Assume Θ \theta Θ Is An Acute Angle. Sin ⁡ Θ = 13 16 Tan ⁡ Θ = [ ? ] [ ? ] \begin{array}{c} \sin \theta = \frac{13}{16} \\ \tan \theta = \frac{[?]}{\sqrt{[?]}} \end{array} Sin Θ = 16 13 ​ Tan Θ = [ ?] ​ [ ?] ​ ​ Fill In The Missing Values For Tan ⁡ Θ \tan \theta Tan Θ .

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Introduction

In trigonometry, the tangent of an angle in a right 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 sine of an angle, we can use the Pythagorean identity to find the cosine of the angle, and then use the definitions of sine and cosine to find the tangent of the angle. In this article, we will use the given information about the sine of an angle to find the missing values for the tangent of the angle.

The Pythagorean Identity

The Pythagorean identity states that for any angle θ\theta in a right triangle, the following equation holds:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

We can use this identity to find the cosine of the angle, given the sine of the angle.

Finding the Cosine of the Angle

Given that sin⁡θ=1316\sin \theta = \frac{13}{16}, we can use the Pythagorean identity to find the cosine of the angle:

cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta

cos⁡2θ=1−(1316)2\cos^2 \theta = 1 - \left(\frac{13}{16}\right)^2

cos⁡2θ=1−169256\cos^2 \theta = 1 - \frac{169}{256}

cos⁡2θ=87256\cos^2 \theta = \frac{87}{256}

Taking the square root of both sides, we get:

cos⁡θ=±87256\cos \theta = \pm \sqrt{\frac{87}{256}}

Since θ\theta is an acute angle, the cosine of the angle is positive. Therefore, we can write:

cos⁡θ=87256\cos \theta = \sqrt{\frac{87}{256}}

Finding the Tangent of the Angle

Now that we have the sine and cosine of the angle, we can use the definitions of sine and cosine to find the tangent of the angle:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

tan⁡θ=131687256\tan \theta = \frac{\frac{13}{16}}{\sqrt{\frac{87}{256}}}

To simplify the expression, we can multiply the numerator and denominator by 256\sqrt{256}:

tan⁡θ=1316⋅25687256⋅256\tan \theta = \frac{\frac{13}{16} \cdot \sqrt{256}}{\sqrt{\frac{87}{256}} \cdot \sqrt{256}}

tan⁡θ=1316⋅1687\tan \theta = \frac{\frac{13}{16} \cdot 16}{\sqrt{87}}

tan⁡θ=1387\tan \theta = \frac{13}{\sqrt{87}}

Simplifying the Expression

To simplify the expression further, we can rationalize the denominator by multiplying the numerator and denominator by 87\sqrt{87}:

tan⁡θ=1387⋅8787\tan \theta = \frac{13}{\sqrt{87}} \cdot \frac{\sqrt{87}}{\sqrt{87}}

tan⁡θ=138787\tan \theta = \frac{13\sqrt{87}}{87}

Conclusion

In this article, we used the given information about the sine of an angle to find the missing values for the tangent of the angle. We first used the Pythagorean identity to find the cosine of the angle, and then used the definitions of sine and cosine to find the tangent of the angle. The final expression for the tangent of the angle is 138787\frac{13\sqrt{87}}{87}.

Final Answer

The final answer is: 138787\boxed{\frac{13\sqrt{87}}{87}}

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states that for any angle θ\theta in a right triangle, the following equation holds:

sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

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

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

A: To find the cosine of an angle, given the sine of the angle, you can use the Pythagorean identity:

cos⁡2θ=1−sin⁡2θ\cos^2 \theta = 1 - \sin^2 \theta

Taking the square root of both sides, you get:

cos⁡θ=±1−sin⁡2θ\cos \theta = \pm \sqrt{1 - \sin^2 \theta}

Since θ\theta is an acute angle, the cosine of the angle is positive. Therefore, you can write:

cos⁡θ=1−sin⁡2θ\cos \theta = \sqrt{1 - \sin^2 \theta}

Q: How do I find the tangent of an angle, given the sine and cosine of the angle?

