Given That $\tan (t)=\frac{5}{12}$, And $0\ \textless \ T\ \textless \ \frac{\pi}{2}$, Complete The Steps To Find $\cos (\theta$\].Which Identity Would Be Best To Start With? $\square$

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Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their relationships. In this article, we will explore how to solve a trigonometric equation involving the tangent function, and we will use this equation to find the cosine of an angle.

The Problem

Given that tan⁡(t)=512\tan (t)=\frac{5}{12}, and 0 \textless t \textless π20\ \textless \ t\ \textless \ \frac{\pi}{2}, we need to find cos⁡(θ)\cos (\theta). To do this, we will use the trigonometric identity that relates the tangent and cosine functions.

Choosing the Right Identity

When dealing with trigonometric equations, it is essential to choose the right identity to start with. In this case, we are given the tangent function, so we need to find an identity that relates the tangent and cosine functions.

One of the most useful identities in trigonometry is the Pythagorean identity:

sin⁡2(t)+cos⁡2(t)=1\sin^2 (t) + \cos^2 (t) = 1

However, this identity does not directly relate the tangent and cosine functions. A more suitable identity is the tangent half-angle formula:

tan⁡(t2)=1−cos⁡(t)1+cos⁡(t)\tan \left(\frac{t}{2}\right) = \frac{1 - \cos (t)}{1 + \cos (t)}

But this is not the best choice for this problem. A better choice is the identity that relates the tangent and cosine functions:

tan⁡(t)=sin⁡(t)cos⁡(t)\tan (t) = \frac{\sin (t)}{\cos (t)}

This identity is more suitable for this problem because it directly relates the tangent and cosine functions.

Using the Identity

Now that we have chosen the right identity, we can use it to find the cosine of the angle. We are given that tan⁡(t)=512\tan (t)=\frac{5}{12}, so we can substitute this value into the identity:

tan⁡(t)=sin⁡(t)cos⁡(t)\tan (t) = \frac{\sin (t)}{\cos (t)}

512=sin⁡(t)cos⁡(t)\frac{5}{12} = \frac{\sin (t)}{\cos (t)}

To solve for cos⁡(t)\cos (t), we can multiply both sides of the equation by cos⁡(t)\cos (t):

512cos⁡(t)=sin⁡(t)\frac{5}{12} \cos (t) = \sin (t)

Now, we can use the Pythagorean identity to express sin⁡(t)\sin (t) in terms of cos⁡(t)\cos (t):

sin⁡2(t)+cos⁡2(t)=1\sin^2 (t) + \cos^2 (t) = 1

sin⁡(t)=1−cos⁡2(t)\sin (t) = \sqrt{1 - \cos^2 (t)}

Substituting this expression into the previous equation, we get:

512cos⁡(t)=1−cos⁡2(t)\frac{5}{12} \cos (t) = \sqrt{1 - \cos^2 (t)}

Squaring both sides of the equation, we get:

25144cos⁡2(t)=1−cos⁡2(t)\frac{25}{144} \cos^2 (t) = 1 - \cos^2 (t)

Expanding the right-hand side of the equation, we get:

25144cos⁡2(t)=1−cos⁡2(t)\frac{25}{144} \cos^2 (t) = 1 - \cos^2 (t)

25144cos⁡2(t)+cos⁡2(t)=1\frac{25}{144} \cos^2 (t) + \cos^2 (t) = 1

Combining like terms, we get:

169144cos⁡2(t)=1\frac{169}{144} \cos^2 (t) = 1

Dividing both sides of the equation by 169144\frac{169}{144}, we get:

cos⁡2(t)=144169\cos^2 (t) = \frac{144}{169}

Taking the square root of both sides of the equation, we get:

cos⁡(t)=±144169\cos (t) = \pm \sqrt{\frac{144}{169}}

Since 0 \textless t \textless π20\ \textless \ t\ \textless \ \frac{\pi}{2}, we know that cos⁡(t)\cos (t) is positive, so we can discard the negative solution:

cos⁡(t)=144169\cos (t) = \sqrt{\frac{144}{169}}

Simplifying the expression, we get:

cos⁡(t)=1213\cos (t) = \frac{12}{13}

Conclusion

In this article, we have shown how to solve a trigonometric equation involving the tangent function, and we have used this equation to find the cosine of an angle. We have chosen the right identity to start with, and we have used it to find the cosine of the angle. The final answer is cos⁡(t)=1213\cos (t) = \frac{12}{13}.

Additional Resources

For more information on trigonometric equations and identities, please see the following resources:

References

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their relationships. In this article, we will provide a Q&A guide to help you understand and solve trigonometric equations.

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent.

Q: What are the most common trigonometric functions?

A: The most common trigonometric functions are:

  • Sine (sin)
  • Cosine (cos)
  • Tangent (tan)
  • Cotangent (cot)
  • Secant (sec)
  • Cosecant (csc)

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you need to use the following steps:

  1. Identify the trigonometric function involved in the equation.
  2. Use the appropriate trigonometric identity to simplify the equation.
  3. Solve for the variable.
  4. Check your solution to make sure it is valid.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • Pythagorean identity: sin⁡2(t)+cos⁡2(t)=1\sin^2 (t) + \cos^2 (t) = 1
  • Tangent half-angle formula: tan⁡(t2)=1−cos⁡(t)1+cos⁡(t)\tan \left(\frac{t}{2}\right) = \frac{1 - \cos (t)}{1 + \cos (t)}
  • Sine and cosine sum and difference formulas: sin⁡(a+b)=sin⁡(a)cos⁡(b)+cos⁡(a)sin⁡(b)\sin (a + b) = \sin (a) \cos (b) + \cos (a) \sin (b) and cos⁡(a+b)=cos⁡(a)cos⁡(b)−sin⁡(a)sin⁡(b)\cos (a + b) = \cos (a) \cos (b) - \sin (a) \sin (b)

Q: How do I use the Pythagorean identity to solve a trigonometric equation?

A: To use the Pythagorean identity to solve a trigonometric equation, you need to follow these steps:

  1. Identify the trigonometric function involved in the equation.
  2. Use the Pythagorean identity to express the other trigonometric function in terms of the first one.
  3. Solve for the variable.
  4. Check your solution to make sure it is valid.

Q: How do I use the tangent half-angle formula to solve a trigonometric equation?

A: To use the tangent half-angle formula to solve a trigonometric equation, you need to follow these steps:

  1. Identify the trigonometric function involved in the equation.
  2. Use the tangent half-angle formula to express the other trigonometric function in terms of the first one.
  3. Solve for the variable.
  4. Check your solution to make sure it is valid.

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

  • Not using the correct trigonometric identity.
  • Not checking the solution to make sure it is valid.
  • Not using the correct trigonometric function.
  • Not simplifying the equation correctly.

Conclusion

In this article, we have provided a Q&A guide to help you understand and solve trigonometric equations. We have covered the most common trigonometric functions, identities, and mistakes to avoid. By following the steps outlined in this article, you should be able to solve trigonometric equations with confidence.

Additional Resources

For more information on trigonometric equations and identities, please see the following resources:

References