If $\cos \theta = \frac{3}{5}$, Find $\tan \theta$.

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Introduction

In trigonometry, the tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Given the cosine of an angle, we can use the Pythagorean identity to find the sine of the angle, and then use the definition of tangent to find the value of $\tan \theta$.

The Pythagorean Identity

The Pythagorean identity states that for any angle $\theta$ in a right-angled triangle, the sum of the squares of the lengths of the two sides adjacent to the angle is equal to the square of the length of the hypotenuse. Mathematically, this can be expressed as:

$$\sin^2 \theta + \cos^2 \theta = 1$$

Finding $\sin \theta$

Given that $\cos \theta = \frac{3}{5}$, we can substitute this value into the Pythagorean identity to solve for $\sin \theta$.

$$\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1$$

$$\sin^2 \theta + \frac{9}{25} = 1$$

$$\sin^2 \theta = 1 - \frac{9}{25}$$

$$\sin^2 \theta = \frac{16}{25}$$

Taking the square root of both sides, we get:

$$\sin \theta = \pm \sqrt{\frac{16}{25}}$$

$$\sin \theta = \pm \frac{4}{5}$$

Since the sine of an angle is positive in the first and second quadrants, we take the positive value:

$$\sin \theta = \frac{4}{5}$$

Finding $\tan \theta$

Now that we have the values of $\sin \theta$ and $\cos \theta$, we can use the definition of tangent to find the value of $\tan \theta$.

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

$$\tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}}$$

$$\tan \theta = \frac{4}{5} \times \frac{5}{3}$$

$$\tan \theta = \frac{4}{3}$$

Therefore, the value of $\tan \theta$ is $\boxed{\frac{4}{3}}$.

Conclusion

In this article, we used the Pythagorean identity to find the value of $\sin \theta$ given the value of $\cos \theta$, and then used the definition of tangent to find the value of $\tan \theta$. This demonstrates the importance of the Pythagorean identity in trigonometry and how it can be used to find the values of trigonometric functions.

Real-World Applications

The value of $\tan \theta$ has many real-world applications, such as:

• Surveying: The tangent of an angle can be used to calculate the height of a building or the distance between two points.
• Navigation: The tangent of an angle can be used to calculate the direction of a ship or a plane.
• Physics: The tangent of an angle can be used to calculate the velocity of an object.

Final Thoughts

In conclusion, the value of $\tan \theta$ is a fundamental concept in trigonometry that has many real-world applications. By using the Pythagorean identity and the definition of tangent, we can find the value of $\tan \theta$ given the value of $\cos \theta$. This demonstrates the importance of trigonometry in mathematics and its many real-world applications.

Additional Resources

• Trigonometry for Dummies: A comprehensive guide to trigonometry, including the Pythagorean identity and the definition of tangent.
• Trigonometry: A First Course: A textbook on trigonometry that covers the basics of the subject, including the Pythagorean identity and the definition of tangent.
• Trigonometry: A Second Course: A textbook on trigonometry that covers more advanced topics, including the Pythagorean identity and the definition of tangent.

References

• "Trigonometry" by Michael Corral: A textbook on trigonometry that covers the basics of the subject, including the Pythagorean identity and the definition of tangent.
• "Trigonometry: A First Course" by Charles P. McKeague: A textbook on trigonometry that covers the basics of the subject, including the Pythagorean identity and the definition of tangent.
• "Trigonometry: A Second Course" by Charles P. McKeague: A textbook on trigonometry that covers more advanced topics, including the Pythagorean identity and the definition of tangent.
  Q&A: If $\cos \theta = \frac{3}{5}$, find $\tan \theta$ =====================================================

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states that for any angle $\theta$ in a right-angled triangle, the sum of the squares of the lengths of the two sides adjacent to the angle is equal to the square of the length of the hypotenuse. Mathematically, this can be expressed as:

$$\sin^2 \theta + \cos^2 \theta = 1$$

Q: How do I find $\sin \theta$ given $\cos \theta$?

