Table of contents
- 0. Review of College Algebra4h 43m
- 1. Measuring Angles39m
- 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
Inverse Sine, Cosine, & Tangent
3:21 minutes
Problem 80
Textbook Question
Textbook QuestionIn Exercises 63–82, use a sketch to find the exact value of each expression. cot (csc⁻¹ 8)
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cosecant and Its Inverse
The cosecant function, denoted as csc, is the reciprocal of the sine function. The inverse cosecant function, csc⁻¹, gives the angle whose cosecant is a given value. In this case, csc⁻¹(8) represents the angle θ such that csc(θ) = 8, which implies sin(θ) = 1/8.
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Cotangent Function
The cotangent function, denoted as cot, is the reciprocal of the tangent function. It can be expressed as cot(θ) = cos(θ)/sin(θ). To find cot(csc⁻¹(8)), we need to determine the cosine and sine values of the angle θ derived from csc⁻¹(8) and then compute their ratio.
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Right Triangle Relationships
In trigonometry, right triangles are fundamental for understanding the relationships between angles and side lengths. Given csc(θ) = 8, we can visualize a right triangle where the hypotenuse is 8 and the opposite side is 1 (since sin(θ) = 1/8). Using the Pythagorean theorem, we can find the adjacent side, which is essential for calculating cot(θ).
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