To determine whether there are values of [tex]\( t \)[/tex] such that [tex]\( \sin t = 0.6 \)[/tex] and [tex]\( \cos t = 0.8 \)[/tex], we can use the Pythagorean identity. The Pythagorean identity states that for any angle [tex]\( t \)[/tex]:
[tex]\[
\sin^2 t + \cos^2 t = 1
\][/tex]
Given:
[tex]\[
\sin t = 0.6 \quad \text{and} \quad \cos t = 0.8
\][/tex]
We will use these given values to check if they satisfy the Pythagorean identity.
First, compute [tex]\(\sin^2 t\)[/tex]:
[tex]\[
\sin^2 t = (0.6)^2 = 0.36
\][/tex]
Next, compute [tex]\(\cos^2 t\)[/tex]:
[tex]\[
\cos^2 t = (0.8)^2 = 0.64
\][/tex]
Now, add these two values together:
[tex]\[
\sin^2 t + \cos^2 t = 0.36 + 0.64 = 1.0
\][/tex]
Since the sum of [tex]\(\sin^2 t\)[/tex] and [tex]\(\cos^2 t\)[/tex] equals 1, the values [tex]\( \sin t = 0.6 \)[/tex] and [tex]\( \cos t = 0.8 \)[/tex] do indeed satisfy the Pythagorean identity.
Therefore, the statement that there are values of [tex]\( t \)[/tex] such that [tex]\( \sin t = 0.6 \)[/tex] and [tex]\( \cos t = 0.8 \)[/tex] is:
A. True
