution when solved.

Why is it important to understand the different identities in one equation?

Understanding the different identities in one equation is important because it allows us to manipulate and simplify equations in different ways, making problem-solving more efficient. It also helps us to recognize patterns and connections between seemingly unrelated equations.

What are some examples of different identities in one equation?

Examples of different identities in one equation include the Pythagorean theorem, which has three different forms (a^2 + b^2 = c^2, c^2 - a^2 = b^2, and c^2 - b^2 = a^2), and the quadratic formula, which has two forms (x = (-b ± √(b^2 - 4ac)) / 2a and x = (-2c) / (b ± √(b^2 - 4ac))).

How can knowing the different identities in one equation help in problem-solving?

Knowing the different identities in one equation can help in problem-solving by allowing us to choose the most efficient form of the equation to work with. It also enables us to recognize and utilize transformations or substitutions that can simplify the equation and make it easier to solve.

Are there any drawbacks to using different identities in one equation?

One potential drawback of using different identities in one equation is that it can be confusing or overwhelming, especially for those who are new to mathematics. It may also require more time and effort to learn and understand the various identities and their applications. Additionally, some identities may only be applicable in certain situations, so it is important to carefully consider which form of the equation is most appropriate for a given problem.

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