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Chapter 7: Problem 17

Find \(\sin \theta\) $$\csc \theta=-\frac{2}{4}$$

Short Answer

Expert verified
The value of \(\text{sin} \theta\) is \(-2\).

Step by step solution

01

Simplify \(\frac{-2}{4}\)

The given \(\frac{-2}{4}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor. \(\frac{-2}{4} = \frac{-1}{2}\).
02

Recall the relationship between \(\text{csc} \theta\) and \(\text{sin} \theta\)

The cosecant function is the reciprocal of the sine function. Therefore, \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). Given that \(\text{csc} \theta = \frac{-1}{2}\), it follows that \(\text{sin} \theta = \frac{1}{\text{csc} \theta}\).
03

Evaluate \(\text{sin} \theta\)

Now substitute the given value for \(\text{csc} \theta\): \(\text{sin} \theta = \frac{1}{\frac{-1}{2}} = -2\).

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Key Concepts

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

sine function
The sine function, often written as \(\text{sin} \theta\), is a fundamental trigonometric function that describes the relationship between the angle of a right triangle and the ratio of the length of the side opposite the angle to the length of the hypotenuse. Simply put, if you have a right triangle:
  • One of its angles, labeled \(\theta\), will guide you in determining specific side lengths.
  • The side opposite this angle is called the **opposite side**.
  • The longest side, opposite the right angle, is the **hypotenuse**.
Using these, the sine of \(\theta\) is given by: \(\text{sin} \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\). Sine is crucial in many aspects of geometry, physics, engineering, and many fields involving periodic functions.
cosecant function
The cosecant function, denoted as \(\text{csc} \theta\), is the reciprocal of the sine function. This means it is the inverse of the sine function with respect to multiplication. Therefore, \(\text{csc} \theta\) is defined as: \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\). Because it is a reciprocal, if the sine of \(\theta\) is known, the cosecant can be easily calculated. In practical terms:
  • When \(\text{sin} \theta\) is small, \(\text{csc} \theta\) can become large.
  • Conversely, when \(\text{sin} \theta\) is large, \(\text{csc} \theta\) can be small.
  • Keep in mind, \(\text{csc} \theta\) is undefined when \(\text{sin} \theta = 0\) because division by zero is undefined.
Understanding cosecant helps to round out one's grasp of the trigonometric functions, especially in the context of solving equations and understanding their behavior.
reciprocal relationships
Reciprocal relationships are a key concept in understanding trigonometric functions. A reciprocal of a number is essentially its 'flip.' For example, the reciprocal of 2 is \(\frac{1}{2}\), and the reciprocal of -\frac{1}{2} is -2. In trigonometry, this principle is employed extensively:
  • The sine function \(\text{sin} \theta\) and the cosecant function \(\text{csc} \theta\) are reciprocals of each other. So, \(\text{csc} \theta = \frac{1}{\text{sin} \theta}\) and \(\text{sin} \theta = \frac{1}{\text{csc} \theta}\).
  • Similarly, other trigonometric pairs like cosine \(\text{cos} \theta\) and secant \(\text{sec} \theta\), and tangent \(\text{tan} \theta\) and cotangent \(\text{cot} \theta\), follow this reciprocal relationship.
To solve problems involving these functions, you should be comfortable flipping between a function and its reciprocal. In our exercise:
  • We began with \(\text{csc} \theta = -\frac{1}{2}\).
  • By applying the reciprocal relationship, we found \(\text{sin} \theta = -2\).
Mastering these relationships is fundamental in trigonometry.

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Most popular questions from this chapter

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \([0,2 \pi)\) and your work leads to \(\frac{1}{2} x=\frac{\pi}{16}, \frac{5 \pi}{12}, \frac{5 \pi}{8} .\) What are the corresponding values of \(x ?\)

Verify that each equation is an identity. $$(\cos 2 x-\sin 2 x)^{2}=1-\sin 4 x$$

(This discussion applies to functions of both angles and real numbers.) Consider the following. $$\begin{aligned}\cos (&\left.180^{\circ}-\theta\right) \\\&=\cos 180^{\circ} \cos \theta+\sin 180^{\circ} \sin \theta \\\&=(-1) \cos \theta+(0) \sin \theta \\\&=-\cos \theta\end{aligned}$$ \(\cos \left(180^{\circ}-\theta\right)=-\cos \theta\) is an example of a reduction formula, which is an identity that reduces a function of a quadrantal angle plus or minus \(\theta\) to a function of \(\theta\) alone. Another example of a reduction formula is \(\cos \left(270^{\circ}+\theta\right)=\sin \theta\) Here is an interesting method for quickly determining a reduction formula for a trigonometric function \(f\) of the form \(f(Q \pm \theta),\) where \(Q\) is a quadrantal angle. There are two cases to consider, and in each case, think of \(\boldsymbol{\theta}\) as a small positive angle in order to determine the quadrant in which \(Q \pm \theta\) will lie. Case 1 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(x\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that \(\operatorname{sign}, f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Case 2 Suppose that \(Q\) is a quadrantal angle whose terminal side lies along the \(y\) -axis. Determine the quadrant in which \(Q \pm \theta\) will lie for a small positive angle \(\theta .\) If the given function \(f\) is positive in that quadrant, use a \(+\) sign on the reduced form. If \(f\) is negative in that quadrant, use a - sign. The reduced form will have that sign, the cofunction of \(f\) as the function, and \(\theta\) as the argument. For example: CAN'T COPY THE GRAPH Use these ideas to write reduction formulas for each of the following. $$\tan \left(270^{\circ}-\theta\right)$$

Verify that each equation is an identity. $$\frac{\sin (s-t)}{\sin t}+\frac{\cos (s-t)}{\cos t}=\frac{\sin s}{\sin t \cos t}$$

Answer each question. Suppose you are solving a trigonometric equation for solutions over the interval \(\left[0^{\circ}, 360^{\circ}\right),\) and your work leads to \(3 \theta=180^{\circ}, 630^{\circ}, 720^{\circ}, 930^{\circ} .\) What are the cor- responding values of \(\theta ?\)
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