Answered

At Westonci.ca, we make it easy for you to get the answers you need from a community of knowledgeable individuals. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.

Select all the correct answers.

If the measure of angle [tex][tex]$\theta$[/tex][/tex] is [tex][tex]$\frac{\pi}{4}$[/tex][/tex], which statements are true?

A. [tex][tex]$\tan (\theta)=1$[/tex][/tex]
B. The measure of the reference angle is [tex][tex]$30^{\circ}$[/tex][/tex]
C. The measure of the reference angle is [tex][tex]$45^{\circ}$[/tex][/tex]
D. The measure of the reference angle is [tex][tex]$60^{\circ}$[/tex][/tex]
E. [tex][tex]$\cos (\theta)=\frac{\sqrt{2}}{2}$[/tex][/tex]
F. [tex][tex]$\sin (\theta)=\frac{\sqrt{2}}{2}$[/tex][/tex]

Sagot :

GinnyAnswer
To solve the problem, we need to evaluate the trigonometric functions for the angle [tex]\(\theta = \frac{\pi}{4}\)[/tex] and determine the reference angle.

### Step-by-Step Solution:

1. Calculate [tex]\(\tan(\theta)\)[/tex]:
[tex]\[ \tan\left(\frac{\pi}{4}\right) = 1 \][/tex]
Thus, the statement "[tex]\(\tan (\theta) = 1\)[/tex]" is true.

2. Determine the Reference Angle:
The reference angle for [tex]\(\theta = \frac{\pi}{4}\)[/tex] can be converted to degrees:
[tex]\[ \frac{\pi}{4} \text{ radians} = 45^{\circ} \][/tex]
So, the statements "[tex]\(\text{The measure of the reference angle is } 45^{\circ}\)[/tex]" is true.
The other potential measures of reference angles 30°, and 60° are not correct in this case.

3. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
Thus, the statement "[tex]\(\cos (\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is true.

4. Calculate [tex]\(\sin(\theta)\)[/tex]:
[tex]\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \approx 0.7071 \][/tex]
Thus, the statement "[tex]\(\sin (\theta) = \frac{\sqrt{2}}{2}\)[/tex]" is true.

### Summary:
Based on the calculations and evaluations, the true statements are:
- [tex]\(\cos (\theta) = \frac{\sqrt{2}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]
- [tex]\(\sin (\theta) = \frac{\sqrt{2}}{2}\)[/tex]

Thus the correct answers are:

- [tex]\(\cos (\theta)=\frac{\sqrt{2}}{2}\)[/tex]
- The measure of the reference angle is [tex]\(45^{\circ}\)[/tex]
- [tex]\(\sin (\theta)=\frac{\sqrt{2}}{2}\)[/tex]

Therefore, the answers are [tex]\( [5, 3] \)[/tex].
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.