Textbook Question
Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.
3 tan 3x = √3
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Linear Trigonometric Equations
Problem 6.3.37
Textbook Question
Solve each equation (x in radians and θ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. Write answers using the least possible nonnegative angle measures.
cos θ/2 = 1
Verified step by step guidance1
Identify the given equation: \(\cos \frac{\theta}{2} = 1\). We want to find all values of \(\theta\) (in degrees) that satisfy this equation.
Recall that \(\cos x = 1\) when \(x = 2k\pi\) for any integer \(k\), where \(x\) is in radians. Since the argument here is \(\frac{\theta}{2}\), set \(\frac{\theta}{2} = 2k\pi\).
Solve for \(\theta\) by multiplying both sides by 2: \(\theta = 4k\pi\). This gives the general solution in radians.
Convert the general solution to degrees by using the conversion \(180^\circ = \pi\) radians: \(\theta = 4k\pi \times \frac{180^\circ}{\pi} = 720k^\circ\).
Write the least possible nonnegative angle measures by choosing integer values of \(k\) starting from 0, which gives \(\theta = 0^\circ, 720^\circ, 1440^\circ, \ldots\). Since angles are often expressed within \(0^\circ\) to \(360^\circ\), the principal solution is \(\theta = 0^\circ\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Basic Trigonometric Equations
Solving trigonometric equations involves finding all angle values that satisfy the given equation. For cosine equations like cos(θ/2) = 1, you identify angles where the cosine function equals 1, then solve for θ considering the function's periodicity.
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Understanding Angle Measures and Units
Angles can be measured in degrees or radians, and converting between these units is essential. The problem requires solutions in both radians and degrees, so knowing that 180° equals π radians helps in converting and expressing answers correctly.
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Using the Unit Circle and Periodicity of Cosine
The unit circle shows cosine values for angles around a circle, with cos(θ) = 1 at θ = 0° (or 0 radians) and repeating every 360° (2π radians). Recognizing this periodicity allows finding all solutions by adding multiples of the period to the principal solution.
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