The original question is "find all radian solutions to sinx + cosx = 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°.
icon
Related questions
Topic Video
Question
100%

Please see attached image.  I see what bartleby did and that makes sense but I did t a different way and got a different answer.  The original question is "find all radian solutions to sinx + cosx = 0"

Please show me where I went wrong

---

**Problem:**

\[ 43. \, ( \sin x + \cos x )^2 = (\cos^2 x - 1 - \sin^2 x) \]

**Solution Steps:**

1. Expand the left-hand side:
   \[
   \sin^2 x + 2 \sin x \cos x + \cos^2 x = 0
   \]

2. Recognize and simplify:
   \[
   \sin^2 x + \cos^2 x = 1
   \]
   \[
   1 + 2 \sin x \cos x = 0
   \]
   \[
   1 + \sin 2x = 0
   \]

3. Solve for \(\sin 2x\):
   \[
   \sin 2x = -1
   \]

4. Solve equation:
   \[
   2x = \frac{3\pi}{2} + 2k\pi
   \]
   \[
   x = \frac{3\pi}{4} + k\pi
   \]

---

**Explanation:**

The student states:

"I'm using \(\sin\), not \(\tan\), and got \(2k\pi\) instead of \(k\pi\) like Bartleby, and \(\sin\) is only \(-1\) at one angle, \(\frac{3\pi}{2}\). Please show me where I went wrong in the above calculation."

This indicates the student's approach might have misunderstood the periodicity or the angles at which these functions take specific values. Verification of solutions against standard trigonometric identities or using a calculator for specific angles could clarify any discrepancies.

---
Transcribed Image Text:--- **Problem:** \[ 43. \, ( \sin x + \cos x )^2 = (\cos^2 x - 1 - \sin^2 x) \] **Solution Steps:** 1. Expand the left-hand side: \[ \sin^2 x + 2 \sin x \cos x + \cos^2 x = 0 \] 2. Recognize and simplify: \[ \sin^2 x + \cos^2 x = 1 \] \[ 1 + 2 \sin x \cos x = 0 \] \[ 1 + \sin 2x = 0 \] 3. Solve for \(\sin 2x\): \[ \sin 2x = -1 \] 4. Solve equation: \[ 2x = \frac{3\pi}{2} + 2k\pi \] \[ x = \frac{3\pi}{4} + k\pi \] --- **Explanation:** The student states: "I'm using \(\sin\), not \(\tan\), and got \(2k\pi\) instead of \(k\pi\) like Bartleby, and \(\sin\) is only \(-1\) at one angle, \(\frac{3\pi}{2}\). Please show me where I went wrong in the above calculation." This indicates the student's approach might have misunderstood the periodicity or the angles at which these functions take specific values. Verification of solutions against standard trigonometric identities or using a calculator for specific angles could clarify any discrepancies. ---
Expert Solution
steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Fundamentals of Trigonometric Identities
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, trigonometry and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Trigonometry (11th Edition)
Trigonometry (11th Edition)
Trigonometry
ISBN:
9780134217437
Author:
Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:
PEARSON
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781305652224
Author:
Charles P. McKeague, Mark D. Turner
Publisher:
Cengage Learning
Algebra and Trigonometry
Algebra and Trigonometry
Trigonometry
ISBN:
9781938168376
Author:
Jay Abramson
Publisher:
OpenStax
Trigonometry (MindTap Course List)
Trigonometry (MindTap Course List)
Trigonometry
ISBN:
9781337278461
Author:
Ron Larson
Publisher:
Cengage Learning