In Exercises 63–84, use an identity to solve each equation on the interval [0, 2𝝅). cos 2x + 5 cos x + 3 = 0
Table of contents
- 0. Review of College Algebra4h 43m
- 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
- OLD 1. Angles and the Trigonometric Functions Coming soon
- OLD 2. Trigonometric Functions graphs, Inverse Trigonometric Functions Coming soon
- OLD 3. Trigonometric Identities and Equations Coming soon
- OLD 4. Laws of Sines, Cosines and Vectors Coming soon
- OLD 5. Complex Numbers, Polar Coordinates and Parametric Equations Coming soon
- NEW (not used) 7. Laws of Sines, Cosines and Vectors Coming soon
- NEW (not used) 8. Vectors Coming soon
- NEW(not used) 9. Polar equations Coming soon
- NEW (not used) 11. Graphing Complex Numbers Coming soon
6. Trigonometric Identities and More Equations
Solving Trigonometric Equations Using Identities
Problem 6
Textbook Question
Find one solution for each equation. Assume all angles involved are acute angles. cos(3θ + 11°) = sin( 7θ + 40°) 5 10
Verified step by step guidance1
Recall the co-function identity in trigonometry: \(\cos A = \sin B\) implies that either \(A = B\) or \(A = 90^\circ - B\) (considering acute angles).
Set up the first equation by equating the angles directly: \(3\theta + 11^\circ = 7\theta + 40^\circ\).
Solve the equation from step 2 for \(\theta\) by isolating \(\theta\) on one side.
Set up the second equation using the complementary angle relationship: \(3\theta + 11^\circ = 90^\circ - (7\theta + 40^\circ)\).
Solve the equation from step 4 for \(\theta\) by simplifying and isolating \(\theta\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Relationship Between Sine and Cosine Functions
Sine and cosine functions are co-functions, meaning sin(α) = cos(90° - α). This identity allows us to rewrite equations involving sine and cosine in terms of each other, which is essential for solving equations like cos(3θ + 11°) = sin(7θ + 40°). Recognizing this relationship simplifies the problem by converting it into a single trigonometric function.
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Solving Linear Trigonometric Equations
Solving equations like cos(A) = cos(B) or sin(A) = sin(B) involves finding angles A and B that satisfy the equality, considering the periodicity and symmetry of trigonometric functions. For acute angles, solutions are restricted to values between 0° and 90°, which narrows down possible solutions and helps identify valid angle measures.
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Domain Restrictions for Acute Angles
Acute angles are angles between 0° and 90°. When solving trigonometric equations with this restriction, only solutions within this interval are valid. This constraint is crucial because trigonometric functions are periodic and can have multiple solutions, but the problem limits the solution set to acute angles, simplifying the solution process.
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