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Magazine Chapter 11: Problem 86
Find the exact value: \(\tan \left[\cos ^{-1}\left(-\frac{3}{7}\right)\right]\)
Short Answer
Expert verified
The exact value is \(-\frac{2\sqrt{10}}{3}\).
Step by step solution
01
Understand the Given Function
The problem requires finding \(\tan [\cos^{-1}(-\frac{3}{7})]\). Begin by letting \(\theta = \cos^{-1}(-\frac{3}{7})\). This implies that \(\cos(\theta) = -\frac{3}{7}\) and \(\theta\) is in the range of \([0, \pi]\).
02
Use Trigonometric Identities
Using the Pythagorean identity, \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Since \(\cos(\theta) = -\frac{3}{7}\), we need to find \(\sin(\theta)\) next.
03
Find \(\sin(\theta)\)
Use the Pythagorean identity \[ \sin^2(\theta) + \cos^2(\theta) = 1. \] Substitute \(\cos(\theta) = -\frac{3}{7}\) into the equation: \[ \sin^2(\theta) + \left(-\frac{3}{7}\right)^2 = 1 \] Simplify to get: \[ \sin^2(\theta) + \frac{9}{49} = 1 \] Subtract \(\frac{9}{49}\) from both sides: \[ \sin^2(\theta) = 1 - \frac{9}{49} = \frac{49}{49} - \frac{9}{49} = \frac{40}{49} \] Therefore, \[ \sin(\theta) = \pm\frac{\sqrt{40}}{7} = \pm\frac{2\sqrt{10}}{7} \].
04
Determine the Correct Sign for \(\sin(\theta)\)
Since \(\theta\) is within \([0, \pi]\) and \(\cos(\theta) = -\frac{3}{7}\) places it in the second quadrant where sine is positive, we have: \[ \sin(\theta) = \frac{2\sqrt{10}}{7} \]
05
Calculate \(\tan(\theta)\)
Now use \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Substitute \(\sin(\theta) = \frac{2\sqrt{10}}{7}\) and \(\cos(\theta) = -\frac{3}{7}\): \[ \tan(\theta) = \frac{\frac{2\sqrt{10}}{7}}{-\frac{3}{7}} = \frac{2\sqrt{10}}{-3} = -\frac{2\sqrt{10}}{3} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
trigonometric identities
Trigonometric identities are essential formulas that connect various trigonometric functions. They help us to simplify and solve complex trigonometric equations. In the given exercise, we utilize the identity for tangent: \ \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \ \] Knowing this identity is crucial for finding the value of \tan(\theta) when given \cos(\theta). We also frequently use the Pythagorean identity in trigonometry, which links sine and cosine functions. It's important to understand and remember these identities as they are fundamental tools for solving trigonometric problems.
Pythagorean identity
The Pythagorean identity is a fundamental trigonometric identity that states: \ \[ \sin^2(\theta) + \cos^2(\theta) = 1 \ \] This identity is derived from the Pythagorean theorem applied to a right-angled triangle. In our exercise, we used this identity to find \sin(\theta) given \cos(\theta). To demonstrate: \ \[ \sin^2(\theta) + (-\frac{3}{7})^2 = 1 \ \] Simplifying this, we found that \sin^2(\theta) = \frac{40}{49}, leading to \sin(\theta) = \pm \frac{2\sqrt{10}}{7}\. This identity allows us to determine one trigonometric function if we know the other.
tangent function
The tangent function is another fundamental trigonometric function. It's defined as the ratio of the sine and cosine of the same angle: \ \[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \ \] In our problem, we used the calculated values of \sin(\theta) and \cos(\theta) to find \tan(\theta). This is a common approach in solving trigonometric problems. Given \cos(\theta) = -\frac{3}{7} and \sin(\theta) = \frac{2\sqrt{10}}{7}, we calculated: \ \[ \tan(\theta) = \frac{\frac{2\sqrt{10}}{7}}{-\frac{3}{7}} = -\frac{2\sqrt{10}}{3} \ \] Understanding the tangent function's relationship with sine and cosine helps in solving many trigonometric equations.
cosine function
The cosine function, often written as \cos(\theta), is one of the core trigonometric functions. It measures the horizontal coordinate of a point on the unit circle. In our exercise, we dealt with the inverse cosine function, \cos^{-1}(x). The key idea is to know \cos(\theta), you can find \theta within specific ranges. Given \cos(\theta) = -\frac{3}{7}, we work within the domain \[ 0, \pi \]. The value \theta = \cos^{-1}(-\frac{3}{7}) returns an angle whose cosine is \frac{-3}{7}. Knowing your angle's quadrant is crucial as it helps determine the possible values of sine and cosine functions, and ensures accurate computation of other trigonometric functions like tangent.
- \cos(\theta): Measures horizontal distance
- Inverse Function \cos^{-1}(x): Finds angle given cosine value
- Careful of Quadrants: Determines sign/value of other trigonometric functions
