Evaluate trig functions Name ma Find the exact value of the trig function. Don't forget to rationalize any denominators. 1) sin 8 (4.-16)
Evaluate trig functions Name ma Find the exact value of the trig function. Don't forget to rationalize any denominators. 1) sin 8 (4.-16)
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
![**Evaluate Trig Functions**
**Instructions:** Find the exact value of the trigonometric function. Don’t forget to rationalize any denominators.
**Name:** [Student's Name Redacted]
1. \( \sin \theta \)
**Explanation:**
- To find the value of \( \sin \theta \), we use the given point \((4, -16)\) on the Cartesian plane in the illustration.
- There is a marked angle \( \theta \) in the diagram.
**Graph Description:**
- The graph depicts the Cartesian coordinate plane with the x-axis and y-axis neatly labeled.
- A point, labeled \((4, -16)\), is plotted in the fourth quadrant.
- From this point, a vector is drawn perpendicular to the x-axis (upward towards the x-axis) and another vector to the y-axis (towards the origin).
- A red arc suggests the angle \( \theta \) formed counterclockwise from the positive x-axis to the vector pointing toward the point \((4, -16)\).
**To Find:**
- We need to calculate \( \sin \theta \).
- The sine of an angle in the coordinate system can be calculated using:
\[
\sin \theta = \frac{y}{r}
\]
where \( y \) is the y-coordinate, and \( r \) is the distance from the origin to the point \((4, -16)\).
- First, compute \( r \):
\[
r = \sqrt{x^2 + y^2} = \sqrt{4^2 + (-16)^2} = \sqrt{16 + 256} = \sqrt{272} = 4\sqrt{17}
\]
- Next, plug the values into the sine function:
\[
\sin \theta = \frac{-16}{4\sqrt{17}} = \frac{-4}{\sqrt{17}}
\]
- Rationalize the denominator:
\[
\sin \theta = \frac{-4\sqrt{17}}{17}
\]
So, the exact value of \( \sin \theta \) is:
\[
\sin \theta = \frac{-4\sqrt{17}}{17}
\]
Administrator's Note: Ensure to clarify and perfect each step for students to easily follow](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24a6a4c4-f26c-45ad-82dd-c449c9817221%2Fbeb47dfa-b1f8-4601-b843-470ef30b42bc%2Forggbik_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluate Trig Functions**
**Instructions:** Find the exact value of the trigonometric function. Don’t forget to rationalize any denominators.
**Name:** [Student's Name Redacted]
1. \( \sin \theta \)
**Explanation:**
- To find the value of \( \sin \theta \), we use the given point \((4, -16)\) on the Cartesian plane in the illustration.
- There is a marked angle \( \theta \) in the diagram.
**Graph Description:**
- The graph depicts the Cartesian coordinate plane with the x-axis and y-axis neatly labeled.
- A point, labeled \((4, -16)\), is plotted in the fourth quadrant.
- From this point, a vector is drawn perpendicular to the x-axis (upward towards the x-axis) and another vector to the y-axis (towards the origin).
- A red arc suggests the angle \( \theta \) formed counterclockwise from the positive x-axis to the vector pointing toward the point \((4, -16)\).
**To Find:**
- We need to calculate \( \sin \theta \).
- The sine of an angle in the coordinate system can be calculated using:
\[
\sin \theta = \frac{y}{r}
\]
where \( y \) is the y-coordinate, and \( r \) is the distance from the origin to the point \((4, -16)\).
- First, compute \( r \):
\[
r = \sqrt{x^2 + y^2} = \sqrt{4^2 + (-16)^2} = \sqrt{16 + 256} = \sqrt{272} = 4\sqrt{17}
\]
- Next, plug the values into the sine function:
\[
\sin \theta = \frac{-16}{4\sqrt{17}} = \frac{-4}{\sqrt{17}}
\]
- Rationalize the denominator:
\[
\sin \theta = \frac{-4\sqrt{17}}{17}
\]
So, the exact value of \( \sin \theta \) is:
\[
\sin \theta = \frac{-4\sqrt{17}}{17}
\]
Administrator's Note: Ensure to clarify and perfect each step for students to easily follow
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