Solving a Trigonometric Equation to Hit a Moving Target

In summary, a trigonometric equation is an equation that involves trigonometric functions and is used to find the value of a variable that makes the equation true. When trying to hit a moving target, trigonometric equations are important as they help determine the angle or direction to aim. The steps to solve a trigonometric equation for this purpose include identifying the values, setting up a trigonometric ratio, solving the equation, and checking the solution. Some tips for solving these equations include drawing a diagram, using special right triangle ratios, considering domain and range restrictions, and double checking the solution. Real-life applications of solving trigonometric equations to hit a moving target can be seen in sports, military operations, and navigation.
  • #1
peanut648
1
0
So the orginial question asked what angle to fire a gun to hit a moving target, and i got all the way to 40sinθcosθ-10sinθ-1 = 0, but we aren't supposed to use calculators so how would you solve this by hand. Thanks!
 
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  • #2
Welcome to PF!

Hi peanut648! Welcome to PF! :smile:

Either use cosθ = √(1 - sin2θ), or write everything in terms of tan(θ/2). :wink:
 
  • #3


I would suggest using the trigonometric identities and basic algebraic techniques to solve this equation by hand. First, we can rewrite the equation as 40sinθcosθ-10sinθ-1 = 0 as (40sinθ-10)(cosθ-1) = 0. This means that either 40sinθ-10 = 0 or cosθ-1 = 0.

To solve for θ, we can use the inverse trigonometric functions. For the first case, 40sinθ-10 = 0, we can use the inverse sine function to find the value of θ. This would give us sinθ = 10/40 = 1/4. Using a trigonometric table or basic trigonometric ratios, we can find that θ = sin^-1(1/4) = 14.48 degrees.

For the second case, cosθ-1 = 0, we can use the inverse cosine function to find the value of θ. This would give us cosθ = 1, which means that θ = cos^-1(1) = 0 degrees.

Therefore, the two solutions for θ are 14.48 degrees and 0 degrees. However, we must consider the practicality of these solutions. Since we are dealing with a moving target, it is highly unlikely that the gun would need to be fired at a 0 degree angle. Therefore, the more practical solution would be 14.48 degrees.

In conclusion, by using trigonometric identities and inverse trigonometric functions, we can solve the given equation by hand without the use of a calculator. However, it is important to also consider the practicality of the solutions in a real-world scenario.
 

What is a trigonometric equation?

A trigonometric equation is an equation that contains at least one trigonometric function, such as sine, cosine, or tangent. The goal of solving a trigonometric equation is to find the value of the variable that makes the equation true.

Why is it important to solve a trigonometric equation to hit a moving target?

When trying to hit a moving target, it is important to know the angle or direction at which to aim. Trigonometric equations can help us determine this angle by using the properties of triangles and the relationships between the sides and angles.

What are the steps to solve a trigonometric equation to hit a moving target?

The steps to solve a trigonometric equation to hit a moving target are:

  1. Identify the given values and the unknown value in the equation.
  2. Use the given information to set up a trigonometric ratio (such as sine, cosine, or tangent) that includes the unknown value.
  3. Solve the equation for the unknown value using algebraic manipulation.
  4. Check your solution by plugging it back into the original equation.

What are some tips for solving trigonometric equations to hit a moving target?

Some tips for solving trigonometric equations to hit a moving target are:

  • Draw a diagram to visualize the situation and help identify which trigonometric function to use.
  • Remember the special right triangle ratios (30-60-90 and 45-45-90) to help solve for unknown values.
  • Be aware of possible domain and range restrictions for trigonometric functions and adjust your solution accordingly.
  • Double check your calculations and make sure your solution makes sense in the context of the problem.

What are some real-life applications of solving trigonometric equations to hit a moving target?

Solving trigonometric equations to hit a moving target has many real-life applications, including:

  • In sports, such as basketball, baseball, and golf, players use trigonometric equations to determine the angle and force needed to hit a moving target (the hoop, the ball, or the hole).
  • In military operations, such as aiming a missile or a cannon, trigonometric equations are used to calculate the trajectory and direction of the target.
  • In navigation, such as using a compass or a GPS, trigonometric equations are used to determine the direction and distance between two points.

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