secccos" (3))
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
How do I find the exact value of the equations ?
![The image contains the mathematical expression:
\[ \sec(\cos^{-1}(\frac{1}{2})) \]
Explanation:
- \(\cos^{-1}(\frac{1}{2})\) represents the inverse cosine (arc cosine) function, which finds the angle whose cosine is \(\frac{1}{2}\).
- \(\sec(\theta)\) is the secant function, defined as \(\frac{1}{\cos(\theta)}\).
In a right triangle example, this could involve finding the angle, then determining the secant based on the cosine value of \(\frac{1}{2}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5445541-3bd8-48a0-bfe5-5b2c37f58de6%2F02afd67a-897b-45ae-9eca-8b00e3b6b803%2F940l1ef_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains the mathematical expression:
\[ \sec(\cos^{-1}(\frac{1}{2})) \]
Explanation:
- \(\cos^{-1}(\frac{1}{2})\) represents the inverse cosine (arc cosine) function, which finds the angle whose cosine is \(\frac{1}{2}\).
- \(\sec(\theta)\) is the secant function, defined as \(\frac{1}{\cos(\theta)}\).
In a right triangle example, this could involve finding the angle, then determining the secant based on the cosine value of \(\frac{1}{2}\).
![The image contains the mathematical expression:
\[ \sin\left[\tan^{-1}(-1)\right] \]
This expression involves two functions:
1. **Inverse Tangent Function (\(\tan^{-1}\))**: This function computes the angle whose tangent is \(-1\).
2. **Sine Function (\(\sin\))**: This function takes the angle found from the inverse tangent and computes its sine.
To evaluate this expression, you determine an angle \(\theta\) such that \(\tan(\theta) = -1\) and find \(\sin(\theta)\).
### Explanation of Concepts:
- **Inverse Trigonometric Functions**: These functions are used to derive an angle from a specific trigonometric ratio. In this case, \(\tan^{-1}(-1)\) refers to the angle \(\theta\) for which the tangent value is \(-1\).
- **Sine Function**: This is a basic trigonometric function that provides the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
### Solving the Expression:
1. Identify the angle \(\theta\) that satisfies \(\tan(\theta) = -1\). Typically, this occurs at \(\theta = -\frac{\pi}{4}\) or \(\theta = \frac{3\pi}{4}\) in radians, depending on the context.
2. Calculate \(\sin(\theta)\) for the angle found above.
The specific interval and context determine the correct angle for the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5445541-3bd8-48a0-bfe5-5b2c37f58de6%2F02afd67a-897b-45ae-9eca-8b00e3b6b803%2Fjyz86ps_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains the mathematical expression:
\[ \sin\left[\tan^{-1}(-1)\right] \]
This expression involves two functions:
1. **Inverse Tangent Function (\(\tan^{-1}\))**: This function computes the angle whose tangent is \(-1\).
2. **Sine Function (\(\sin\))**: This function takes the angle found from the inverse tangent and computes its sine.
To evaluate this expression, you determine an angle \(\theta\) such that \(\tan(\theta) = -1\) and find \(\sin(\theta)\).
### Explanation of Concepts:
- **Inverse Trigonometric Functions**: These functions are used to derive an angle from a specific trigonometric ratio. In this case, \(\tan^{-1}(-1)\) refers to the angle \(\theta\) for which the tangent value is \(-1\).
- **Sine Function**: This is a basic trigonometric function that provides the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle.
### Solving the Expression:
1. Identify the angle \(\theta\) that satisfies \(\tan(\theta) = -1\). Typically, this occurs at \(\theta = -\frac{\pi}{4}\) or \(\theta = \frac{3\pi}{4}\) in radians, depending on the context.
2. Calculate \(\sin(\theta)\) for the angle found above.
The specific interval and context determine the correct angle for the solution.
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