In Exercises 23–34, find the exact value of each of the remaining trigonometric functions of θ. tan θ = -2/3, sin θ > 0
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- 0. Review of College Algebra4h 45m
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- 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
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- 11. Graphing Complex Numbers1h 7m
2. Trigonometric Functions on Right Triangles
Trigonometric Functions on Right Triangles
Problem 36
Textbook Question
In Exercises 33–42, let sin t = a, cos t = b, and tan t = c. Write each expression in terms of a, b, and c. 3 cos(-t) - cos t
Verified step by step guidance1
Recall the even-odd properties of trigonometric functions: cosine is an even function, so \(\cos(-t) = \cos t\).
Substitute \(\cos(-t)\) with \(\cos t\) in the expression: \(3 \cos(-t) - \cos t\) becomes \(3 \cos t - \cos t\).
Combine like terms: \(3 \cos t - \cos t = (3 - 1) \cos t = 2 \cos t\).
Since \(\cos t = b\), rewrite the expression in terms of \(b\): \$2b$.
Thus, the expression \(3 \cos(-t) - \cos t\) simplifies to \$2b$.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Even and Odd Properties of Trigonometric Functions
Cosine is an even function, meaning cos(-t) = cos t. This property allows simplification of expressions involving negative angles by replacing cos(-t) with cos t, which is essential for rewriting the given expression in terms of a, b, and c.
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Even and Odd Identities
Basic Trigonometric Identities
Understanding the fundamental identities such as sin²t + cos²t = 1 and tan t = sin t / cos t helps relate the variables a, b, and c. These identities are crucial for expressing trigonometric expressions consistently in terms of the given variables.
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Fundamental Trigonometric Identities
Expression Simplification Using Given Variables
The problem requires rewriting trigonometric expressions using the variables a = sin t, b = cos t, and c = tan t. This involves substituting and simplifying expressions by replacing trigonometric functions with their corresponding variables to achieve the desired form.
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Solve Trig Equations Using Identity Substitutions
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