Given $\cos \theta = \frac{12}{13}$, Find $\sin \theta$.A. $\frac{12}{5}$ B. $\frac{5}{12}$ C. $\frac{13}{12}$ D. $\frac{5}{13}$

Introduction

In trigonometry, the sine and cosine functions are fundamental in describing the relationships between the angles and side lengths of triangles. Given the cosine value of an angle, we can use the Pythagorean identity to find the sine value. In this article, we will explore how to find the sine of an angle given the cosine value.

The Pythagorean Identity

The Pythagorean identity states that for any angle $\theta$, the following equation holds:

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

This identity can be used to find the sine value of an angle given the cosine value.

Finding Sine Value

Given $\cos \theta = \frac{12}{13}$, we can use the Pythagorean identity to find the sine value.

Step 1: Square the Cosine Value

First, we square the given cosine value:

$$\left(\frac{12}{13}\right)^2 = \frac{144}{169}$$

Step 2: Subtract from 1

Next, we subtract the squared cosine value from 1:

$$1 - \frac{144}{169} = \frac{25}{169}$$

Step 3: Take the Square Root

Finally, we take the square root of the result to find the sine value:

$$\sin \theta = \sqrt{\frac{25}{169}} = \frac{5}{13}$$

Conclusion

In this article, we used the Pythagorean identity to find the sine value of an angle given the cosine value. We squared the cosine value, subtracted it from 1, and then took the square root of the result to find the sine value. The final answer is $\frac{5}{13}$.

Answer

The correct answer is:

• D. $\frac{5}{13}$

Additional Information

• The Pythagorean identity can be used to find the sine value of an angle given the cosine value.
• The sine value can be found by squaring the cosine value, subtracting it from 1, and then taking the square root of the result.
• The final answer is $\frac{5}{13}$.

References

• [1] "Trigonometry" by Michael Corral, 2018.
• [2] "Mathematics for the Nonmathematician" by Morris Kline, 1967.

Related Topics

• Finding cosine value given sine value
• Using the Pythagorean identity to find sine and cosine values
• Trigonometric identities and formulas
  Frequently Asked Questions (FAQs) =====================================

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental equation in trigonometry that states:

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

This identity can be used to find the sine value of an angle given the cosine value, and vice versa.

Q: How do I find the sine value of an angle given the cosine value?

A: To find the sine value of an angle given the cosine value, you can use the following steps:

1. Square the cosine value.
2. Subtract the squared cosine value from 1.
3. Take the square root of the result.

Q: What if the cosine value is negative?

A: If the cosine value is negative, you can use the following steps to find the sine value:

1. Square the absolute value of the cosine value.
2. Subtract the squared absolute value of the cosine value from 1.
3. Take the square root of the result.

Q: Can I use the Pythagorean identity to find the cosine value of an angle given the sine value?

A: Yes, you can use the Pythagorean identity to find the cosine value of an angle given the sine value. Simply square the sine value, subtract it from 1, and then take the square root of the result.

Q: What if I have a right triangle with a hypotenuse of length 13 and a leg of length 12?

A: If you have a right triangle with a hypotenuse of length 13 and a leg of length 12, you can use the Pythagorean identity to find the sine and cosine values of the angle opposite the leg.

First, find the length of the other leg using the Pythagorean theorem:

$$a^2 + b^2 = c^2$$

where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

In this case, $a = 12$, $c = 13$, and $b$ is unknown. Plugging in these values, we get:

$$12^2 + b^2 = 13^2$$

Simplifying, we get:

$$144 + b^2 = 169$$

Subtracting 144 from both sides, we get:

$$b^2 = 25$$

Taking the square root of both sides, we get:

$$b = 5$$

Now that we have the lengths of both legs, we can use the Pythagorean identity to find the sine and cosine values of the angle opposite the leg.

First, find the sine value:

$$\sin \theta = \frac{a}{c} = \frac{12}{13}$$

Next, find the cosine value:

$$\cos \theta = \frac{b}{c} = \frac{5}{13}$$

Q: Can I use the Pythagorean identity to find the sine and cosine values of an angle given the lengths of the legs of a right triangle?

A: Yes, you can use the Pythagorean identity to find the sine and cosine values of an angle given the lengths of the legs of a right triangle. Simply use the following formulas:

$$\sin \theta = \frac{a}{c}$$

$$\cos \theta = \frac{b}{c}$$

where $a$ and $b$ are the lengths of the legs, and $c$ is the length of the hypotenuse.

Q: What if I have a right triangle with a hypotenuse of length 5 and a leg of length 12?

A: If you have a right triangle with a hypotenuse of length 5 and a leg of length 12, you cannot use the Pythagorean identity to find the sine and cosine values of the angle opposite the leg. This is because the length of the hypotenuse is less than the length of the leg, which is not possible in a right triangle.

Conclusion

In this article, we answered some frequently asked questions about the Pythagorean identity and how to use it to find the sine and cosine values of an angle given the cosine value, and vice versa. We also discussed how to use the Pythagorean identity to find the sine and cosine values of an angle given the lengths of the legs of a right triangle.