If t=sin^-1(-2 / √(5)), find the exact values of cost, tant cott, sect, and csct.

Name: Problem 57, Chapter 1: Calculus, Early Transcendentals
Uploaded: 2021-08-12T17:18:00-08:00
Duration: 1 min 9 s
Channel: Carson Merrill
Description: If t=sin^-1(-2 / √(5)), find the exact values of cost, tant cott, sect, and csct.

step 1 For the following problem, we see that if t equals the inverse sign of negative 2 over root 5, t

If t=sin^-1(-2 / √(5)), find the exact values of cost, tant...

Key Concepts

Trigonometric Function Definitions

Knowing the definitions of the basic trigonometric functions—sine, cosine, tangent—and their reciprocal functions—cosecant, secant, cotangent—is crucial for solving problems that require conversion between these functions. This understanding provides a framework for computing unknown ratios from known ones.

Inverse Trigonometric Functions

Inverse trigonometric functions allow one to determine an angle given a trigonometric ratio. Understanding these functions involves knowing their principal value ranges, which ensures that each value of the inverse function corresponds to a unique angle. This is fundamental when transitioning from a ratio value to an angle in problems involving sin?¹, cos?¹, etc.

Pythagorean Identity

The Pythagorean identity, expressed as sin²? + cos²? = 1, is essential in linking the sine and cosine of the same angle. It provides a method to derive one trigonometric function given another, making it a critical tool in problems where only one ratio is provided and the others need to be computed.

Right Triangle Relationships

Using the geometry of right triangles helps visualize and compute trigonometric ratios. When one ratio is known, a right triangle can be constructed with proportional sides, allowing for the determination of all other trigonometric functions by applying basic geometric principles and the Pythagorean theorem.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions, such as secant, cosecant, and cotangent, are defined as the reciprocals of cosine, sine, and tangent respectively. They are important in trigonometry because certain problems are more easily solved by shifting between a function and its reciprocal, depending on the given information.

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