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Chapter 1: Problem 38

In Exercises \(37-40\) , use the given information to find the values of the six trigonometric functions at the angle \(\theta\) . Give exact answers. $$\theta=\tan ^{-1}\left(-\frac{5}{12}\right)$$

Short Answer

Expert verified
The values of the six trigonometric functions at the angle \( \theta = \tan^{-1}(-5/12) \) are: \( \sin(\theta) = -5/13 \), \( \cos(\theta) = 12/13 \), \( \tan(\theta) = -5/12 \), \( \csc(\theta) = -13/5 \), \( \sec(\theta) = 13/12 \), and \( \cot(\theta) = -12/5 \).

Step by step solution

01

Determine the hypotenuse

Using the Pythagorean theorem for the right triangle, the hypotenuse \( r \) is \( \sqrt{(12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \) .
02

Compute sine value

Now, the sine of the angle \( \theta \), \( \sin(\theta) \), is determined by the ratio of the opposite side to the hypotenuse: \( \sin(\theta) = \frac{-5}{13} \) .
03

Compute cosine value

The cosine of the angle \( \theta \), \( \cos(\theta) \), is determined by the ratio of the adjacent side to the hypotenuse: \( \cos(\theta) = \frac{12}{13} \).
04

Compute tangent value

The tangent of the angle \( \theta \), \( \tan(\theta) \), is determined by the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{-5}{12} \), which is given already.
05

Compute the remaining trigonometric functions

For the remaining functions, cosecant \( \csc(\theta) \) is the reciprocal of sin(\( \theta \)), secant \( \sec(\theta) \) is the reciprocal of cos(\( \theta \)), and cotangent \( \cot(\theta) \) is the reciprocal of tan(\( \theta \)). So, \( \csc(\theta) = \frac{-13}{5} \), \( \sec(\theta) = \frac{13}{12} \), \( \cot(\theta) = \frac{-12}{5} \).

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Key Concepts

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

Pythagorean Theorem
The Pythagorean Theorem is a fundamental concept in geometry. It is used to find the length of the sides in a right triangle. According to this theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In mathematical terms: \[ c^2 = a^2 + b^2\]where \(c\) is the hypotenuse, and \(a\) and \(b\) are the other two sides.
  • This theorem is particularly helpful when you have two sides of a triangle and need to find the third. It is widely used in trigonometry to calculate distances and angles.
  • In this specific exercise, we use it to find the hypotenuse of a triangle. With given sides 12 and \(-5\), the hypotenuse is found to be 13 by computing:\[ \sqrt{12^2 + (-5)^2} = 13\]
The theorem not only aids in solving right triangles but also lays the groundwork for understanding trigonometric functions.
Inverse Trigonometric Functions
Inverse trigonometric functions are essential for determining angles within right triangles. These functions 'undo' the trigonometric functions such as sine, cosine, and tangent. These inverse operations allow us to find an angle when we know two sides of a triangle. Here are some key points:
  • The inverse tangent, or \(\tan^{-1}\), is used when the ratio of the opposite to adjacent sides is known. It helps find the angle when the tangent value is given. In our exercise, \(\tan^{-1}\left(-\frac{5}{12}\right)\) is used to find angle \(\theta\).
  • The notation \(\sin^{-1}(x)\), \(\cos^{-1}(x)\), and \(\tan^{-1}(x)\) are used to represent inverse trigonometric functions, commonly called arcsin, arccos, and arctan, respectively.
  • These functions are restricted to specific ranges to maintain output as a unique angle.
Understanding inverses is crucial as they link known side ratios back to the angles in trigonometric equations.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions provide a deeper understanding of sine, cosine, and tangent by expressing them as ratios and extending their applications: cosecant, secant, and cotangent.
  • Cosecant \(\csc(\theta)\) is the reciprocal of sine \(\sin(\theta)\). If \(\sin(\theta) = \frac{-5}{13}\), then \(\csc(\theta) = \frac{-13}{5}\).
  • Secant \(\sec(\theta)\) is the reciprocal of cosine \(\cos(\theta)\). For \(\cos(\theta) = \frac{12}{13}\), \(\sec(\theta) = \frac{13}{12}\).
  • Cotangent \(\cot(\theta)\) is the reciprocal of tangent \(\tan(\theta)\). Given \(\tan(\theta) = \frac{-5}{12}\), then \(\cot(\theta) = \frac{-12}{5}\).
These reciprocal functions are useful for solving equations and scenarios where division by a trigonometric value is involved. Recognizing these relationships allows one to convert between different trigonometric functions easily.

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Most popular questions from this chapter

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