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Chapter 6: Problem 59

Find the solutions of the equation that are in the interval \([0,2 \pi)\). $$1-\sin t=\sqrt{3} \cos t$$

Short Answer

Expert verified
The solutions are \(t = \frac{2\pi}{3}\) and \(t = 2\pi\).

Step by step solution

01

Rearrange the Equation

Start by rearranging the given equation, \[1 - \sin t = \sqrt{3} \cos t\]so that it is set equal to zero:\[1 - \sin t - \sqrt{3} \cos t = 0.\]
02

Use Trigonometric Identities

We know that if \( \sin \theta = \frac{a}{\sqrt{a^2 + b^2}} \) and \( \cos \theta = \frac{b}{\sqrt{a^2 + b^2}} \), then the expression \( a \sin t + b \cos t \) can be simplified using trigonometric identities. Rearranging the equation gives:\[1 - \sin t - \sqrt{3} \cos t = 0\]This can be expressed as \[\left(\frac{1}{2} \right) - \left(\sin t \cdot \frac{1}{2} + \cos t \cdot \frac{\sqrt{3}}{2}\right) = 0.\]
03

Recognize and Apply a Known Angle

Recognize that \( \frac{1}{2} \) is \( \cos 60^\circ \) or \( \cos \frac{\pi}{3} \), and \( \frac{\sqrt{3}}{2} \) is \( \sin 60^\circ \) or \( \sin \frac{\pi}{3} \). This suggests the identity:\[ \sin t \cdot \frac{1}{2} + \cos t \cdot \frac{\sqrt{3}}{2} = \cos(t - \frac{\pi}{3}), \]which means \[\cos(t - \frac{\pi}{3}) = \frac{1}{2}. \]
04

Solve the Trigonometric Equation

Solving for \( t \) when \( \cos(t - \frac{\pi}{3}) = \frac{1}{2} \):The cosine function equals \( \frac{1}{2} \) at standard angles \( t = \frac{\pi}{3} \) and \( t = \frac{5\pi}{3} \).Thus,\[ t - \frac{\pi}{3} = \frac{\pi}{3} + 2k\pi, \]\[ t - \frac{\pi}{3} = \frac{5\pi}{3} + 2k\pi, \]for integer \( k \).
05

Find Solutions in the Given Interval

To find solutions in \([0, 2\pi)\), solve:\[ t = (\frac{\pi}{3} + \frac{\pi}{3}) = \frac{2\pi}{3}, \]\[ t = (\frac{5\pi}{3} + \frac{\pi}{3}) = 2\pi,\]since these t-values are within the interval \([0, 2\pi)\). The solutions are:\[ t = \frac{2\pi}{3} \text{ or } t = 2\pi.\]
06

Verify Solutions

Double-check these solutions by substituting back into the original equation. For \( t = \frac{2\pi}{3} \) and \( t = 2\pi \), you get:\(1 - \sin \frac{2\pi}{3} = \sqrt{3}\cos\frac{2\pi}{3}\) holds and \(1 - \sin 2\pi = \sqrt{3} \cos 2\pi\) holds, confirming these are valid solutions.

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Key Concepts

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

Step-by-Step Solutions
Solving trigonometric equations might seem challenging at first, but with systematic steps, these problems become manageable. Let's break down the process for tackling such equations.
Start by rearranging the equation in a way that sets it equal to zero. This facilitates the application of trigonometric identities or methods.
In our example, the original equation \(1 - \sin t = \sqrt{3} \cos t\) was rearranged to \(1 - \sin t - \sqrt{3} \cos t = 0\), paving the way for further transformation.
Once the equation is in a friendly format, applying known identities or recognizing patterns becomes more intuitive. Remember: a neat arrangement unlocks the equation's potential for solutions.
By carefully following each step, you ensure that no detail is overlooked and your solution remains error-free.
Trigonometric Identities
Trigonometric identities are powerful tools in solving equations because they allow us to simplify complex expressions. Two commonly used identities are the Pythagorean identities and angle addition formulas.
In the given equation, recognizing parts of the equation as components of trigonometric identities transforms the equation into a more manageable form.
For instance, using the generic transformation, \(a \sin t + b \cos t= R \sin(t + \theta) \) (where \( R = \sqrt{a^2 + b^2}\)), the equation can be rewritten with recognizable sinusoidal patterns. By aligning the equation into a known trigonometric identity, you transition to finding explicit values for angles, greatly simplifying the process of solving it.
Angle Recognition
Recognizing angles is crucial when solving trigonometric equations because it links trigonometric values to specific angles that are well-known. This skill hinges on understanding which angles within the unit circle correspond to fundamental sine and cosine values.
In this exercise, \( \frac{1}{2} \) was identified as \( \cos 60^\circ \) or \( \cos \frac{\pi}{3} \), and \( \frac{\sqrt{3}}{2} \) as \( \sin 60^\circ \) or \( \sin \frac{\pi}{3} \). Such recognition allows us to explore sine and cosine functions as they relate to these standard angles
By associating these coefficients with known angles, solving the trigonometric equation becomes significantly straightforward, leading to an efficient resolution.
Standard Angles
Standard angles are those regularly encountered in trigonometry, such as \(0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), and their corresponding degree measures.These angles have sine and cosine values that are frequently used for solving equations.
In our problem, the cosine function equaled \( \frac{1}{2} \) at the standard angles \(t = \frac{\pi}{3}\) and \(t = \frac{5\pi}{3}\). Recognizing these values quickly is beneficial for solving equations accurately within a given interval like \([0, 2\pi)\).
By leveraging the simplicity of standard angles, we can find solutions efficiently and confirm they lie within the correct interval, thus ensuring a thorough and precise solution.

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