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Magazine Chapter 7: Problem 38
Find the remaining five trigonometric functions of \(\theta .\) $$\cos \theta=-\frac{1}{4}, \sin \theta>0$$
Short Answer
Expert verified
\( \sin \theta = \frac{\sqrt{15}}{4} \), \( \tan \theta = -\sqrt{15} \), \( \csc \theta = \frac{4\sqrt{15}}{15} \), \( \sec \theta = -4 \), \( \cot \theta = -\frac{\sqrt{15}}{15} \).
Step by step solution
01
- Determine the Quadrant
Given that \(\theta\) is such that \(\cos \theta = -\frac{1}{4}\) and \(\sin \theta > 0\), the angle \(\theta\) is in the second quadrant because cosine is negative and sine is positive only in this quadrant.
02
- Use the Pythagorean Identity
According to the Pythagorean identity, \(\sin^2 \theta + \cos^2 \theta = 1\). Substitute \(\cos \theta = -\frac{1}{4}\):\[\begin{align*}\sin^2 \theta + \(\frac{-1}{4}\)^2 &= 1 \sin^2 \theta + \frac{1}{16} &= 1 \sin^2 \theta &= 1 - \frac{1}{16} \sin^2 \theta &= \frac{16}{16} - \frac{1}{16} \sin^2 \theta &= \frac{15}{16}\]Then find \(\sin \theta\) by taking the positive square root (since \(\sin \theta > 0\)):\[\sin \theta = \sqrt{\frac{15}{16}} = \frac{\sqrt{15}}{4}\]
03
- Find the Tangent Function
The tangent function is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Substitute the values found:\[\tan \theta = \frac{\frac{\sqrt{15}}{4}}{\frac{-1}{4}} = -\sqrt{15}\]
04
- Find the Cosecant Function
The cosecant function is the reciprocal of the sine function: \(\csc \theta = \frac{1}{\sin \theta}\). Substitute the value found:\[\csc \theta = \frac{1}{\frac{\sqrt{15}}{4}} = \frac{4}{\sqrt{15}} = \frac{4\sqrt{15}}{15}\]
05
- Find the Secant Function
The secant function is the reciprocal of the cosine function: \(\sec \theta = \frac{1}{\cos \theta}\). Substitute the value given:\[\sec \theta = \frac{1}{-\frac{1}{4}} = -4\]
06
- Find the Cotangent Function
The cotangent function is the reciprocal of the tangent function: \(\cot \theta = \frac{1}{\tan \theta}\). Substitute the value found:\[\cot \theta = \frac{1}{-\sqrt{15}} = -\frac{1}{\sqrt{15}} = -\frac{\sqrt{15}}{15}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosine
Cosine is one of the primary trigonometric functions. It is abbreviated as \(\text{cos}\). In a right triangle, cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse. The cosine function is defined as:
\[ \text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Cosine values range between -1 and 1. Understanding the cosine function is essential for solving many trigonometric problems, especially those involving angles in different quadrants.
\[ \text{cos} \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Cosine values range between -1 and 1. Understanding the cosine function is essential for solving many trigonometric problems, especially those involving angles in different quadrants.
Sine
Sine is another fundamental trigonometric function, abbreviated as \(\text{sin}\). In a right triangle, sine of an angle is the ratio of the length of the opposite side to the hypotenuse. The sine function is expressed as:
\[ \text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
Sine values also range from -1 to 1. It's particularly useful when working with the Pythagorean identity and determining values of other trigonometric functions.
\[ \text{sin} \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
Sine values also range from -1 to 1. It's particularly useful when working with the Pythagorean identity and determining values of other trigonometric functions.
Quadrants
The coordinate plane is divided into four quadrants. Each quadrant has specific characteristics for the signs of trigonometric functions:
This knowledge helps in identifying the correct trigonometric functions values based on the given conditions.
- Quadrant I: Both sine and cosine are positive.
- Quadrant II: Sine is positive and cosine is negative.
- Quadrant III: Both sine and cosine are negative.
- Quadrant IV: Sine is negative and cosine is positive.
This knowledge helps in identifying the correct trigonometric functions values based on the given conditions.
Pythagorean Identity
The Pythagorean identity is a fundamental relation in trigonometry which shows the connection between sine and cosine. It is given by:
\[ \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \]
This identity is derived from the Pythagorean Theorem applied to a unit circle. It is crucial because it allows you to find \(\text{sin} \theta\) given \(\text{cos} \theta\) and vice versa.
\[ \text{sin}^2 \theta + \text{cos}^2 \theta = 1 \]
This identity is derived from the Pythagorean Theorem applied to a unit circle. It is crucial because it allows you to find \(\text{sin} \theta\) given \(\text{cos} \theta\) and vice versa.
Tangent
Tangent is a trigonometric function defined as the ratio of sine to cosine. It is abbreviated as \(\text{tan}\). The tangent function is expressed as:
\[ \text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \]
Tangent values can range from negative to positive infinity. Knowing tangent is useful for calculating angles and solving trigonometric equations.
\[ \text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta} \]
Tangent values can range from negative to positive infinity. Knowing tangent is useful for calculating angles and solving trigonometric equations.
Cosecant
Cosecant, abbreviated as \(\text{csc}\), is the reciprocal of the sine function. It is expressed as:
\[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \]
Since sine values range between -1 and 1 (excluding 0), cosecant will vary from negative to positive infinity. Understanding cosecant helps in many trigonometric calculations.
\[ \text{csc} \theta = \frac{1}{\text{sin} \theta} \]
Since sine values range between -1 and 1 (excluding 0), cosecant will vary from negative to positive infinity. Understanding cosecant helps in many trigonometric calculations.
Secant
Secant, abbreviated as \(\text{sec}\), is the reciprocal of the cosine function. It is given by:
\[ \text{sec} \theta = \frac{1}{\text{cos} \theta} \]
Secant values range from -infinity to -1 and 1 to infinity. It provides a convenient way to find values when cosine is known.
\[ \text{sec} \theta = \frac{1}{\text{cos} \theta} \]
Secant values range from -infinity to -1 and 1 to infinity. It provides a convenient way to find values when cosine is known.
Cotangent
Cotangent is the reciprocal of the tangent function, abbreviated as \(\text{cot}\). It can be defined as:
\[ \text{cot} \theta = \frac{1}{\text{tan} \theta} \]
Cotangent is also expressed as the ratio of cosine to sine:
\[ \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \] Cotangent helps to complement the set of trigonometric functions for solving various types of trigonometric problems.
\[ \text{cot} \theta = \frac{1}{\text{tan} \theta} \]
Cotangent is also expressed as the ratio of cosine to sine:
\[ \text{cot} \theta = \frac{\text{cos} \theta}{\text{sin} \theta} \] Cotangent helps to complement the set of trigonometric functions for solving various types of trigonometric problems.
