3. 2 sin? x - cos x -1=0
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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Find all solutions in the interval from 0 < theta
![### Equation Explanation
The following is a trigonometric equation that can be explored in the context of solving or simplifying trigonometric identities:
\[ 2\sin^2 x - \cos x - 1 = 0 \]
#### Components:
- **\(2\sin^2 x\):** This term represents two times the square of the sine function. In trigonometric identities, \(\sin^2 x\) can often be replaced using the Pythagorean identity \(\sin^2 x = 1 - \cos^2 x\).
- **\(- \cos x\):** This is the cosine function with a negative sign, which plays a critical role in finding the values of \(x\) that satisfy the equation.
- **\(-1:\)** A constant term that shifts the entire function downwards.
### Analysis:
This equation is a quadratic in form and can be solved using substitutions or known identities to factor or use the quadratic formula. The goal is often to find the values of \(x\) that satisfy the equation in a given interval, typically \([0, 2\pi]\) or \([0^\circ, 360^\circ]\).
### Methods of Solution:
1. **Substitution:** Use a trigonometric identity to transform the equation, such as replacing \(\sin^2 x\) with \(1 - \cos^2 x\), transforming it into a quadratic equation in terms of \(\cos x\).
2. **Quadratic Formula:** Once transformed, solve for \(\cos x\) using the quadratic formula or by factoring.
3. **Graphical Interpretation:** Visualizing the functions \(2\sin^2 x\) and \(\cos x + 1\) graphically can provide intersection points to signify solutions.
This equation and its solutions are commonly encountered in calculus and algebra courses, emphasizing practice with identities and algebraic manipulation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F838d6200-dd8b-4de4-b6ca-eaa9f9f43bab%2Fa3d0a880-1fc8-4e20-a84f-8ca6bd0dc338%2Fohz32x_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Equation Explanation
The following is a trigonometric equation that can be explored in the context of solving or simplifying trigonometric identities:
\[ 2\sin^2 x - \cos x - 1 = 0 \]
#### Components:
- **\(2\sin^2 x\):** This term represents two times the square of the sine function. In trigonometric identities, \(\sin^2 x\) can often be replaced using the Pythagorean identity \(\sin^2 x = 1 - \cos^2 x\).
- **\(- \cos x\):** This is the cosine function with a negative sign, which plays a critical role in finding the values of \(x\) that satisfy the equation.
- **\(-1:\)** A constant term that shifts the entire function downwards.
### Analysis:
This equation is a quadratic in form and can be solved using substitutions or known identities to factor or use the quadratic formula. The goal is often to find the values of \(x\) that satisfy the equation in a given interval, typically \([0, 2\pi]\) or \([0^\circ, 360^\circ]\).
### Methods of Solution:
1. **Substitution:** Use a trigonometric identity to transform the equation, such as replacing \(\sin^2 x\) with \(1 - \cos^2 x\), transforming it into a quadratic equation in terms of \(\cos x\).
2. **Quadratic Formula:** Once transformed, solve for \(\cos x\) using the quadratic formula or by factoring.
3. **Graphical Interpretation:** Visualizing the functions \(2\sin^2 x\) and \(\cos x + 1\) graphically can provide intersection points to signify solutions.
This equation and its solutions are commonly encountered in calculus and algebra courses, emphasizing practice with identities and algebraic manipulation.
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