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Chapter 8: Problem 72

Solve each equation on the interval \(0 \leq \theta<2 \pi\) \(\cos \theta-\sin (-\theta)=0\)

Short Answer

Expert verified
The solutions are \(\theta = \frac{3\pi}{4}\) and \(\theta = \frac{7\pi}{4}\).

Step by step solution

01

Simplify the Equation Using Trigonometric Identities

First, simplify \(\sin(-\theta)\). Recall that \(\sin(-\theta) = -\sin(\theta)\). Hence, the equation becomes: \(\cos(\theta) - (-\sin(\theta)) = 0\), which simplifies to \(\cos(\theta) + \sin(\theta) = 0\).
02

Isolate One Trigonometric Function

Rearrange the equation to isolate \(\cos(\theta)\): \(\cos(\theta) = -\sin(\theta)\).
03

Divide Both Sides by \(\cos(\theta)\)

Divide both sides of the equation by \(\cos(\theta)\) to obtain: \(1 = -\tan(\theta)\), or equivalently, \(\tan(\theta) = -1\).
04

Find Solutions for \(\tan(\theta) = -1\)

Determine the angles \(\theta\) on the interval \(0 \leq \theta < 2 \pi\) for which \(\tan(\theta) = -1\). The tangent function is negative in the second and fourth quadrants. Hence, the solutions are at: \(\theta = \frac{3\pi}{4}\) and \(\theta = \frac{7\pi}{4}\).

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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 equations involving trigonometric functions that are true for all values of the variables involved. These identities are very useful in simplifying and solving equations. For instance, in our problem, we use the identity for sine: \(\sin(-\theta) = -\sin(\theta)\). This allows us to transform the original equation \(\cos(\theta) - \-\sin(-\theta) = 0\) into a more manageable form: \(\cos(\theta) + \sin(\theta) = 0\). This step is crucial as it simplifies the problem and makes it easier to solve.
tangent function
The tangent function, denoted as \(\rex\tan(\theta)\), is one of the primary trigonometric functions. In terms of sine and cosine, it is defined as: \(\rex\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\).
In our problem, we use this relationship to further simplify the equation. By isolating \(\cos(\theta)\) and dividing both sides by \(\cos(\theta)\), we obtain: \(1 = -\tan(\theta)\), which can be rewritten as: \(\tan(\theta) = -1\).
Understanding the properties of the tangent function is key to solving trigonometric equations. For example, knowing that the tangent function is periodic with period \(\pi\) helps us identify solutions within a specific interval.
solving trigonometric equations
Solving trigonometric equations involves isolating the trigonometric function and finding the angles that satisfy the equation. Here is how we approach it step-by-step:
  • First, simplify the equation using trigonometric identities.
  • Next, isolate one trigonometric function to reduce the complexity.
  • Then, express the equation in terms of a single function.
  • Finally, find the solutions within the given interval.
For our problem, after simplifying and isolating, we got: \(\tan(\theta) = -1\).
The next step is determining \(\theta\) values within \(0 \leq \theta < 2 \pi\) for which this is true. The tangent function is negative in the second and fourth quadrants, thus:
  • \(\theta = \frac{3\pi}{4} \) in the second quadrant
  • \(\theta = \frac{7\pi}{4} \) in the fourth quadrant.
These steps illustrate the logical progression needed to solve trigonometric equations effectively.

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