Math 170 Exam 3 Practice 1 Verify The Identity Sin Θ Tan Θ C
Math 170 Exam 3 Practice1 Verify The Identity Sin Θ Tan Θ Cos Θ
Math 170 Exam 3 Practice 1. Verify the identity: sin θ tan θ + cos θ = sec θ. 2. Verify the identity: cos(α + β) cos α sin β = cot β − tan α. 3. Verify the identity: 2 sin θ csc(2θ) = sec θ. 4. Verify the identity: 1 − sin θ cos θ = cos θ / (1 + sin θ). 5. Write cos⁴ θ that does not involve powers of sine or cosine greater than 1. 6. Verify the identity: cos θ + cos(3θ) = 2 cos(2θ) cos θ. 7. Solve the equation on the interval 0 ≤ θ
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The set of exercises and problems provided for verification of trigonometric identities and solving triangle-related equations are fundamental to mastering trigonometry and its applications. These tasks combine algebraic manipulation, application of identities, and problem-solving in a practical context, making them highly relevant for students preparing for exams or applying trigonometry in real-world scenarios.
Verification of Identities
The initial exercises focus on verifying various trigonometric identities, a core skill for understanding the behavior and interrelationships of different functions. For example, verifying that \(\sin \theta \tan \theta + \cos \theta = \sec \theta\) requires expressing \(\tan \theta\) as \(\frac{\sin \theta}{\cos \theta}\) and simplifying to reach the right-hand side. Similarly, confirming that \(\cos (α + β) \cos α \sin β = \cot β - \tan α\) involves applying angle addition formulas and algebraic manipulations. These exercises promote a deeper understanding of how identities interconnect and reinforce algebraic skills necessary for more complex derivations.
Other identities such as \(2 \sin θ \csc(2θ) = \sec θ\) leverage reciprocal identities and double-angle formulas, key concepts in simplifying and transforming trigonometric expressions. Writing \(\cos^4 θ\) without powers greater than 1 in sine or cosine involves using power-reduction formulas, which convert higher powers into linear combinations of basic functions, important for integration and solving equations.
Solving Trigonometric Equations
The subsequent problems involve solving trigonometric equations within specified intervals, a critical skill for analytical and applied mathematics. For example, solving \(\cos(2θ) + \cos(4θ) = 0\) over \(0 \leq θ
Similarly, quadratic trigonometric equations such as \(\cos(2θ) + 5 \cos θ + 3 = 0\) require substitution techniques (like setting \(x = \cos θ\)) to reduce the problem to a quadratic form, then solving for the roots within the physical bounds of cosine. These exercises exemplify the importance of algebraic methods in trigonometry and understanding the periodic nature of functions.
Applying Law of Sines and Cosines
The problems involving solving triangles given different sets of data utilize the Law of Sines and Law of Cosines to find unknown sides and angles. For example, given side \(c = 10\), \(A = 40^\circ\), and right angle \(C = 90^\circ\), we can employ trigonometric ratios and the Pythagorean theorem to determine other sides and angles accurately, round to two decimal places for precision.
In scenarios where two sides and an angle are known, such as \(a=3\), \(b=2\), \(A=40^\circ\), the Law of Sines allows calculating the remaining angles and sides. These problems reinforce understanding of triangle properties, emphasizing the importance of choosing the right method based on what data is given.
Area and Oscillation Models
Questions about calculating the area of a triangle with given sides and included angles involve employing formulas like \( \frac{1}{2}ab \sin C \). For example, with \(a=8\), \(b=6\), and \(C=30^\circ\), the area is obtained by plugging into the formula, providing insights into geometric and trigonometric integration.
The harmonic motion and pendulum problems introduce the application of trigonometry in physics. The maximum displacement in simple harmonic motion, identified by the amplitude, is straightforward from the given sinusoidal equation. The period and frequency are derived from the mathematical form of the function \(d = -3 \sin \frac{1}{2} t\), illustrating how trigonometric functions model oscillations.
The pendulum problem involves understanding damping effects and differential equations to model real-world damping processes. Using provided data, one can develop exponential decay models combined with sinusoidal functions to approximate oscillation behavior, which is vital for mechanical and engineering applications.
Conclusion
Overall, these exercises encapsulate key principles of trigonometry, including identities, equation solving, triangle analysis, and application to physics. Mastery of these topics not only prepares students for examinations but also enhances problem-solving skills applicable in scientific and engineering contexts.
References
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