A: To find the tangent of an angle, given the sine and cosine of the angle, you can use the definition of tangent:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Q: What is the final expression for the tangent of the angle?

A: The final expression for the tangent of the angle is:

tan⁡θ=138787\tan \theta = \frac{13\sqrt{87}}{87}

Q: Can I simplify the expression for the tangent of the angle further?

A: Yes, you can simplify the expression for the tangent of the angle further by rationalizing the denominator. To do this, you can multiply the numerator and denominator by 87\sqrt{87}:

tan⁡θ=138787⋅8787\tan \theta = \frac{13\sqrt{87}}{87} \cdot \frac{\sqrt{87}}{\sqrt{87}}

tan⁡θ=13⋅8787⋅87\tan \theta = \frac{13 \cdot 87}{87 \cdot \sqrt{87}}

tan⁡θ=1387\tan \theta = \frac{13}{\sqrt{87}}

Q: Why do I need to rationalize the denominator?

A: Rationalizing the denominator is necessary to eliminate any radicals in the denominator. In this case, the denominator is 87\sqrt{87}, which is a radical. By multiplying the numerator and denominator by 87\sqrt{87}, we can eliminate the radical in the denominator and simplify the expression.

Q: What is the significance of the Pythagorean identity in trigonometry?

A: The Pythagorean identity is a fundamental concept in trigonometry that allows us to find the cosine of an angle, given the sine of the angle. This identity is used extensively in trigonometry to solve problems involving right triangles.

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

A: Yes, you can use the Pythagorean identity to find the sine of an angle, given the cosine of the angle. To do this, you can rearrange the Pythagorean identity to solve for sin⁡θ\sin \theta:

sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta

Taking the square root of both sides, you get:

sin⁡θ=±1−cos⁡2θ\sin \theta = \pm \sqrt{1 - \cos^2 \theta}

Since θ\theta is an acute angle, the sine of the angle is positive. Therefore, you can write:

sin⁡θ=1−cos⁡2θ\sin \theta = \sqrt{1 - \cos^2 \theta}

Q: What is the final expression for the sine of the angle?

A: The final expression for the sine of the angle is:

sin⁡θ=1−cos⁡2θ\sin \theta = \sqrt{1 - \cos^2 \theta}

sin⁡θ=1−(138787)2\sin \theta = \sqrt{1 - \left(\frac{13\sqrt{87}}{87}\right)^2}

sin⁡θ=1−132⋅87872\sin \theta = \sqrt{1 - \frac{13^2 \cdot 87}{87^2}}

sin⁡θ=1−132⋅87872\sin \theta = \sqrt{1 - \frac{13^2 \cdot 87}{87^2}}

sin⁡θ=872−132⋅87872\sin \theta = \sqrt{\frac{87^2 - 13^2 \cdot 87}{87^2}}

sin⁡θ=87(87−132)872\sin \theta = \sqrt{\frac{87(87 - 13^2)}{87^2}}

sin⁡θ=87(87−169)872\sin \theta = \sqrt{\frac{87(87 - 169)}{87^2}}

sin⁡θ=87(−82)872\sin \theta = \sqrt{\frac{87(-82)}{87^2}}

sin⁡θ=−8287\sin \theta = \sqrt{\frac{-82}{87}}

sin⁡θ=−8287\sin \theta = \frac{\sqrt{-82}}{87}

sin⁡θ=82i87\sin \theta = \frac{\sqrt{82}i}{87}

Conclusion

In this article, we have answered some frequently asked questions about solving for tan⁡θ\tan \theta. We have discussed the Pythagorean identity, how to find the cosine of an angle, given the sine of the angle, and how to find the tangent of an angle, given the sine and cosine of the angle. We have also simplified the expression for the tangent of the angle and discussed the significance of the Pythagorean identity in trigonometry.