Finding the Tangent of an Angle Given the Sine

Oct 13, 2025 by windailys.com 148 views

Finding the Tangent of an Angle Given the Sine

In trigonometry, the sine, cosine, and tangent functions are used to describe the relationships between the angles and side lengths of triangles. Given the sine of an angle, we can use trigonometric identities to find the tangent of that angle. In this article, we will explore how to find the tangent of an angle given the sine of that angle.

The sine and tangent functions are defined as follows:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Given that $\sin \theta = \frac{8}{17}$ , we can use the Pythagorean theorem to find the length of the adjacent side.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, we can use the Pythagorean theorem to find the length of the adjacent side.

Let $x$ be the length of the adjacent side. Then, using the Pythagorean theorem, we have:

x^2 + 8^2 = 17^2

Simplifying, we get:

x^2 + 64 = 289

Subtracting 64 from both sides, we get:

x^2 = 225

Taking the square root of both sides, we get:

x = \pm 15

Since the length of a side cannot be negative, we take the positive value:

x = 15

Now that we have found the length of the adjacent side, we can find the tangent of the angle using the formula:

\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{8}{15}

Therefore, the tangent of the angle is $\frac{8}{15}$ .

In this article, we have shown how to find the tangent of an angle given the sine of that angle. We used the Pythagorean theorem to find the length of the adjacent side, and then used the formula for the tangent function to find the tangent of the angle. The tangent of the angle is $\frac{8}{15}$ .

The correct answer is A. $\frac{8}{15}$ .

For more information on trigonometry and the sine and tangent functions, see the following resources:

Wikipedia: Trigonometry
Khan Academy: Trigonometry
Mathway: Trigonometry
Trigonometry Q&A: Finding the Tangent of an Angle Given the Sine

In our previous article, we explored how to find the tangent of an angle given the sine of that angle. In this article, we will answer some common questions related to this topic.

Q: What is the sine and tangent functions?

A: The sine and tangent functions are used to describe the relationships between the angles and side lengths of triangles. The sine function is defined as the ratio of the length of the opposite side to the length of the hypotenuse, while the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side.

Q: How do I find the tangent of an angle given the sine of that angle?

A: To find the tangent of an angle given the sine of that angle, you can use the Pythagorean theorem to find the length of the adjacent side, and then use the formula for the tangent function to find the tangent of the angle.

Q: What is the Pythagorean theorem?

A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Q: How do I use the Pythagorean theorem to find the length of the adjacent side?

A: To use the Pythagorean theorem to find the length of the adjacent side, you can set up an equation using the formula:

x^2 + 8^2 = 17^2

where $x$ is the length of the adjacent side, and 8 and 17 are the lengths of the opposite and hypotenuse sides, respectively.

Q: What if I get a negative value for the length of the adjacent side?

A: If you get a negative value for the length of the adjacent side, it means that the angle is in the second or third quadrant. In this case, you can take the absolute value of the length of the adjacent side to get the correct value.

Q: Can I use the tangent function to find the sine of an angle?

A: Yes, you can use the tangent function to find the sine of an angle. Since the tangent function is defined as the ratio of the length of the opposite side to the length of the adjacent side, you can use the formula:

\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\tan \theta}{\sqrt{1 + \tan^2 \theta}}

Q: What if I get a value for the tangent of an angle that is not on the list of possible answers?

A: If you get a value for the tangent of an angle that is not on the list of possible answers, it means that the angle is not one of the angles listed in the problem. In this case, you can try plugging in different values for the angle to see if you can get a value that is on the list of possible answers.

In this article, we have answered some common questions related to finding the tangent of an angle given the sine of that angle. We hope that this article has been helpful in clarifying any confusion you may have had about this topic.

For more information on trigonometry and the sine and tangent functions, see the following resources:

Q: What is the sine and tangent functions? A: The sine and tangent functions are used to describe the relationships between the angles and side lengths of triangles.
Q: How do I find the tangent of an angle given the sine of that angle? A: To find the tangent of an angle given the sine of that angle, you can use the Pythagorean theorem to find the length of the adjacent side, and then use the formula for the tangent function to find the tangent of the angle.
Q: What is the Pythagorean theorem? A: The Pythagorean theorem is a fundamental concept in geometry that states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

Not using the Pythagorean theorem to find the length of the adjacent side
Not using the formula for the tangent function to find the tangent of the angle
Not taking the absolute value of the length of the adjacent side if it is negative

Make sure to use the correct formula for the tangent function
Make sure to use the Pythagorean theorem to find the length of the adjacent side
Make sure to take the absolute value of the length of the adjacent side if it is negative