4. 4 cos? x- 4 sin x - 5 = 0 ,2
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°.
Related questions
Question
Find all solutions in the interval from 0 to 2pi
![**Equation Analysis**
The given trigonometric equation is:
\[ 4\cos^2 x - 4\sin x - 5 = 0 \]
**Explanation**
This equation involves trigonometric functions: cosine (cos) and sine (sin). It combines both squared and linear terms of the trigonometric functions.
**Key Points for Solution:**
1. **Identify Trigonometric Identities:**
- Recall that \(\cos^2 x + \sin^2 x = 1\). This identity can help in transforming the equation entirely in terms of one trig function.
2. **Convert and Simplify:**
- You may substitute \(\cos^2 x\) as \(1 - \sin^2 x\) to solve in terms of \(\sin x\), or vice versa, depending on the approach.
3. **Form a Quadratic Equation:**
- Converting the entire equation in terms of one trigonometric function often leads to a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula.
**Purpose**
Understanding how to manipulate and solve trigonometric equations like this one is crucial for higher mathematics, particularly in calculus and analytical geometry. It enhances problem-solving skills applicable in various scientific fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F838d6200-dd8b-4de4-b6ca-eaa9f9f43bab%2Fa18ba13f-f2c4-436d-9e96-0466fbe20e86%2Fmuay9oq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Equation Analysis**
The given trigonometric equation is:
\[ 4\cos^2 x - 4\sin x - 5 = 0 \]
**Explanation**
This equation involves trigonometric functions: cosine (cos) and sine (sin). It combines both squared and linear terms of the trigonometric functions.
**Key Points for Solution:**
1. **Identify Trigonometric Identities:**
- Recall that \(\cos^2 x + \sin^2 x = 1\). This identity can help in transforming the equation entirely in terms of one trig function.
2. **Convert and Simplify:**
- You may substitute \(\cos^2 x\) as \(1 - \sin^2 x\) to solve in terms of \(\sin x\), or vice versa, depending on the approach.
3. **Form a Quadratic Equation:**
- Converting the entire equation in terms of one trigonometric function often leads to a quadratic equation, which can be solved by factoring, completing the square, or using the quadratic formula.
**Purpose**
Understanding how to manipulate and solve trigonometric equations like this one is crucial for higher mathematics, particularly in calculus and analytical geometry. It enhances problem-solving skills applicable in various scientific fields.
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