If sin x = 3/4, x in quadrant I, then find (without finding x) : sin(2x) cos(2x) tan(2x)

Given \(\sin(x) = \frac{3}{4}\) and \(x\) is in the first quadrant, we need to find \(\sin(2x)\), \(\cos(2x)\), and \(\tan(2x)\) without finding \(x\).

First, we'll find \(\cos(x)\). Since \(x\) is in the first quadrant, both sine and cosine are positive. Using the Pythagorean identity:

\[
\sin^2(x) + \cos^2(x) = 1
\]

Substitute \(\sin(x) = \frac{3}{4}\):

\[
\left(\frac{3}{4}\right)^2 + \cos^2(x) = 1
\]

\[
\frac{9}{16} + \cos^2(x) = 1
\]

\[
\cos^2(x) = 1 - \frac{9}{16} = \frac{16}{16} - \frac{9}{16} = \frac{7}{16}
\]

\[
\cos(x) = \sqrt{\frac{7}{16}} = \frac{\sqrt{7}}{4}
\]

Next, we use the double-angle identities to find \(\sin(2x)\), \(\cos(2x)\), and \(\tan(2x)\):

1. **\(\sin(2x)\)**:
   
   \[
   \sin(2x) = 2\sin(x)\cos(x)
   \]

   Substitute \(\sin(x) = \frac{3}{4}\) and \(\cos(x) = \frac{\sqrt{7}}{4}\):

   \[
   \sin(2x) = 2 \cdot \frac{3}{4} \cdot \frac{\sqrt{7}}{4} = 2 \cdot \frac{3\sqrt{7}}{16} = \frac{3\sqrt{7}}{8}
   \]

2. **\(\cos(2x)\)**:
   
   \[
   \cos(2x) = \cos^2(x) - \sin^2(x)
   \]

   Substitute \(\cos(x) = \frac{\sqrt{7}}{4}\) and \(\sin(x) = \frac{3}{4}\):

   \[
   \cos(2x) = \left(\frac{\sqrt{7}}{4}\right)^2 - \left(\frac{3}{4}\right)^2 = \frac{7}{16} - \frac{9}{16} = -\frac{2}{16} = -\frac{1}{8}
   \]

3. **\(\tan(2x)\)**:
   
   \[
   \tan(2x) = \frac{\sin(2x)}{\cos(2x)}
   \]

   Substitute \(\sin(2x) = \frac{3\sqrt{7}}{8}\) and \(\cos(2x) = -\frac{1}{8}\):

   \[
   \tan(2x) = \frac{\frac{3\sqrt{7}}{8}}{-\frac{1}{8}} = \frac{3\sqrt{7}}{8} \cdot \left(-8\right) = -3\sqrt{7}
   \]

Therefore, the values are:
- \(\sin(2x) = \frac{3\sqrt{7}}{8}\)
- \(\cos(2x) = -\frac{1}{8}\)
- \(\tan(2x) = -3\sqrt{7}\)

Would you like more details or have any questions?

Here are some related questions:

1. How do you find \(\sin(x)\) if \(\cos(x) = \frac{4}{5}\) and \(x\) is in the first quadrant?
2. What is the double-angle formula for \(\cos(2x)\)?
3. How do you derive \(\sin(2x)\) from the angle addition formula?
4. What is the relationship between \(\sin(x)\) and \(\cos(x)\) in different quadrants?
5. How do you find \(\cos(x)\) given \(\sin(x) = \frac{5}{13}\) in the first quadrant?
6. How can the Pythagorean identity be used to find \(\sin(x)\) or \(\cos(x)\)?
7. What are the trigonometric values of angles in the first quadrant?
8. How do double-angle identities help simplify trigonometric expressions?

**Tip:** Always use the appropriate identity for the function you're trying to find, and consider the quadrant to determine the correct signs of the trigonometric values.

Learn how to find sin(2x), cos(2x), and tan(2x) given sin(x) = 3/4 and x in the first quadrant without solving for x. This problem explores the application of trigonometric identities such as double-angle formulas and the Pythagorean identity.

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