A: To find $\sin \theta$ given $\cos \theta$, you can use the Pythagorean identity. Substitute the value of $\cos \theta$ into the equation and solve for $\sin \theta$.

For example, if $\cos \theta = \frac{3}{5}$, you can substitute this value into the Pythagorean identity to solve for $\sin \theta$.

$$\sin^2 \theta + \left(\frac{3}{5}\right)^2 = 1$$

$$\sin^2 \theta + \frac{9}{25} = 1$$

$$\sin^2 \theta = 1 - \frac{9}{25}$$

$$\sin^2 \theta = \frac{16}{25}$$

Taking the square root of both sides, you get:

$$\sin \theta = \pm \sqrt{\frac{16}{25}}$$

$$\sin \theta = \pm \frac{4}{5}$$

Since the sine of an angle is positive in the first and second quadrants, you take the positive value:

$$\sin \theta = \frac{4}{5}$$

Q: How do I find $\tan \theta$ given $\sin \theta$ and $\cos \theta$?

A: To find $\tan \theta$ given $\sin \theta$ and $\cos \theta$, you can use the definition of tangent. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Mathematically, this can be expressed as:

$$\tan \theta = \frac{\sin \theta}{\cos \theta}$$

For example, if $\sin \theta = \frac{4}{5}$ and $\cos \theta = \frac{3}{5}$, you can substitute these values into the definition of tangent to solve for $\tan \theta$.

$$\tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}}$$

$$\tan \theta = \frac{4}{5} \times \frac{5}{3}$$

$$\tan \theta = \frac{4}{3}$$

Q: What are some real-world applications of $\tan \theta$?

A: The value of $\tan \theta$ has many real-world applications, such as:

• Surveying: The tangent of an angle can be used to calculate the height of a building or the distance between two points.
• Navigation: The tangent of an angle can be used to calculate the direction of a ship or a plane.
• Physics: The tangent of an angle can be used to calculate the velocity of an object.

Q: What are some common mistakes to avoid when working with $\tan \theta$?

A: Some common mistakes to avoid when working with $\tan \theta$ include:

• Not using the correct definition of tangent: Make sure to use the definition of tangent as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
• Not checking the quadrant: Make sure to check the quadrant of the angle to ensure that the value of $\tan \theta$ is positive or negative.
• Not using the correct values of $\sin \theta$ and $\cos \theta$: Make sure to use the correct values of $\sin \theta$ and $\cos \theta$ when calculating $\tan \theta$.

Q: What are some tips for memorizing the values of $\tan \theta$?

A: Some tips for memorizing the values of $\tan \theta$ include:

• Using flashcards: Create flashcards with the values of $\tan \theta$ and the corresponding angles.
• Practicing problems: Practice problems that involve calculating the values of $\tan \theta$.
• Using visual aids: Use visual aids such as diagrams or graphs to help visualize the values of $\tan \theta$.

Additional Resources

• Trigonometry for Dummies: A comprehensive guide to trigonometry, including the Pythagorean identity and the definition of tangent.
• Trigonometry: A First Course: A textbook on trigonometry that covers the basics of the subject, including the Pythagorean identity and the definition of tangent.
• Trigonometry: A Second Course: A textbook on trigonometry that covers more advanced topics, including the Pythagorean identity and the definition of tangent.

References

• "Trigonometry" by Michael Corral: A textbook on trigonometry that covers the basics of the subject, including the Pythagorean identity and the definition of tangent.
• "Trigonometry: A First Course" by Charles P. McKeague: A textbook on trigonometry that covers the basics of the subject, including the Pythagorean identity and the definition of tangent.
• "Trigonometry: A Second Course" by Charles P. McKeague: A textbook on trigonometry that covers more advanced topics, including the Pythagorean identity and the definition of tangent.