Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover a wealth of knowledge from professionals across various disciplines on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.

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're committed to providing you with the best information available. Return anytime for more. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.