Answered

Looking for answers? Westonci.ca is your go-to Q&A platform, offering quick, trustworthy responses from a community of experts. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.

Find the exact value without a calculator.

[tex]\[
\tan \frac{7 \pi}{12} = -\sqrt{\frac{? + \sqrt{\square}}{-\sqrt{\square}}}
\][/tex]

Double-Angle Formulas:
[tex]\[
\sin(2 \theta) = 2 \sin \theta \cos \theta
\][/tex]
[tex]\[
\cos(2 \theta) = \cos^2 \theta - \sin^2 \theta
\][/tex]
[tex]\[
\tan(2 \theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}
\][/tex]

Half-Angle Formulas:
[tex]\[
\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}
\][/tex]
[tex]\[
\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}
\][/tex]
[tex]\[
\tan \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}
\][/tex]

Sagot :

GinnyAnswer
To find the exact value of [tex]\(\tan\left(\frac{7\pi}{12}\right)\)[/tex] using a trigonometric approach, we'll follow these steps.

Let's break down the problem systematically.

Step 1: Express [tex]\(\frac{7\pi}{12}\)[/tex] in terms of familiar angles.

[tex]\[ \frac{7\pi}{12} = \frac{3\pi}{4} - \frac{\pi}{3} \][/tex]

So, we need to find [tex]\(\tan\left(\frac{3\pi}{4} - \frac{\pi}{3}\right)\)[/tex].

Step 2: Use the tangent subtraction formula.

The formula for the tangent of the difference of two angles is:

[tex]\[ \tan(a - b) = \frac{\tan(a) - \tan(b)}{1 + \tan(a)\tan(b)} \][/tex]

Substituting [tex]\(a = \frac{3\pi}{4}\)[/tex] and [tex]\(b = \frac{\pi}{3}\)[/tex]:

[tex]\[ \tan\left(\frac{3\pi}{4}\right) = -1 \quad \text{(because \(\tan(135^\circ) = -1\))} \][/tex]
[tex]\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \quad \text{(because \(\tan(60^\circ) = \sqrt{3}\))} \][/tex]

Step 3: Substitute the known values into the tangent subtraction formula.

[tex]\[ \tan\left(\frac{3\pi}{4} - \frac{\pi}{3}\right) = \frac{\tan\left(\frac{3\pi}{4}\right) - \tan\left(\frac{\pi}{3}\right)}{1 + \tan\left(\frac{3\pi}{4}\right)\tan\left(\frac{\pi}{3}\right)} = \frac{-1 - \sqrt{3}}{1 + (-1)(\sqrt{3})} = \frac{-1 - \sqrt{3}}{1 - \sqrt{3}} \][/tex]

Step 4: Simplify the expression.

To simplify [tex]\(\frac{-1 - \sqrt{3}}{1 - \sqrt{3}}\)[/tex], multiply the numerator and the denominator by the conjugate of the denominator:

[tex]\[ \frac{-1 - \sqrt{3}}{1 - \sqrt{3}} \cdot \frac{1 + \sqrt{3}}{1 + \sqrt{3}} = \frac{(-1 - \sqrt{3})(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \][/tex]

Calculate the denominator:

[tex]\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2 \][/tex]

Calculate the numerator:

[tex]\[ (-1 - \sqrt{3})(1 + \sqrt{3}) = (-1)(1) + (-1)(\sqrt{3}) + (-\sqrt{3})(1) + (-\sqrt{3})(\sqrt{3}) = -1 - \sqrt{3} - \sqrt{3} - 3 = -4 - 2\sqrt{3} \][/tex]

So the expression becomes:

[tex]\[ \frac{-1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{-4 - 2\sqrt{3}}{-2} = 2 + \sqrt{3} \][/tex]

Rewriting it for the final exact form:

[tex]\[ \tan\left(\frac{7\pi}{12}\right) = -\sqrt{\frac{-4 + 2\sqrt{3}}{-2}} = 2 + \sqrt{3} \][/tex]

Hence the identity for [tex]\(\tan\left(\frac{7\pi}{12}\right) = 2 + \sqrt{3}\)[/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.