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Magazine Chapter 5: Problem 13
\(\sin \theta=\frac{48}{73} ; \cos \theta<0\)
Short Answer
Expert verified
\( \cos \theta = -\frac{55}{73} \)
Step by step solution
01
Understanding the Given Information
We are given \( \sin \theta = \frac{48}{73} \) and \( \cos \theta < 0 \). This suggests that \( \theta \) is in either the second or third quadrant because the sine is positive and cosine is negative in these quadrants.
02
Using the Pythagorean Identity
Recall the Pythagorean identity: \( \sin^2 \theta + \cos^2 \theta = 1 \). We can use this to find \( \cos \theta \). Substitute \( \sin \theta = \frac{48}{73} \) into the identity:\[ \left( \frac{48}{73} \right)^2 + \cos^2 \theta = 1 \]
03
Calculating \( \cos^2 \theta \)
First, calculate \( \left( \frac{48}{73} \right)^2 \):\[ \frac{48^2}{73^2} = \frac{2304}{5329} \]Substitute \( \frac{2304}{5329} \) into the identity:\[ \frac{2304}{5329} + \cos^2 \theta = 1 \]
04
Simplifying the Equation
Rearrange the equation to find \( \cos^2 \theta \):\[ \cos^2 \theta = 1 - \frac{2304}{5329} = \frac{5329 - 2304}{5329} = \frac{3025}{5329} \]
05
Finding \( \cos \theta \)
Since \( \cos^2 \theta = \frac{3025}{5329} \), take the square root to find \( \cos \theta \). Remember \( \cos \theta < 0 \):\[ \cos \theta = -\sqrt{\frac{3025}{5329}} = -\frac{55}{73} \]
06
Conclusion
We've calculated \( \cos \theta \) considering the sign in the given conditions. Thus, \( \cos \theta = -\frac{55}{73} \), which is consistent with our quadrant analysis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Pythagorean Identity
The Pythagorean Identity is a fundamental relationship in trigonometry that holds true for any angle \( \theta \). It states that the square of the sine of an angle plus the square of the cosine of that angle equals 1. Mathematically, this is expressed as: \[\sin^2 \theta + \cos^2 \theta = 1\]This identity is derived from the Pythagorean Theorem and the unit circle where the radius equals 1.
In a right triangle, this relation also helps us when we know one trigonometric value, such as sine, and we need to find the cosine. We just rearrange the identity to solve for the missing value.
In a right triangle, this relation also helps us when we know one trigonometric value, such as sine, and we need to find the cosine. We just rearrange the identity to solve for the missing value.
- Example: If \( \sin \theta = \frac{48}{73} \), substitute into the identity to find \( \cos \theta \).
- The identity is especially useful in solving trigonometric equations and analyzing trigonometric functions.
Sine Function
The sine function is a fundamental component of trigonometry, associated with the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
Symbolically, it is denoted as \( \sin \theta \), where \( \theta \) is an angle. In our original exercise, we have \( \sin \theta = \frac{48}{73} \).
When using the unit circle, the sine of an angle is interpreted as the y-coordinate of the point on the circle.
Symbolically, it is denoted as \( \sin \theta \), where \( \theta \) is an angle. In our original exercise, we have \( \sin \theta = \frac{48}{73} \).
When using the unit circle, the sine of an angle is interpreted as the y-coordinate of the point on the circle.
- The sine function is periodic with a period of \( 2\pi \).
- It ranges between -1 and 1 for all angles \( \theta \).
- The function is positive in the first and second quadrants, as mentioned in the exercise when identifying the possible quadrants for \( \theta \).
Cosine Function
The cosine function complements the sine function and is equally essential in trigonometry. It represents the ratio of the length of the adjacent side to the hypotenuse in a right-angled triangle.
It is represented as \( \cos \theta \), where \( \theta \) is the angle of interest. In the exercise, \( \cos \theta < 0 \), meaning the cosine value is negative.
Referring to the unit circle, the cosine of an angle is the x-coordinate where the terminal side intersects. Some key features are:
It is represented as \( \cos \theta \), where \( \theta \) is the angle of interest. In the exercise, \( \cos \theta < 0 \), meaning the cosine value is negative.
Referring to the unit circle, the cosine of an angle is the x-coordinate where the terminal side intersects. Some key features are:
- Like sine, it has a range of -1 to 1.
- The cosine function is periodic with a period of \( 2\pi \).
- It is positive in the first and fourth quadrants, negative in the second and third.
Quadrants of Angles
Understanding quadrants is crucial for solving trigonometric problems since the sign (positive or negative) of the sine and cosine functions depends on the quadrant of the angle. The coordinate plane is divided into four quadrants:
Identifying the correct quadrant helps determine the correct signs of trigonometric functions, which is crucial in their applications.
- First Quadrant (0 to \( \frac{\pi}{2} \)): Sine and cosine are both positive.
- Second Quadrant (\( \frac{\pi}{2} \) to \( \pi \)): Sine is positive, cosine is negative.
- Third Quadrant (\( \pi \) to \( \frac{3\pi}{2} \)): Sine and cosine are both negative.
- Fourth Quadrant (\( \frac{3\pi}{2} \) to \( 2\pi \)): Sine is negative, cosine is positive.
Identifying the correct quadrant helps determine the correct signs of trigonometric functions, which is crucial in their applications.
