cos(30°))² – (sin(30°) ||
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
![On this educational page, we explore the trigonometric expression:
\[
(\cos(30^\circ))^2 - (\sin(30^\circ))^2 = \, \text{[Input Box]}
\]
This expression requires the application of trigonometric identities to simplify. The angle, 30 degrees, is part of common angles in trigonometry where cosine and sine values can be memorized or calculated.
**Explanation of Terms:**
- \(\cos(30^\circ)\) refers to the cosine of 30 degrees, a standard angle in trigonometry. The cosine of 30 degrees is \(\frac{\sqrt{3}}{2}\).
- \(\sin(30^\circ)\) refers to the sine of 30 degrees. The sine of 30 degrees is \(\frac{1}{2}\).
The expression is squared, meaning each trigonometric function is multiplied by itself. The goal is to determine the difference between these squared values.
By substituting the known values:
\[
(\cos(30^\circ))^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}
\]
\[
(\sin(30^\circ))^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]
\[
(\cos(30^\circ))^2 - (\sin(30^\circ))^2 = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
\]
Thus, the expression simplifies to \(\frac{1}{2}\).
This is an excellent opportunity to practice the use of trigonometric identities and the Pythagorean identity, reinforcing the foundational concepts of trigonometry.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F871d5c06-5fd6-4f49-949d-34968a02c382%2F45440d62-51cf-4ab2-978c-a7291ea64e9e%2Fa0zw62_processed.png&w=3840&q=75)
Transcribed Image Text:On this educational page, we explore the trigonometric expression:
\[
(\cos(30^\circ))^2 - (\sin(30^\circ))^2 = \, \text{[Input Box]}
\]
This expression requires the application of trigonometric identities to simplify. The angle, 30 degrees, is part of common angles in trigonometry where cosine and sine values can be memorized or calculated.
**Explanation of Terms:**
- \(\cos(30^\circ)\) refers to the cosine of 30 degrees, a standard angle in trigonometry. The cosine of 30 degrees is \(\frac{\sqrt{3}}{2}\).
- \(\sin(30^\circ)\) refers to the sine of 30 degrees. The sine of 30 degrees is \(\frac{1}{2}\).
The expression is squared, meaning each trigonometric function is multiplied by itself. The goal is to determine the difference between these squared values.
By substituting the known values:
\[
(\cos(30^\circ))^2 = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4}
\]
\[
(\sin(30^\circ))^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]
\[
(\cos(30^\circ))^2 - (\sin(30^\circ))^2 = \frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}
\]
Thus, the expression simplifies to \(\frac{1}{2}\).
This is an excellent opportunity to practice the use of trigonometric identities and the Pythagorean identity, reinforcing the foundational concepts of trigonometry.
Transcribed Image Text:The text in the image reads:
"Without using a calculator, evaluate"
This directive is typically used in educational contexts to challenge students to solve problems manually, encouraging a deeper understanding of arithmetic processes and improving their mental math skills.
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