Are there values of [tex]$t[tex]$[/tex] such that [tex]$[/tex]\sin t = 0.35[tex]$[/tex] and [tex]$[/tex]\cos t = 0.6$[/tex]?

A. True
B. False



Answer :

To determine whether there are values of \( t \) such that \( \sin t = 0.35 \) and \( \cos t = 0.6 \), we need to check if these values satisfy the trigonometric identity:

[tex]\[ \sin^2 t + \cos^2 t = 1 \][/tex]

Let's substitute the given values into the identity:

[tex]\[ (0.35)^2 + (0.6)^2 \][/tex]

First, we calculate the squares:

[tex]\[ (0.35)^2 = 0.1225 \][/tex]
[tex]\[ (0.6)^2 = 0.36 \][/tex]

Next, we add these results together:

[tex]\[ 0.1225 + 0.36 = 0.4825 \][/tex]

This sum does not equal 1, given that:

[tex]\[ 0.4825 \neq 1 \][/tex]

Since the fundamental trigonometric identity \( \sin^2 t + \cos^2 t \) must equal 1 for all values of \( t \), and this equation does not hold with the given values, it concludes that there are no values of \( t \) such that both \( \sin t = 0.35 \) and \( \cos t = 0.6 \) at the same time.

Therefore, the correct answer is:

B. False