Problem 49 In Exercises $49-58$, use the ... [FREE SOLUTION]

Chapter 10: Problem 49

In Exercises $49-58$, use the given information about $\theta$ to find the exact values of \- $\sin (2 \theta)$ \- $\cos (2 \theta)$ \- $\tan (2 \theta)$ $\sin \left(\frac{\theta}{2}\right)$ - $\cos \left(\frac{\theta}{2}\right)$ - $\tan \left(\frac{\theta}{2}\right)$ $$ \sin (\theta)=-\frac{7}{25} \text { where } \frac{3 \pi}{2}<\theta<2 \pi $$

Short Answer

Expert verified

$\sin(2\theta) = -\frac{336}{625}$, $\cos(2\theta) = \frac{527}{625}$, $\tan(2\theta) = -\frac{336}{527}$, $\sin(\frac{\theta}{2}) = \frac{\sqrt{2}}{10}$, $\cos(\frac{\theta}{2}) = \frac{7\sqrt{2}}{10}$, $\tan(\frac{\theta}{2}) = \frac{1}{7}$.

Step by step solution

Determine the Quadrant

The angle $\theta$ lies between $\frac{3\pi}{2}$ and $2\pi$, which places it in the fourth quadrant. In this quadrant, sine is negative, cosine is positive, and tangent is negative.

Calculate $\cos(\theta)$

Use the Pythagorean identity $\sin^2(\theta) + \cos^2(\theta) = 1$. Given that $\sin(\theta) = -\frac{7}{25}$, we find $\cos(\theta)$ as follows:\[\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \left(-\frac{7}{25}\right)^2 = 1 - \frac{49}{625} = \frac{576}{625}\]Since we're in the fourth quadrant where cosine is positive:\[\cos(\theta) = \frac{24}{25}\]

Calculate $\sin(2\theta)$

Use the double angle identity for sine: $\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$.Substituting the values we have:\[\sin(2\theta) = 2 \left(-\frac{7}{25}\right) \left(\frac{24}{25}\right) = -\frac{336}{625}\]

Calculate $\cos(2\theta)$

Use the double angle identity for cosine: $\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$.\[\cos(2\theta) = \left(\frac{24}{25}\right)^2 - \left(-\frac{7}{25}\right)^2 = \frac{576}{625} - \frac{49}{625} = \frac{527}{625}\]

Calculate $\tan(2\theta)$

Use the relationship $\tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)}$.\[\tan(2\theta) = \frac{-\frac{336}{625}}{\frac{527}{625}} = -\frac{336}{527}\]

Calculate $\sin(\frac{\theta}{2})$

Use the half-angle identity for sine: $\sin(\frac{\theta}{2}) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$.In the fourth quadrant, $\frac{\theta}{2}$ lies in the second quadrant where sine is positive. Therefore:\[\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \frac{24}{25}}{2}} = \sqrt{\frac{1}{50}} = \frac{1}{\sqrt{50}} = \frac{\sqrt{50}}{50} = \frac{\sqrt{2}\sqrt{25}}{50} = \frac{5\sqrt{2}}{50} = \frac{\sqrt{2}}{10}\]

Calculate $\cos(\frac{\theta}{2})$

Use the half-angle identity for cosine: $\cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$.In the second quadrant, $\frac{\theta}{2}$ makes cosine positive:\[\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{24}{25}}{2}} = \sqrt{\frac{49}{50}} = \frac{7}{\sqrt{50}} = \frac{7\sqrt{2}}{10}\]

Calculate $\tan(\frac{\theta}{2})$

Use the relationship $\tan(\frac{\theta}{2}) = \frac{\sin(\frac{\theta}{2})}{\cos(\frac{\theta}{2})}$.\[\tan\left(\frac{\theta}{2}\right) = \frac{\frac{\sqrt{2}}{10}}{\frac{7\sqrt{2}}{10}} = \frac{1}{7}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Double Angle Formulas

The double angle formulas are essential tools in trigonometry. They allow you to express trigonometric functions of angles like $2\theta$ in terms of functions of $\theta$ itself. These formulas include:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$
$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$
$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$

For our problem, knowing $\sin(\theta) = -\frac{7}{25}$ and using these double angle identities, we found $\sin(2\theta)$, $\cos(2\theta)$, and $\tan(2\theta)$. Understanding these formulas helps unravel complex trigonometric problems by converting them to more manageable forms.

Half-Angle Identities

Half-angle identities are derived from double angle formulas and are valuable for finding trigonometric functions of $\frac{\theta}{2}$. These identities include:

$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}}$
$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}}$
$\tan\left(\frac{\theta}{2}\right) = \frac{\sin(\theta)}{1 + \cos(\theta)}$

Choosing the correct sign in these identities is crucial, depending on the quadrant of $\frac{\theta}{2}$. In our case, the positive sign was chosen because $\frac{\theta}{2}$ lies in the second quadrant for $\theta$ in the fourth quadrant. Once the right identities are applied, the simplification gets easier.

Pythagorean Identity

The Pythagorean identity is the backbone of many trigonometric solutions. It is given by the equation:\[\sin^2(\theta) + \cos^2(\theta) = 1\]This identity reflects the fundamental property of trigonometric functions related to the unit circle. When $\sin(\theta) = -\frac{7}{25}$, we used the Pythagorean identity to find $\cos(\theta)$. This helps in solving for other functions like $\cos(2\theta)$ using the double angle formula, ensuring that calculations are accurate.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are essential to understanding angles and their properties:

$\sin(\theta)$ measures the opposite side over the hypotenuse in a right triangle.
$\cos(\theta)$ represents the adjacent side over the hypotenuse.
$\tan(\theta)$ is the ratio of $\sin(\theta)$ to $\cos(\theta)$.

These functions are interrelated and useful for providing a deeper understanding of trigonometry. In problems involving $\theta$, knowing one function aids in determining the others, especially with identities and formulas linked to them.

Angle Quadrants

Understanding angle quadrants helps determine the sign of trigonometric functions. The four quadrants are:

First Quadrant: All functions are positive.
Second Quadrant: Sine positive, cosine and tangent negative.
Third Quadrant: Tangent positive, sine and cosine negative.
Fourth Quadrant: Cosine positive, sine and tangent negative.

In our example, $\theta$ is in the fourth quadrant, hence $\sin(\theta)$ is negative, and $\cos(\theta)$ is positive. This classification helps pick the correct signs when applying identities and formulas, ensuring that the calculated values are accurate and consistent with the trigonometric principles.

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Short Answer

Step by step solution

Determine the Quadrant

Calculate \(\cos(\theta)\)

Calculate \(\sin(2\theta)\)

Calculate \(\cos(2\theta)\)

Calculate \(\tan(2\theta)\)

Calculate \(\sin(\frac{\theta}{2})\)

Calculate \(\cos(\frac{\theta}{2})\)

Calculate \(\tan(\frac{\theta}{2})\)

Key Concepts

Double Angle Formulas

Half-Angle Identities

Pythagorean Identity

Trigonometric Functions

Angle Quadrants

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