In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cos² x - 3 = 0
Table of contents
- 0. Review of College Algebra4h 45m
- 1. Measuring Angles40m
- 2. Trigonometric Functions on Right Triangles2h 5m
- 3. Unit Circle1h 19m
- 4. Graphing Trigonometric Functions1h 19m
- 5. Inverse Trigonometric Functions and Basic Trigonometric Equations1h 41m
- 6. Trigonometric Identities and More Equations2h 34m
- 7. Non-Right Triangles1h 38m
- 8. Vectors2h 25m
- 9. Polar Equations2h 5m
- 10. Parametric Equations1h 6m
- 11. Graphing Complex Numbers1h 7m
5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Linear Trigonometric Equations
Problem 6.2.49
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.
3 csc x― 2√3 = 0
Verified step by step guidance1
Start by isolating the cosecant function in the equation: 3 \(\csc\) x - 2\(\sqrt{3}\) = 0. Add 2\(\sqrt{3}\) to both sides to get 3 \(\csc\) x = 2\(\sqrt{3}\).
Divide both sides of the equation by 3 to solve for \(\csc\) x: \(\csc\) x = \(\frac{2\sqrt{3}\)}{3}.
Recall that \(\csc\) x is the reciprocal of \(\sin\) x, so rewrite the equation as \(\sin\) x = \(\frac{3}{2\sqrt{3}\)}. Simplify this expression to find the exact value of \(\sin\) x.
Determine all angles x (in radians) where \(\sin\) x equals the simplified value, considering the unit circle and the range of sine function. Use the least possible nonnegative angle measures and include all solutions within one full rotation (0 to 2\(\pi\)).
For solutions in degrees (\(\theta\)), convert the radian solutions to degrees by multiplying by \(\frac{180}{\pi}\), then round approximate answers to the nearest tenth of a degree.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Reciprocal Trigonometric Functions
The cosecant function (csc x) is the reciprocal of the sine function, defined as csc x = 1/sin x. Understanding this relationship allows you to rewrite equations involving csc x in terms of sin x, which is often easier to solve.
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Introduction to Trigonometric Functions
Solving Trigonometric Equations
Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the specified domain. Since sine and cosecant are periodic, solutions repeat every 2π radians or 360 degrees, so all solutions must be expressed using general solution formulas.
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How to Solve Linear Trigonometric Equations
Angle Measurement and Conversion
Angles can be measured in radians or degrees, and problems may require answers in either unit. Knowing how to convert between radians and degrees and how to express angles within the least nonnegative measure (0 to 2π or 0° to 360°) is essential for providing correct and standardized solutions.
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