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Magazine Chapter 10: Problem 55
Use the given information about \(\theta\) to find the exact values of \- \(\sin (2 \theta)\) \- \(\cos (2 \theta)\) \- \(\tan (2 \theta)\) \(\sin \left(\frac{\theta}{2}\right)\) - \(\cos \left(\frac{\theta}{2}\right)\) - \(\tan \left(\frac{\theta}{2}\right)\) $$ \cos (\theta)=\frac{12}{13} \text { where } \frac{3 \pi}{2}<\theta<2 \pi $$
Short Answer
Expert verified
\(\sin(2\theta) = -\frac{120}{169}, \cos(2\theta) = \frac{119}{169}, \tan(2\theta) = -\frac{120}{119}, \sin\left(\frac{\theta}{2}\right) = -\frac{1}{\sqrt{26}}, \cos\left(\frac{\theta}{2}\right) = \frac{5}{\sqrt{26}}, \tan\left(\frac{\theta}{2}\right) = \frac{1}{5}\).
Step by step solution
01
Determine Quadrant of Angle
Since \(\frac{3\pi}{2} < \theta < 2\pi\), \(\theta\) is in the fourth quadrant, where cosine is positive, but sine and tangent are negative.
02
Find \(\sin(\theta)\)
Using the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) and substituting \(\cos(\theta) = \frac{12}{13}\), we have:\[ \sin^2(\theta) + \left(\frac{12}{13}\right)^2 = 1 \]\[ \sin^2(\theta) = 1 - \frac{144}{169} = \frac{25}{169} \]\[ \sin(\theta) = -\frac{5}{13} \]due to being in the fourth quadrant.
03
Calculate \(\sin(2\theta)\)
Use the double angle identity \(\sin(2\theta) = 2\sin(\theta)\cos(\theta)\):\[ \sin(2\theta) = 2 \cdot -\frac{5}{13} \cdot \frac{12}{13} = -\frac{120}{169} \]
04
Calculate \(\cos(2\theta)\)
Use the double angle identity \(\cos(2\theta) = 2\cos^2(\theta) - 1\):\[ \cos(2\theta) = 2 \left( \frac{12}{13} \right)^2 - 1 = \frac{288}{169} - 1 = \frac{119}{169} \]
05
Calculate \(\tan(2\theta)\)
Use the double angle identity \(\tan(2\theta) = \frac{\sin(2\theta)}{\cos(2\theta)}\):\[ \tan(2\theta) = \frac{-\frac{120}{169}}{\frac{119}{169}} = -\frac{120}{119} \]
06
Find \(\sin\left(\frac{\theta}{2}\right)\)
Using the identity:\[ \sin\left(\frac{\theta}{2}\right) = -\sqrt{\frac{1 - \cos(\theta)}{2}} \]For \(\frac{3\pi}{2} < \theta < 2\pi\), \(\sin\left(\frac{\theta}{2}\right)\) must be negative as it reflects the fourth quadrant condition after half-angle calculation:\[ \sin\left(\frac{\theta}{2}\right) = -\sqrt{\frac{1 - \frac{12}{13}}{2}} = -\frac{1}{\sqrt{26}} \]
07
Find \(\cos\left(\frac{\theta}{2}\right)\)
Using the identity:\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos(\theta)}{2}} \]\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \frac{12}{13}}{2}} = \frac{5}{\sqrt{26}} \]
08
Find \(\tan\left(\frac{\theta}{2}\right)\)
Using the identity:\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos(\theta)}{\sin(\theta)} \]\[ \tan\left(\frac{\theta}{2}\right) = \frac{1 - \frac{12}{13}}{-\frac{5}{13}} = \frac{1}{5} \]
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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 are used to find the trigonometric functions of an angle that is twice another angle. These formulas can simplify complex trigonometric expressions. Let's look into these formulas for sine, cosine, and tangent.
- The double angle formula for sine is given by: \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \). This formula allows you to express the sine of double an angle in terms of the sine and cosine of the single angle.
- The double angle formula for cosine is: \( \cos(2\theta) = 2\cos^2(\theta) - 1 \). Alternatively, it can be expressed as \( \cos(2\theta) = 1 - 2\sin^2(\theta) \). Both forms are useful depending on which trigonometric function you know initially.
- The double angle formula for tangent simplifies to: \( \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \). It's a little more complex but useful for calculating the tangent of a double angle.
Half Angle Formulas
The half angle formulas are another set of essential identities in trigonometry. They help calculate the trigonometric functions for half of a given angle. These identities are particularly useful when dealing with integrals and other trigonometric transformations.
- For sine, the half angle formula is: \( \sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{2}} \). The choice of the positive or negative root depends on the quadrant in which the angle \( \frac{\theta}{2} \) resides.
- For cosine, the formula becomes: \( \cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos(\theta)}{2}} \). Again, the sign depends on the quadrant of the angle \( \frac{\theta}{2} \).
- For tangent, it is: \( \tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}} \). This identity can be expressed in different forms, but it is crucial to consider the sign based on the quadrant.
Trigonometric Functions
Trigonometric functions serve as the backbone of trigonometry, relating angles to ratios of sides in right-angled triangles. The primary trigonometric functions include sine, cosine, and tangent. Each has a distinct relationship based on an angle in the context of a unit circle.
- **Sine** (\( \sin \theta \)) corresponds to the ratio of the length of the opposite side to the hypotenuse.
- **Cosine** (\( \cos \theta \)) relates to the ratio of the adjacent side to the hypotenuse.
- **Tangent** (\( \tan \theta \)) is the ratio of the opposite side to the adjacent side, which can also be expressed as the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
Fourth Quadrant
In the realm of trigonometry, understanding the placement of angles in quadrants is crucial. The fourth quadrant is particularly interesting. This quadrant is where angles range from \(\frac{3\pi}{2} \) to \(2\pi\), or 270掳 to 360掳.
- **Cosine** (\( \cos \theta \)) is positive in the fourth quadrant. This is as the x-coordinate (adjacent side) remains positive.
- **Sine** (\( \sin \theta \)) is negative because the y-coordinate (opposite side) is below the horizontal axis.
- **Tangent** (\( \tan \theta \)) combines the characteristics of sine and cosine, leading it to be negative in this quadrant: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).
