If Sin ⁡ X = 5 8 \sin X=\frac{5}{8} Sin X = 8 5 ​ And X X X Is In Quadrant I, Then Find (without Finding X X X ): Sin ⁡ ( 2 X ) = □ Cos ⁡ ( 2 X ) = □ Tan ⁡ ( 2 X ) = □ \begin{array}{l} \sin (2x) = \square \\ \cos (2x) = \square \\ \tan (2x) = \square \end{array} Sin ( 2 X ) = □ Cos ( 2 X ) = □ Tan ( 2 X ) = □ ​

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Introduction

In this article, we will explore the trigonometric functions of double angles, specifically $\sin (2x)$, $\cos (2x)$, and $\tan (2x)$, given that $\sin x = \frac{5}{8}$ and $x$ is in quadrant I. We will use the double angle formulas to find the values of these trigonometric functions without finding the value of $x$.

Double Angle Formulas

The double angle formulas for sine, cosine, and tangent are:

• $\sin (2x) = 2\sin x \cos x$
• $\cos (2x) = \cos^2 x - \sin^2 x$
• $\tan (2x) = \frac{\sin (2x)}{\cos (2x)}$

Finding $\sin (2x)$

To find $\sin (2x)$, we can use the double angle formula:

$\sin (2x) = 2\sin x \cos x$

Since $\sin x = \frac{5}{8}$, we need to find $\cos x$ to substitute into the formula. We can use the Pythagorean identity:

$\sin^2 x + \cos^2 x = 1$

Substituting $\sin x = \frac{5}{8}$, we get:

$\left(\frac{5}{8}\right)^2 + \cos^2 x = 1$

Simplifying, we get:

$\frac{25}{64} + \cos^2 x = 1$

$\cos^2 x = 1 - \frac{25}{64}$

$\cos^2 x = \frac{39}{64}$

Since $x$ is in quadrant I, $\cos x$ is positive. Therefore, we can take the square root of both sides:

$\cos x = \sqrt{\frac{39}{64}}$

$\cos x = \frac{\sqrt{39}}{8}$

Now that we have found $\cos x$, we can substitute it into the double angle formula for $\sin (2x)$:

$\sin (2x) = 2\sin x \cos x$

$\sin (2x) = 2\left(\frac{5}{8}\right)\left(\frac{\sqrt{39}}{8}\right)$

$\sin (2x) = \frac{5\sqrt{39}}{32}$

Finding $\cos (2x)$

To find $\cos (2x)$, we can use the double angle formula:

$\cos (2x) = \cos^2 x - \sin^2 x$

Substituting $\cos x = \frac{\sqrt{39}}{8}$ and $\sin x = \frac{5}{8}$, we get:

$\cos (2x) = \left(\frac{\sqrt{39}}{8}\right)^2 - \left(\frac{5}{8}\right)^2$

$\cos (2x) = \frac{39}{64} - \frac{25}{64}$

$\cos (2x) = \frac{14}{64}$

$\cos (2x) = \frac{7}{32}$

Finding $\tan (2x)$

To find $\tan (2x)$, we can use the double angle formula:

$\tan (2x) = \frac{\sin (2x)}{\cos (2x)}$

Substituting $\sin (2x) = \frac{5\sqrt{39}}{32}$ and $\cos (2x) = \frac{7}{32}$, we get:

$\tan (2x) = \frac{\frac{5\sqrt{39}}{32}}{\frac{7}{32}}$

$\tan (2x) = \frac{5\sqrt{39}}{7}$

Conclusion

In this article, we used the double angle formulas to find the values of $\sin (2x)$, $\cos (2x)$, and $\tan (2x)$ given that $\sin x = \frac{5}{8}$ and $x$ is in quadrant I. We found that:

• $\sin (2x) = \frac{5\sqrt{39}}{32}$
• $\cos (2x) = \frac{7}{32}$
• $\tan (2x) = \frac{5\sqrt{39}}{7}$

Introduction

In our previous article, we explored the double angle formulas for sine, cosine, and tangent, and used them to find the values of these trigonometric functions given that $\sin x = \frac{5}{8}$ and $x$ is in quadrant I. In this article, we will answer some common questions related to double angle formulas and trigonometry.

Q: What are the double angle formulas for sine, cosine, and tangent?

A: The double angle formulas for sine, cosine, and tangent are:

• $\sin (2x) = 2\sin x \cos x$
• $\cos (2x) = \cos^2 x - \sin^2 x$
• $\tan (2x) = \frac{\sin (2x)}{\cos (2x)}$

Q: How do I use the double angle formulas to find the values of trigonometric functions?

A: To use the double angle formulas, you need to know the values of $\sin x$ and $\cos x$. You can then substitute these values into the formulas to find the values of $\sin (2x)$, $\cos (2x)$, and $\tan (2x)$.

Q: What is the difference between the double angle formulas and the sum and difference formulas?

A: The double angle formulas and the sum and difference formulas are two different sets of formulas used in trigonometry. The double angle formulas are used to find the values of trigonometric functions of double angles, while the sum and difference formulas are used to find the values of trigonometric functions of sums and differences of angles.

Q: Can I use the double angle formulas to find the values of trigonometric functions of any angle?

A: Yes, you can use the double angle formulas to find the values of trigonometric functions of any angle. However, you need to know the values of $\sin x$ and $\cos x$ for the given angle.

Q: How do I find the values of $\sin x$ and $\cos x$ for a given angle?

A: To find the values of $\sin x$ and $\cos x$ for a given angle, you can use the unit circle or the Pythagorean identity:

$\sin^2 x + \cos^2 x = 1$

Q: What is the unit circle and how is it used in trigonometry?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is used to define the trigonometric functions and to find the values of $\sin x$ and $\cos x$ for any angle.

Q: Can I use the double angle formulas to find the values of trigonometric functions of negative angles?

A: Yes, you can use the double angle formulas to find the values of trigonometric functions of negative angles. However, you need to be careful when substituting the values of $\sin x$ and $\cos x$ into the formulas, as the signs of these values may change for negative angles.

Q: How do I use the double angle formulas to find the values of trigonometric functions of angles in different quadrants?

A: To use the double angle formulas to find the values of trigonometric functions of angles in different quadrants, you need to know the values of $\sin x$ and $\cos x$ for the given angle and quadrant. You can then substitute these values into the formulas to find the values of $\sin (2x)$, $\cos (2x)$, and $\tan (2x)$.

Conclusion

In this article, we answered some common questions related to double angle formulas and trigonometry. We hope that this article has provided a clear understanding of the double angle formulas and how to use them to find the values of trigonometric functions. If you have any further questions, please don't hesitate to ask.