Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Lial12th EditionTrigonometryISBN: 9780136552161, 9780135440780, 9780136881117, 9780135440827, 9780135924891Non è quello che usi tu?Cambia libro di testo

Lial 12th Edition

Ch. 6 - Inverse Circular Functions and Trigonometric Equations

Problema 6.40

Capitolo 7, Problema 6.40

Solve each equation for all exact solutions, in degrees.
2√3 cos (θ/2) = -3

Guida verificata passo dopo passo

Start by isolating the cosine term in the equation: \(2\sqrt{3} \cos\left(\frac{\theta}{2}\right) = -3\). Divide both sides by \(2\sqrt{3}\) to get \(\cos\left(\frac{\theta}{2}\right) = \frac{-3}{2\sqrt{3}}\).

Simplify the right-hand side by rationalizing the denominator if needed. This will give you a simplified exact value for \(\cos\left(\frac{\theta}{2}\right)\).

Use the inverse cosine function to find the principal value(s) of \(\frac{\theta}{2}\): \(\frac{\theta}{2} = \cos^{-1}(\text{value})\). Remember that cosine is positive in the first and fourth quadrants and negative in the second and third quadrants, so consider all angles where cosine equals this value.

Write the general solutions for \(\frac{\theta}{2}\) using the cosine periodicity: \(\frac{\theta}{2} = 360^\circ k \pm \alpha\), where \(\alpha\) is the reference angle found from the inverse cosine and \(k\) is any integer.

Finally, multiply all parts of the equation by 2 to solve for \(\theta\): \(\theta = 2 \times (360^\circ k \pm \alpha)\). This gives all exact solutions for \(\theta\) in degrees.

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Solving Trigonometric Equations

This involves isolating the trigonometric function and finding all angle values that satisfy the equation within the given domain. For cosine equations, solutions often come in pairs due to the function's periodicity and symmetry.

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How to Solve Linear Trigonometric Equations

Inverse Trigonometric Functions

Inverse cosine (arccos) is used to find the principal angle whose cosine equals a given value. Since cosine is positive in the first and fourth quadrants and negative in the second and third, understanding how to find all solutions requires considering these quadrants.

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Introduction to Inverse Trig Functions

Angle Transformation and Domain Considerations

When the variable is inside a function argument like θ/2, solving requires adjusting the solution by multiplying or dividing accordingly. Additionally, since the problem asks for all solutions in degrees, one must consider the periodicity of cosine (360°) and adjust solutions to cover all possible angles.

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Domain and Range of Function Transformations

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